.^ SOUND, 1 subjectively the sense impression of the organ of 1 " Sound " is an interesting example of the numerous homonymous words in the English language .
.^ In the sense in which it is treated in this article it appears in 'Middle English as soun, and comes through Fr.
son from Lat.
sonus; the
d is a mere
addition, as in the nautical term " bound " (outward, homeward
bound) for the earlier " boun," to make ready, prepare.
.^ In the adjectival meaning, healthy, perfect, complete, chiefly used of a deep undisturbed sleep , or of a wellbased argument or doctrine, or of a person well trained in his profession, the word is in O. Eng.
sund, and appears also
in Ger.
gesund, Du.
.^ It is probably cognate with the Lat.
sonus, healthy, whence the Eng.
sane,
insanity,
sanitation, &c.
.^ Lastly, there is a group of words which etymologists are inclined to treat as being all forms of the word which in 0.
is
sund, meaning "
swimming." These words are for (I) the swim
bladder of a
fish; (2) a narrow stretch of water between an
inland sea and the ocean, or between an island and the mainland,
&c., cf.
.^ Sound,, The , below; (3) to test or measure the depth of anything, particularly the depth of water in lakes or seas (see Sounding , below).
^ When two tones are sounded together with frequencies not very different, " beats " or swellingsout of the sound are heard of frequency equal to the difference of frequencies of the two tones (see below).
.^ As a substantive the term is used of a surgical instrument for the exploration of a wound , cavity, &c., a probe.
.^ In these senses the word has frequently been referred to Lat.
sub unda, under the water; and Fr.
sombre, gloomy, possibly from
sub umbra, beneath the shade, is given as a
parallel.
.^ Any sound (such as that of the human voice) transmitting its rays into the reflector, and communicating vibratory motion to the membrane, will cause the feather to trace a sinuous line on the paper.
.^ The physiological and psychical aspects of sound are treated in the article Hearing .
^ H. von Helmholtz treats the theoretical aspects of sound in his Vorlesungen fiber die mathematischen Principien der Akustik (1898), and the physiological and psychical aspects in his Die Lehre von den Tonempfindungen (1st ed., 1863; 5th ed., 1896), English translation by A. J. Ellis, On the Sensations of Tone (1885).
^ In this article, which covers the science of Acoustics , we shall consider only the physical aspect of sound, that is, the physical phenomena outside ourselves which excite our sense of hearing.
.^ In this article, which covers the science of Acoustics , we shall consider only the physical aspect of sound, that is, the physical phenomena outside ourselves which excite our sense of hearing.
^ The physiological and psychical aspects of sound are treated in the article Hearing .
^ We shall only consider one interesting case of sound diffraction which may be easily observed.
.^ We shall discuss the disturbance which is propagated from the source to the ear , and which there produces sound, and the modes in which various sources vibrate and give rise to the disturbance.
^ The various modes of vibration may also be exhibited.
^ If we study the source producing it we find that there is no regularity of vibration.
Sound is due to Vibrations
.^ We may easily satisfy ourselves that, in every instance in which the sensation of sound is excited, the body whence the sound proceeds must have been thrown, by a blow or other means, into a state of agitation or tremor, implying the existence of a vibratory motion, or motion to and fro, of the particles of which it consists.
^ It is also in many cases possible to follow with the eye the motions of the particles of the sounding body, as, for instance, in the case of a violin string or any string fixed at both ends, when the string will appear through the persistence of visual sensation to occupy at once all the positions which it successively assumes during its vibratory motion.
^ We shall only consider one interesting case of sound diffraction which may be easily observed.
.^ Thus, if a common glass  jar be struck so as to yield an audible sound, the existence of a motion of this kind may be felt by the finger lightly applied to the edge of the glass; and, on increasing the pressure so as to destroy this motion the sound forthwith ceases.
^ We may obtain an excellent representation of the motion of the layers of air in a train of sound waves by means of a device due to Crova and known as " Crova's disk."
^ But the waves on the surface of a liquid, which are not of the sound kind, are both longitudinal and transverse, the compound nature being easily seen in watching the motion of a floating particle.
Small pieces of
cork put in the jar will be found
to
dance about during the
continuance of the sound; water or
spirits of
wine poured into the glass will, under the same
circumstances, exhibit a ruffled surface.
.^ The experiment is usually performed, in a more striking manner, with a bell jar and a number of small light wooden balls suspended by silk strings to a fixed frame above the jar, so as to be just in contact with the widest part of the glass.
^ U 0 =331.37 met./sec.; but the number of experiments over the longer distance was greater, and he appears to have put more confidence in the result from them, viz.
.^ On drawing a violin bow across the edge, the pendulums are thrown off to a considerable distance, and falling back are again repelled, and so on.
^ It draws off from any wave it has sent backward and so spreads the succeeding waves over a longer distance than if it had remained at rest.
.^ It is also in many cases possible to follow with the eye the motions of the particles of the sounding body, as, for instance, in the case of a violin string or any string fixed at both ends, when the string will appear through the persistence of visual sensation to occupy at once all the positions which it successively assumes during its vibratory motion.
^ We may easily satisfy ourselves that, in every instance in which the sensation of sound is excited, the body whence the sound proceeds must have been thrown, by a blow or other means, into a state of agitation or tremor, implying the existence of a vibratory motion, or motion to and fro, of the particles of which it consists.
^ Paper, parchment , or any other thin membrane stretched over a square, circular, &c., frame, when in the vicinity of a sufficiently powerful vibrating body, will, through the medium of the air, be itself made to vibrate in unison, and, by using sand, as in previous instances, the nodal lines will be depicted to the eye, and seen to vary in form, number and position with the tension of the plate and the pitch of the originating sound.
Sound takes Time to Travel
If we
watch a man breaking
stones by the roadside some distance away, we can see the
hammer fall before we hear the
blow. We see the
steam issuing
from the
whistle of a
distant
engine long before we
hear the sound. We see
lightning before we hear the
thunder which spreads out from
the flash, and the more distant the flash the longer the interval
between the two. The wellknown rule of a mile for every five
seconds between flash and peal gives a fair estimate of the
distance of the lightning.
Sound needs a Material Medium to Travel Through
.^ In order that the ear may be affected by a sounding body there must be continuous matter reaching all the way from the body to the ear.
^ In ordinary soundwaves the effect of the particle velocity in affecting the velocity of transmission must be very small.
^ There are shears of the order dry/dx and the simple Young's modulus system can no longer be taken to represent the actual condition (see Rayleigh, Sound, i.
.^ This can be shown by suspending an electric bell in the receiver of an air  pump , the wires conveying the current passing through an airtight cork closing the hole at the top of the receiver.
^ If the bob of the pendulum is iron, and if a coil is placed just below the centre of swing, then, if a current passes through the coil, while and only while the bob is moving towards it, the vibration is maintained.
^ When the wire was heated by an electric current a fine line of vapour descended from each drop.
.^ These wires form a material channel from the bell to the outside air, but if they are fine the sound which they carry is hardly appreciable.
^ If small paper rings are put on a monochord wire they rotate through these vibrations when the wire is bowed.
^ Another form of sensitive jet is very easily made by putting a piece of fine wire gauze 2 or 3 in.
.^ If while the air within the receiver is at atmospheric pressure the bell is set ringing continuously, the sound is very audible.
^ It is obvious from the various experiments that the velocity of sound in dry air at o° C. is not yet known with very great accuracy.
^ This is a consequence of the very unequal audibility of a whisper in front and behind the speaker, a phenomenon which may easily be observed in the open air" ( Sound, ii.
.^ But as the air is withdrawn by the pump the sound decreases, and when the exhaustion is high the.
^ Thus in the airpump experiment, before exhaustion it travels through the glass of the receiver and the base plate.
bell is almost inaudible.
Usually air is the medium through which sound travels, but it
can travel through solids or liquids. Thus in the airpump
experiment, before exhaustion it travels through the glass of the
receiver and the base plate. We may easily realise its transmission
through a solid by putting the ear against a table and scratching
the wood at some distance, and through a liquid by keeping both
ears under water in a
bath and
tapping the side of the bath.
Sound is a Disturbance of the Wave Kind
As sound arises in general from vibrating bodies, as it takes
time to travel, and as the medium which carries it does not on the
whole travel forward, but subsides into its original position when
the sound has passed, we are forced to conclude that the
disturbance is of the wave kind, We can at once gather some idea of
the nature of sound waves in air by considering how they are
produced by a bell.
Let AB (fig. I) be a small portion of a bell which vibrates to
and fro from CD to EF and back.
.^ As AB moves from CD to EF it pushes forward the layer of air in contact with it.
^ As AB again moves from CD towards EF another compression or push is sent out, as it returns from EF towards CD another extension or pull, and so on.
^ That layer C A E presses against and pushes forward the next layer and so on.
That layer C A E
presses against and pushes forward the next layer and so on.
.^ Thus a push or a compression of the X air is transmitted onwards in the direction OX .
^ Thus waves are propagated along OX, each wave consisting of one push and one pull, one wave emanating from each complete vibration to and fro of the source AB. .
As AB
returns from EF towards CD the layer of air next to it follows it
as if it D E F were pulled back by AB. Really, FIG. I of course, it
is pressed into the space made for it by the rest of the air, and
flowing into this space it is extended.
.^ It makes room for the next layer of air to move back and to be extended and so on, and an extension of the air is.
^ As AB moves from CD to EF it pushes forward the layer of air in contact with it.
^ As AB returns from EF towards CD the layer of air next to it follows it as if it D E F were pulled back by AB. Really, FIG. I of course, it is pressed into the space made for it by the rest of the air, and flowing into this space it is extended.
transmitted onwards following the
compression which has already gone out.
.^ As AB again moves from CD towards EF another compression or push is sent out, as it returns from EF towards CD another extension or pull, and so on.
^ As AB moves from CD to EF it pushes forward the layer of air in contact with it.
^ As AB returns from EF towards CD the layer of air next to it follows it as if it D E F were pulled back by AB. Really, FIG. I of course, it is pressed into the space made for it by the rest of the air, and flowing into this space it is extended.
Thus waves
are propagated along OX, each wave consisting of one push and one
pull, one wave emanating from each complete vibration to and fro of
the source AB.
Crova's Disk
We may obtain an excellent representation of the motion of the
layers of air in a
train of
sound waves by means of a
device due to Crova and known as " Crova's
disk." A small circle, say 2 or 3 mm.
radius, is drawn on a card as in fig.
.^ From these points as centres, circles are drawn in succession, each with radius greater than the last by a fixed amount, say 4 or 5 mm.
FIG. 2.
taken.
.^ From these points as centres, circles are drawn in succession, each with radius greater than the last by a fixed amount, say 4 or 5 mm.
^ In the figure the radius of the inner circle is 3 mm.
^ A small circle, say 2 or 3 mm.
.^ In the figure the radius of the inner circle is 3 mm.
^ From these points as centres, circles are drawn in succession, each with radius greater than the last by a fixed amount, say 4 or 5 mm.
and the radii of the circles drawn round it are 12, 16,
20, &c.
.^ If the figure thus drawn is spun round its centre in the right direction in its own plane waves appear to travel out from the centre along any radius.
^ The waves from a source surrounded by a uniform medium at rest spread out as spheres with the source as centre.
^ Suppose that a disturbance is travelling with velocity U unchanged in form along a rod from left to right.
If a second card with a narrow
slit in it is held in front of the first, the slit running from the
centre outwards, the wave motion is still more evident.
.^ If the figure be photographed as a lantern slide which is mounted so as to turn round, the wave motion is excellently shown on the screen , the compressions and extensions being represented by the crowding in and opening out of the lines.
^ As AB again moves from CD towards EF another compression or push is sent out, as it returns from EF towards CD another extension or pull, and so on.
^ If the fork has slightly greater frequency, then a white line will not quite reach the next place while the fork is making its swing ip and out, and the waves will travel against the motion of the cylinder.
.^ Another illustration is afforded by a long spiral of wire with coils, say 2 in.
in diameter
and 2 in. apart.
.^ It may be hung up by threads.
so as to lie
horizontally.
.^ If one end is sharply pressed in, a compression can be seen running along the spring.
^ For instance, if a rope is fixed at one end and held in the hand at the other end, a transverse jerk by the hand will travel as a transverse wave along the rope.
The Disturbance in Sound Waves is Longitudinal
The motion of a particle of air is, as represented in these
illustrations, to and fro in the direction of
propagation,
i.e. the disturbance is " longitudinal." There is no "
transverse " disturbance, that is, there is in air no motion across
the line of propagation, for such motion could only be propagated
from one layer to the next by the " viscous " resistance to
relative motion, and would die away at a very short distance from
the source. But transverse disturbances may be propagated as waves
in solids. For instance, if a
rope is fixed at one end and held
in the hand at the other end, a transverse jerk by the hand will
travel as a transverse wave along the rope. In liquids sound waves
are longitudinal as they are in air. But the waves on the surface
of a liquid, which are not of the sound kind, are both longitudinal
and transverse, the compound nature being easily seen in watching
the motion of a floating particle.
.^ We can represent waves of longitudinal displacement by a curve , and this enables us to draw very important conclusions in a very simple way.
^ Let it be represented by a displacement curve Ahbkc .
^ The displacement curve of the waves from a tuningfork on its resonance box , or from the human voice sounding oo, are nearly smooth and symmetrical, as in fig.
Let a train of
waves be passing from left to right in the direction ABCD (fig. 3).
At every point R N Q FIG. 3.
let a line be drawn perpendicular to
.^ AD and proportional to the displacement of the particle which was at the point before the disturbance began.
^ At the points ABCD there is no displacement, and the line AD through these points is called the axis.
^ It is easily shown that after a time we shall have to superpose on the original displacement a displacement proportional to the square of the particle velocity, and this will introduce just the same set of combination tones.
.^ Thus let the particle which was at L be at to the right or forwards, at a given instant.
^ Let the particle which was at M originally be at m at the given instant, being displaced to the left or backwards.
^ Let R be the receiver at a given instant, R' its position a second later, its velocity being v.
.^ Draw LP upward and some convenient multiple of Ll.
^ Draw NR the same multiple of Nn and upwards.
.^ Let the particle which was at M originally be at m at the given instant, being displaced to the left or backwards.
^ Thus let the particle which was at L be at to the right or forwards, at a given instant.
^ Let MP =y represent the forward displacement of the particle originally at M, and NQ = y +dy that of the particle originally at N. The layer of air originally of thickness dx now has thickness dx+dy, since N is displaced forwards dy more than M. The volume dx, then, has increased to dx+dy or volume I has increased to I +dy/dx and the increase of volume I is dy/dx.
.^ Draw MQ downwards, the same multiple of Mm.
^ Draw NR the same multiple of Nn and upwards.
.^ Let N be displaced forward to n.
.^ Draw NR the same multiple of Nn and upwards.
^ Draw LP upward and some convenient multiple of Ll.
^ Draw MQ downwards, the same multiple of Mm.
.^ If this is done for every point we obtain a continuous curve Apbqcrd , which represents the displacement at every point at the given instant, though by a length at right angles to the actual displacement and on an arbitrary scale.
^ Let it be represented by a displacement curve Ahbkc .
^ Let the particle which was at M originally be at m at the given instant, being displaced to the left or backwards.
.^ At the points ABCD there is no displacement, and the line AD through these points is called the axis.
^ Forward displacement is represented by height above the axis, backward displacement by depth below it.
^ The dotted curve represents the distribution of compression by height above the axis, and of extension by depth below it.
.^ For instance, if we have a wave with displacement curve of form ABC (fig.
^ In ordinary sound waves the displacement is very minute, perhaps of the order 105 cm., so that we multiply it perhaps by ioo,000 in forming the displacement curve.
^ The displacement curve of the waves from a tuningfork on its resonance box , or from the human voice sounding oo, are nearly smooth and symmetrical, as in fig.
Wave Length and Frequency
.^ For instance, if we have a wave with displacement curve of form ABC (fig.
^ If the waves are continuous and each of the same shape they form a " train," and the displacement curve repeats itself.
^ The displacement curve of the waves from a tuningfork on its resonance box , or from the human voice sounding oo, are nearly smooth and symmetrical, as in fig.
.^ The shortest distance in which this repetition occurs is called the wavelength.
^ Let us suppose that a system of stationary waves is formed in the air in a pipe of indefinite length, and let fig.
^ Cldme is merely Ahbkc moved on a distance AC. Then AC =A is the wavelength or period of the curve.
.^ AC = X. If the source makes n vibrations in one second it is said to have " frequency " n.
^ Thus waves are propagated along OX, each wave consisting of one push and one pull, one wave emanating from each complete vibration to and fro of the source AB. .
^ This is generally equal to the number of waves issuing from the source per second, and therefore equal to its frequency of vibration.
.^ It sends out n waves in each second.
^ If each wave travels out from the source with velocity U the n waves emitted in one second must occupy a length U and therefore U = nX. .
If each
wave travels out from the source with velocity U the n waves
emitted in one second must occupy a length U and therefore U =
nX.
Distribution of Compression and Extension in a Wave
Let fig.
.^ In the time dt which the wave takes to travel over MN the particle displacement at N changes by QR, and QR=  udt, so that QR/MN =  u/U. But QR/MN = dy/dx.
^ Let a disturbance once set going travel along unchanged in form from A to B with velocity U. Then move AB from right to left with this velocity, and the disturbance remains fixed in space.
^ Suppose that a disturbance is travelling with velocity U unchanged in form along a rod from left to right.
FIG. 4.
.^ At A the air occupies its original position, while at H it is displaced towards the right or away from A since HP is above the axis.
^ Let MP =y represent the forward displacement of the particle originally at M, and NQ = y +dy that of the particle originally at N. The layer of air originally of thickness dx now has thickness dx+dy, since N is displaced forwards dy more than M. The volume dx, then, has increased to dx+dy or volume I has increased to I +dy/dx and the increase of volume I is dy/dx.
^ Since the curve represents a longitudinal disturbance in air it is always continuous, at a finite distance from the axis, and with only one ordinate for each abscissa .
.^ Between A and H, then, and about H, it is extended.
.^ At L there is also displacement towards B and again compression.
^ At J the displacement is forward, but since the curve at Q is parallel to the axis the displacement is approximately the same for all the points close to J, and the air is neither extended nor compressed, but merely displaced bodily a distance represented by JQ. At B there is no displacement, but at K there is displacement towards B represented by KR, i.e.
^ Forward displacement is represented by height above the axis, backward displacement by depth below it.
.^ At L there is also displacement towards B and again compression.
^ The figure shows that when the curve of displacement slopes down in the direction of propagation there is compression, and the pressure is above the normal, and that when it slopes up there is extension, and the pressure is below the normal.
^ At J the displacement is forward, but since the curve at Q is parallel to the axis the displacement is approximately the same for all the points close to J, and the air is neither extended nor compressed, but merely displaced bodily a distance represented by JQ. At B there is no displacement, but at K there is displacement towards B represented by KR, i.e.
.^ At M, as at J, there is neither extension nor compression.
^ The figure shows that when the curve of displacement slopes down in the direction of propagation there is compression, and the pressure is above the normal, and that when it slopes up there is extension, and the pressure is below the normal.
.^ At N the displacement is away from C and there is extension.
^ The figure shows that when the curve of displacement slopes down in the direction of propagation there is compression, and the pressure is above the normal, and that when it slopes up there is extension, and the pressure is below the normal.
.^ Forward displacement is represented by height above the axis, backward displacement by depth below it.
^ The distribution of velocity then is represented by the dotted curve and is forward when the curve is above the axis and Dackward when it is below.
^ The dotted curve represents the distribution of compression by height above the axis, and of extension by depth below it.
.^ Or we may take it as representing the pressure  excess over the normal pressure in compression, defect from it in extension.
^ Now as each source lets out the wind periodically it affects the pressure in the chest so that we cannot regard this as constant, but may take it as better represented by p+Xa sin ( 27rn i t+e)+µb sin (27rn 2 t+f).
^ The figure shows that when the curve of displacement slopes down in the direction of propagation there is compression, and the pressure is above the normal, and that when it slopes up there is extension, and the pressure is below the normal.
.^ At L there is also displacement towards B and again compression.
^ The figure shows that when the curve of displacement slopes down in the direction of propagation there is compression, and the pressure is above the normal, and that when it slopes up there is extension, and the pressure is below the normal.
^ At M, as at J, there is neither extension nor compression.
Distribution of Velocity in a Wave
.^ If a wave travels on without alteration the travelling may be represented by pushing on the displacement curve.
^ Let it be represented by a displacement curve Ahbkc .
^ The displacement curve of the waves from a tuningfork on its resonance box , or from the human voice sounding oo, are nearly smooth and symmetrical, as in fig.
.^ Let the wave Aqbtc (fig.
^ Let us suppose that a system of stationary waves is formed in the air in a pipe of indefinite length, and let fig.
^ Let a train of waves be passing from left to right in the direction ABCD (fig.
.^ A'QB'TC' in a very short time.
.^ At K it increases by RR' forwards, or the motion is forwards towards B. At L the displacement backward decreases, or the motion is forward FIG. 5.
^ In that short time the displacement at H decreases from HP to HP' or by PP'.
^ The motion of the particle is therefore backwards towards A. At J the displacement remains the same, or the particle is not moving.
At K it increases by RR' forwards, or the motion is
forwards towards B. At L the displacement backward decreases, or
the motion is forward FIG. 5.
.^ At M, as at J, there is no change, and at N it is easily seen that the motion is backward.
^ This means that at the loops while the motion is greatest there are no pressure changes.
^ At the nodes A, B, C, D, E there is no displacement, but there are maximum volume and pressure changes.
.^ Forward displacement is represented by height above the axis, backward displacement by depth below it.
^ The distribution of velocity then is represented by the dotted curve and is forward when the curve is above the axis and Dackward when it is below.
^ The dotted curve represents the distribution of compression by height above the axis, and of extension by depth below it.
Comparing figs.
.^ Since U=n X where U is the velocity of sound, X the wavelength, and n the frequency, it follows that the forward frequency is greater than the backward frequency.
^ The velocity with which waves of longitudinal disturbance travel in air or in any other fluid can be calculated from the resistance to compression and extension and the density of the fluid.
The Relations between Displacement, Compression and
Velocity
.^ The relations shown by figs.
.^ Another form of sensitive jet is very easily made by putting a piece of fine wire gauze 2 or 3 in.
^ If then we resolve Ahbkc into harmonics by Fourier's theorem, we may follow the motion of the separate harmonics, and their superposition will give the form of the string at any instant.
^ The more general case of motion of source, medium and receiver may be treated very easily if the motions are all in the line joining source and receiver.
.^ Let OX (fig.
6) be the direction
P?/R 0 N X FIG. 6.
of travel, and let x be the distance of any point M from a fixed
point O. Let ON. =x+dx.
.^ Let N be displaced forward to n.
^ Let it be represented by a displacement curve Ahbkc .
^ To find the relation of the velocity to displacement and pressure we shall express the fact that the wave travels on carrying all its conditions with it, so that the displacement now at M will arrive at N while the wave travels over MN. Let U be the velocity of the wave and let u be the velocity of the particle originally at N. Let MN = dx = Udt.
.^ In the time dt which the wave takes to travel over MN the particle displacement at N changes by QR, and QR=  udt, so that QR/MN =  u/U. But QR/MN = dy/dx.
^ To find the relation of the velocity to displacement and pressure we shall express the fact that the wave travels on carrying all its conditions with it, so that the displacement now at M will arrive at N while the wave travels over MN. Let U be the velocity of the wave and let u be the velocity of the particle originally at N. Let MN = dx = Udt.
^ Let MP =y represent the forward displacement of the particle originally at M, and NQ = y +dy that of the particle originally at N. The layer of air originally of thickness dx now has thickness dx+dy, since N is displaced forwards dy more than M. The volume dx, then, has increased to dx+dy or volume I has increased to I +dy/dx and the increase of volume I is dy/dx.
.^ Sound takes Time to Travel .
^ In the time dt which the wave takes to travel over MN the particle displacement at N changes by QR, and QR=  udt, so that QR/MN =  u/U. But QR/MN = dy/dx.
^ The maximum velocity of a particle in the wavetrain is the amplitude of dy/dt.
.^ Then u/U =  dy/dx (2) This gives the velocity of any particle in terms of the displacement.
^ The maximum particle velocity is 21rna (where n is the frequency and a the amplitude), or 27raU/X. This gives maximum u=about 8 cm./sec., which would not seriously change the form of the wave in a few wavelengths.
^ But the energy will also be doubled, so that (15) still gives the average excess of pressure.
.^ Generally, if any condition in the wave is carried forward unchanged with velocity U, the change of 4 at a given point in time dt is equal to the change of as we go back along the curve a distance dx = Udt at the beginning of dt.
^ The displacement curve of the waves from a tuningfork on its resonance box , or from the human voice sounding oo, are nearly smooth and symmetrical, as in fig.
^ But a priori we are hardly justified in assuming that waves can be propagated at all, and certainly not justified in assuming that they go on unchanged by the action of the internal forces alone.
 Sounds
differ from each other only in the three respects of loudness,
pitch and quality.
The
loudness of the sound brought by a train of waves
of given wavelength depends on the extent of the to and fro
excursion of the air particles. This is obvious if we consider that
the greater the vibration of the source the greater is the
excursion of the air in the issuing waves, and the louder is the
sound heard. Half the total excursion is called the
amplitude. Thus in fig. 4
QJ is the amplitude. Methods of measuring the amplitude in sound
waves in air have been devised and will be described later.
.^ We may say here that the energy or the intensity of the sound of given wavelength is proportional to the square of the amplitude.
^ With a still shorter wavelength we may have the length.
^ The jet tube, for a reason which will be given when we consider the maintenance of vibrations, must be less than c a9 half the length of the sounding tube.
Missing image
Sound1.jpg
Missing image
Sound2.jpg
The pitch of a sound, the note which we assign to it, depends on
the number of waves received by the ear per second.
.^ This is generally equal to the number of waves issuing from the source per second, and therefore equal to its frequency of vibration.
^ AC = X. If the source makes n vibrations in one second it is said to have " frequency " n.
^ As an example of the application of this siren, suppose that the number of revolutions of the plate, as shown by the indices, amounts to 5400 in a minute, that is, to 90 per second, then the number of vibrations per second of the note heard amounts to 90n, or (if number of holes in each plate = 8) to 720.
.^ All experiments in frequency show that two notes, forming a definite musical interval, have their frequencies always in the same ratio wherever in the musical scale the two notes are situated.
^ The method ensures that the two frequencies shall be exactly the same.
^ We shall now describe some of the methods of determining frequency.
Here it is sufficient to say that the frequencies of a
note, its major third, its fifth and its
octave, are in the ratios of 4: 5: 6: 8.
The
quality or
timbre of sound,
i.e.
that which differentiates a note sounded on one instrument from the
same note on another instrument, depends neither on amplitude nor
on frequency or wavelength.
.^ We can only conclude that it depends on wave form, a conclusion fully borne out by investigation.
^ Rayleigh points out that this clinging of the sound to the surface of a concave wall does not depend on the exactness of the spherical form.
^ We shall investigate the velocity of such plane waves by a method which is only a slight modification of a method given by W. J. M. Rankine ( Phil.
.^ The displacement curve of the waves from a tuningfork on its resonance box , or from the human voice sounding oo, are nearly smooth and symmetrical, as in fig.
^ For instance, if we have a wave with displacement curve of form ABC (fig.
^ Tdx/ds, and when the disturbance is sufficiently small the curve of displacement is so nearly parallel to the axis that /ds = I, and this component is T. The component of T perpendicular to the axis is Tdy/ds=Tdy/dx.
.^ That for the air waves from a violin are probably nearly as in fig.
^ The displacement curve of the waves from a tuningfork on its resonance box , or from the human voice sounding oo, are nearly smooth and symmetrical, as in fig.
^ Let us suppose that a system of stationary waves is formed in the air in a pipe of indefinite length, and let fig.
7b. FIG. 7.
.^ When a wave of sound meets a surface separating two media it is in part reflected, travelling back from the surface into the first medium again with the velocity with which it approached.
^ Since U=n X where U is the velocity of sound, X the wavelength, and n the frequency, it follows that the forward frequency is greater than the backward frequency.
^ The velocity of any part of a wave front relative to the ground will be the normal velocity of sound + the velocity of the wind at that point.

.^ The velocity with which waves of longitudinal disturbance travel in air or in any other fluid can be calculated from the resistance to compression and extension and the density of the fluid.
^ When a wave of sound meets a surface separating two media it is in part reflected, travelling back from the surface into the first medium again with the velocity with which it approached.
^ Suppose that a disturbance is travelling with velocity U unchanged in form along a rod from left to right.
.^ It is convenient to give this calculation before proceeding to describe the experimental determination of the velocity in air, in other gases and in water, since the calculation serves to some extent as a guide in conducting and interpreting the observations.
^ Velocity of Sound in Air and other Gases in Pipes .
^ In a long series of experiments carried out by V. Regnault in the years 1862 to 1866 on the velocity of sound in open air, in air in pipes and in various other gases in pipes, he sought to eliminate personal equaticn by dispensing with the human element in the observations, using electric receivers as observers.
.^ The waves from a source surrounded by a uniform medium at rest spread out as spheres with the source as centre.
^ Meanwhile the waves are spreading out and the value of u is falling in inverse proportion to the distance from the source, so that very soon its effect must become negligible.
^ If each wave travels out from the source with velocity U the n waves emitted in one second must occupy a length U and therefore U = nX. .
.^ With a still shorter wavelength we may have the length.
^ It is sufficient to take a single wavelength.
^ In fact, we may neglect the divergence, and may regard them as " plane waves."
.^ Every particle in the plane will have the same displacement and the same velocity, and these will be perpendicular to the plane and parallel to the line of propagation.
^ They found that the velocity of propagation of different musical sounds was the same.
^ Tdx/ds, and when the disturbance is sufficiently small the curve of displacement is so nearly parallel to the axis that /ds = I, and this component is T. The component of T perpendicular to the axis is Tdy/ds=Tdy/dx.
.^ The waves for some little distance on each side of the plane will be practically of the same size.
^ If we take one of these spheres a distance from the source very great as compared with a single wavelength, and draw a radius to a point on the sphere, then for some little way round that point the sphere may be regarded as a plane perpendicular to the radius or the line of propagation.
^ Thus, if the one note be an octave higher than the other, it will give double the number of waves in the same distance.
.^ In fact, we may neglect the divergence, and may regard them as " plane waves."
^ Experiments may be made with plane and curved mirrors to verify these laws, but it is necessary to use short waves, in order to diminish diffraction effects.
^ If we take one of these spheres a distance from the source very great as compared with a single wavelength, and draw a radius to a point on the sphere, then for some little way round that point the sphere may be regarded as a plane perpendicular to the radius or the line of propagation.
.^ We shall investigate the velocity of such plane waves by a method which is only a slight modification of a method given by W. J. M. Rankine ( Phil.
^ As a preliminary to the investigation of the modes of vibration of certain sources of sound we shall consider the formation of " stationary waves."
^ Subject to a limitation which we shall examine later, the velocity of a longitudinal disturbance along a wire or rod depends only on the material of the rod, and not upon the crosssection.
Trans., 1870, p.
277).
.^ If the velocity U is so chosen that E  poU 2 = o, then X = o, or the wave travels on through the action of the internal forces only, unchanged in form and with velocity U = (E/p).
^ Whatever the form of a wave, we could always force it to travel on with that form unchanged, and with any velocity we chose, if we could apply any " external " force we liked to each particle, in addition to the " internal " force called into play by the compressions or extensions.
^ We shall investigate the external force needed to make a train of plane waves travel on unchanged in form with velocity U. We shall regard the external force as applied in the form of a pressure X per square centimetre parallel to the line of propagation and varied from point to point as required in order to make the disturbance travel on unchanged in form with the specified velocity U. In addition there will be the internal force due to the change in volume, and consequent change in pressure, from point to point.
.^ The form of the curve is evidently as represented in fig.
^ For instance, if we have a wave with displacement curve of form ABC (fig.
^ Let us here suppose that the string AB is displaced into the form AHB (fig.
8), and we require it to
travel FIG. 8.
on in time
.^ This change can always be effected if we can apply whatever force may be needed to produce it.
^ Let the particle which was at M originally be at m at the given instant, being displaced to the left or backwards.
^ In ordinary soundwaves the effect of the particle velocity in affecting the velocity of transmission must be very small.
.^ We shall investigate the external force needed to make a train of plane waves travel on unchanged in form with velocity U. We shall regard the external force as applied in the form of a pressure X per square centimetre parallel to the line of propagation and varied from point to point as required in order to make the disturbance travel on unchanged in form with the specified velocity U. In addition there will be the internal force due to the change in volume, and consequent change in pressure, from point to point.
^ In order that the velocity shall remain unchanged the tension T must remain the same.
^ At the nodes A, B, C, D, E there is no displacement, but there are maximum volume and pressure changes.
.^ For instance, if a rope is fixed at one end and held in the hand at the other end, a transverse jerk by the hand will travel as a transverse wave along the rope.
^ Suppose that a disturbance is travelling with velocity U unchanged in form along a rod from left to right.
^ Let A be a point in the disturbance and B a point in the undisturbed portion.
Let A (fig. 9) be a point fixed in space in the disturbed
region, B a fixed point where the medium is not yet disturbed, the
medium A FIG. 9.
moving through A and B from right to left.
.^ The material between A and B, though continually changing, is always in the same condition, and therefore the momentum within AB is constant.
^ Since the condition of the medium between A and B remains constant, even though the matter is continually changing, the momentum possessed by the matter between A and B is constant.
^ The condition of the matter between A and B remains constant, though fresh matter keeps coming in at B and an equal quantity leaves at A. Hence the angular momentum of the part between A and B remains constant, or as much enters at B as leaves at A. But at B there is no torsion, and no torsion couple of one part of the wire on the next.
.^ P per second from the matter to the right of it.
^ Now momentum is transferred in two ways, viz.
^ The total momentum moving in at B is therefore P+poU 2.
.^ The total momentum moving in at B is therefore P+poU 2.
^ U cubic centimetres move in per second at B, and if the density is po the mass moving in through a square centimetre is po U. But it has velocity U, and therefore momentum poU 2 is carried in.
^ So that no angular momentum enters at B, and therefore on the whole none leaves at A. The transfer of angular momentum through A is of two kinds  first, that due to the passage of rotating matter, and, secondly, that due to the couple with which matter to the right of A acts upon matter to the left of A. The mass of matter moving through A per second is pwa 2 U, where a is the radius of the wire and p is its density.
.^ P per second from the matter to the right of it.
^ In addition there is a pressure between the layers of the medium, and if this pressure in the undisturbed parts of the medium is P, momentum P per second is being transferred from right to left across each square centimetre.
^ At A, if the velocity of the disturbance relative to undisturbed parts of the rod is u from left to right, the velocity relative to A is U  u.
.^ Hence the matter moving in is receiving on this.
account
.^ P per second from the matter to the right of it.
^ In addition there is a pressure between the layers of the medium, and if this pressure in the undisturbed parts of the medium is P, momentum P per second is being transferred from right to left across each square centimetre.
.^ The total momentum moving in at B is therefore P+poU 2.
^ Let us move the rod from right to left, so that the undisturbed parts move with velocity U. Then the disturbance remains fixed in space.
^ Hence the matter moving in is receiving on this.
of pressure due to change of volume be
.^ W, so that the total " internal " pressure is P+&,.
There is also
the " external " applied pressure X, and the total momentum flowing
out per second is XIP4W1p(U  u)2.
.^ Let MP =y represent the forward displacement of the particle originally at M, and NQ = y +dy that of the particle originally at N. The layer of air originally of thickness dx now has thickness dx+dy, since N is displaced forwards dy more than M. The volume dx, then, has increased to dx+dy or volume I has increased to I +dy/dx and the increase of volume I is dy/dx.
^ B a fixed point where the medium is not yet disturbed, the medium A FIG. 9.
^ We have U  u =U(I  u/U) =U(1  v/V), since u/U=  dy/dx= v/V. Also since p(V  v) =poV, or p=po/(I  v/V), then p(U  u)2 = VpoU 2 (i  v/V).
.^ B a fixed point where the medium is not yet disturbed, the medium A FIG. 9.
^ If then we apply a pressure X given by (5) at every point, and move the medium with any uniform velocity U, the disturbance remains fixed in space.
^ Let A be a point in the disturbance and B a point in the undisturbed portion.
.^ If the velocity U is so chosen that E  poU 2 = o, then X = o, or the wave travels on through the action of the internal forces only, unchanged in form and with velocity U = (E/p).
^ We shall investigate the external force needed to make a train of plane waves travel on unchanged in form with velocity U. We shall regard the external force as applied in the form of a pressure X per square centimetre parallel to the line of propagation and varied from point to point as required in order to make the disturbance travel on unchanged in form with the specified velocity U. In addition there will be the internal force due to the change in volume, and consequent change in pressure, from point to point.
^ Whatever the form of a wave, we could always force it to travel on with that form unchanged, and with any velocity we chose, if we could apply any " external " force we liked to each particle, in addition to the " internal " force called into play by the compressions or extensions.
.^ The pressure X is introduced in order to show that a wave can be propagated unchanged in form.
^ But a priori we are hardly justified in assuming that waves can be propagated at all, and certainly not justified in assuming that they go on unchanged by the action of the internal forces alone.
^ When intense sound waves impinged on the wall, the disk moved back through the hole, and by an amount showing a pressure of the order given by the following investigation: Suppose that a train of waves is incident normally on the surface S (fig.
.^ If we omitted it we should have to assume this, and equation (6) would give us the velocity of propagation if the assumption were justified.
^ The maximum particle velocity is 21rna (where n is the frequency and a the amplitude), or 27raU/X. This gives maximum u=about 8 cm./sec., which would not seriously change the form of the wave in a few wavelengths.
^ This is hardly to be explained by equation (I I), for at the very front of the disturbance u =o and the velocity should be normal.
.^ But a priori we are hardly justified in assuming that waves can be propagated at all, and certainly not justified in assuming that they go on unchanged by the action of the internal forces alone.
^ If the velocity U is so chosen that E  poU 2 = o, then X = o, or the wave travels on through the action of the internal forces only, unchanged in form and with velocity U = (E/p).
^ The pressure X is introduced in order to show that a wave can be propagated unchanged in form.
.^ If, however, we put on external forces of the required type X it is obvious that any wave can be propagated with any velocity, and our investigation shows that when U has the value in (6) then and only then X is zero everywhere, and the wave will be propagated with that velocity when once set going.
^ We shall investigate the external force needed to make a train of plane waves travel on unchanged in form with velocity U. We shall regard the external force as applied in the form of a pressure X per square centimetre parallel to the line of propagation and varied from point to point as required in order to make the disturbance travel on unchanged in form with the specified velocity U. In addition there will be the internal force due to the change in volume, and consequent change in pressure, from point to point.
^ Helmholtz investigated the velocity of propagation of sound in pipes, taking into account the viscosity of the air (Rayleigh, Sound, ii.
.^ Let MP =y represent the forward displacement of the particle originally at M, and NQ = y +dy that of the particle originally at N. The layer of air originally of thickness dx now has thickness dx+dy, since N is displaced forwards dy more than M. The volume dx, then, has increased to dx+dy or volume I has increased to I +dy/dx and the increase of volume I is dy/dx.
^ It may be noted that the elasticity E is only constant for small volume changes or for small values of dy/dx.
^ But if y is the displacement at A, dy/dx is the extension at A, and the force acting is a pull across A equal to Y&uodyldx, where Y is Young's modulus of elasticity.
ii., § 8, props. 4849). . dp/dp = k = p/p.
^ He supposed that in air Boyle's law holds in the extensions and compressions, or that p = kp, whence dp/dp = k = p/p.
^ The velocity with which waves of longitudinal disturbance travel in air or in any other fluid can be calculated from the resistance to compression and extension and the density of the fluid.
^ That is to say, instead of using Boyle's law, which supposes that the pressure changes so exceedingly slowly that conduction keeps the temperature constant, we must use the adiabatic relation p = kpy, whence d p /d p = y k p Y 1= yp/p, and U = ( yp/p ) [Laplace's formula].
.^ His value of the velocity in air is therefore U = iJ ( p ip.
^ Both obtained the value for the velocity (U I C RA(21rNp ' where U is the velocity in free air, R is the radius of the pipe, N the frequency, and p the air density.
^ The velocity of sound in air is independent of the pressure, T 'e ' but varies with the temperature, its value at t° C. .
) (Newton's
formula). .^ At the standard pressure of 76 cm.
of mercury or 1,014,000 dynes /
sq. cm., the density of dry air at o° C. being taken as 0.001293,
we get for the velocity in dry air at o° C.
.^ But, as we shall see, all the determinations give a value of Uo in the neighbourhood of 33, 000 cm./sec., or about 1080 ft./sec.
^ U 0 = 28,000 cm.sec.
.^ But, as we shall see, all the determinations give a value of Uo in the neighbourhood of 33, 000 cm./sec., or about 1080 ft./sec.
).
Then or (5) approximately.
.^ Newton found 979 ft./sec.
But, as we
shall see, all the determinations give a value of Uo in the
neighbourhood of 33, 000 cm./sec., or about 1080 ft./sec. This
discrepancy e was not explained till 1816, when Laplace (
Ann.
de chimie, 1816, vol. iii.) pointed out that the compressions
and extensions in sound waves in air alternate so rapidly that
there is no time for the temperature inequalities produced by them
to spread.
.^ That is to say, instead of using Boyle's law, which supposes that the pressure changes so exceedingly slowly that conduction keeps the temperature constant, we must use the adiabatic relation p = kpy, whence d p /d p = y k p Y 1= yp/p, and U = ( yp/p ) [Laplace's formula].
^ U, for in the stationary wave system the pressure change and the amplitude are both double those in either train, so that the same relation holds.
^ He supposed that in air Boyle's law holds in the extensions and compressions, or that p = kp, whence dp/dp = k = p/p.
(8) If we
take y =1.4 we obtain approximately for the velocity in dry air at
0° C.
.^ Uo=33,150 cm./sec., which is closely in accordance with observation.
^ But, as we shall see, all the determinations give a value of Uo in the neighbourhood of 33, 000 cm./sec., or about 1080 ft./sec.
.^ Indeed Sir G. G. Stokes ( Math.
and Phys.
Papers, iii.
.^ In ordinary soundwaves the effect of the particle velocity in affecting the velocity of transmission must be very small.
^ E flat as keynote introduces another flat, and so on, each flat not quite coinciding with a sharp but at a very small interval from it.
^ But there is no doubt that with very loud explosive sounds the normal velocity is quite considerably exceeded.
If we put
p =kp(I }at) in (8) we get the velocity in
a
gas at C.
Ut = yk(i+at)}.
.^ At 0° C.: we have Uo =1/ (yk), and hence U t = Uolt (I +at) =U 0 (I+o o0184t) (for small values of t).
.^ The velocity then should be independent of the barometric pressure, a result confirmed by observation.
^ He found that within wide limits the velocity was independent of the pressure, thus confirming the theory.
^ For two different gases with the same value of y, but with densities at the same pressure and temperature respectively p i and p2, we should have U1/U2 =1 1 (P2/P1), (Io) another result confirmed by observation.
.^ For two different gases with the same value of y, but with densities at the same pressure and temperature respectively p i and p2, we should have U1/U2 =1 1 (P2/P1), (Io) another result confirmed by observation.
^ The velocity then should be independent of the barometric pressure, a result confirmed by observation.
^ Comparing the velocities of sound U i and U2 in two different gases with densities and at the same temperature and pressure, and with ratios of specific heats 'yl, 72, theory gives Ui/U2 = 1/ {71 p 2/72 p i }.
Alteration of Form of the Waves when Pressure Changes are
Considerable
.^ When the value of dyldx is not very small E is no longer constant, but is rather greater in compression and rather less in extension than yP. This can be seen by considering that the relation between p and is given by a curve and not by a straight line.
^ The jet tube, for a reason which will be given when we consider the maintenance of vibrations, must be less than c a9 half the length of the sounding tube.
^ It may be noted that the elasticity E is only constant for small volume changes or for small values of dy/dx.
.^ The consequence is that the compression travels rather faster, and the extension rather slower, than at the speed found above.
^ The velocity with which waves of longitudinal disturbance travel in air or in any other fluid can be calculated from the resistance to compression and extension and the density of the fluid.
^ When the value of dyldx is not very small E is no longer constant, but is rather greater in compression and rather less in extension than yP. This can be seen by considering that the relation between p and is given by a curve and not by a straight line.
.^ We may get some idea of the effect by supposing that for a short time the change in form is negligible.
^ Suppose the two notes to correspond to 200 and 203 vibrations per second; at some instant of time, the air particles, through which the waves are passing, will be similarly displaced by both, and consequently the joint effect will be a sound of some intensity.
^ But when the wind is steady its effect may be eliminated by " reciprocal " observations, that is, by observations of the time of passage of sound in each direction over the measured distance..
In the momentum equation (4)
we may now omit X and it becomes 0.+P(U  u) 2 =poU2.
.^ Let us seek a more'exact value for w.
^ At present we cannot assign a more exact value than Uo = 331 metres per second.
If when P changes to P+w
volume V changes to V  v then (P f w) (V  v)y = PVy, whence w=
P (yv [y(y2 I) V2)
= y V (I 4:y1 } y 2 I J.
.^ We have U  u =U(I  u/U) =U(1  v/V), since u/U=  dy/dx= v/V. Also since p(V  v) =poV, or p=po/(I  v/V), then p(U  u)2 = VpoU 2 (i  v/V).
^ If y is the displacement at A, and if E is the elasticity, substituting for w and u from (2) and (3) we get X  Ed x d +pU2 But since the volume dx with density po has become volume dx+dy with density p p (d = po.
^ We have already found that if V changes to V  v iw= yP + 11 v2 2( d y y + I dy 2 r ( V 2 12) =p0U i  dx + 1 since v/V =  dy/dx.
.^ In the momentum equation (4) we may now omit X and it becomes 0.+P(U  u) 2 =poU2.
^ Equating this to the momentum entering at B and subtracting P' from each X+W+p(U  u)2 =poU 2.
^ U cubic centimetres move in per second at B, and if the density is po the mass moving in through a square centimetre is po U. But it has velocity U, and therefore momentum poU 2 is carried in.
.^ Let us move the rod from right to left, so that the undisturbed parts move with velocity U. Then the disturbance remains fixed in space.
^ If U (yP/po) is the velocity for small disturbances, we may put Uo for U in the small term on the right, and we have y I u U =Uo I + 4 (Jo or U = Uo+4 (y} I) u.
^ Suppose that a disturbance is travelling with velocity U unchanged in form along a rod from left to right.
.^ This investigation is obviously not exact, for it assumes that the form is unchanged, i.e.
.^ B, an assumption no longer tenable when the form changes.
.^ But for very small times the assumption may perhaps be made, and the result at least shows the way in which the velocity is affected by the addition of a small term depending on and changing sign with u.
^ In ordinary soundwaves the effect of the particle velocity in affecting the velocity of transmission must be very small.
^ It implies that the different parts of a wave move on at different rates, so that its form must change.
As we obtained the result on the
supposition of unchanged form, we can of course only apply it for
such short lengths and such short times that the part dealt with
does not appreciably alter.
.^ We see at once that, where u=o, the velocity has its " normal " value, while where u is positive the velocity is in excess, and where u is negative the velocity is in defect of the normal value.
^ Regnault's apparatus, found, that the velocity could be represented by 33 3(1 +C/P), where P is the mean excess of pressure above the normal.
^ Or we may take it as representing the pressure  excess over the normal pressure in compression, defect from it in extension.
If, then, a (fig. io)
represents the displacement curve of a train of waves, will
represent the pressure excess and particle velocity, and from
.^ If the steepness gets very great our investigation ceases to apply, and neither experiment nor theory has yet shown what happens.
^ (II) we see that while the nodal conditions of b, with Co' and u=o, travel with velocity 1/(E/p), the crests exceed that velocity by 1(7 + i)u, and the hollows fall short of it by 1(7 + I)u, with the result that the fronts of the pressure waves become steeper and steeper, and the train b changes into something like c.
^ We shall investigate the external force needed to make a train of plane waves travel on unchanged in form with velocity U. We shall regard the external force as applied in the form of a pressure X per square centimetre parallel to the line of propagation and varied from point to point as required in order to make the disturbance travel on unchanged in form with the specified velocity U. In addition there will be the internal force due to the change in volume, and consequent change in pressure, from point to point.
.^ Probably there is a breakdown of the wave somewhat like the breaking of a waterwave when the crest gains on the next trough.
.^ In ordinary soundwaves the effect of the particle velocity in affecting the velocity of transmission must be very small.
^ In order that the ear may be affected by a sounding body there must be continuous matter reaching all the way from the body to the ear.
^ When a wave of sound meets a surface separating two media it is in part reflected, travelling back from the surface into the first medium again with the velocity with which it approached.
FIG. 10.
.^ Experiments, referred to later, have been made to find the amplituae of swing of the air particles in organ pipes.
^ The disturbance made at the commencement of the blowing will no doubt set the air in the pipe vibrating in its own natural period, just as any irregular air disturbance will set a suspended body swinging in its natural period, but we are to consider how the vibration is maintained when once set going.
^ In the organ pipe  as in the common whistle  a thin sheet of air is forced through a narrow slit at the bottom of the embouchure and impinges against the top edge, which is made very p c.
.^ Thus Mach found an amplitude 0 2 cm.
^ Wien used a telephone plate, of which the amplitude could be determined from the value of the exciting current, and he found that the smallest amplitude audible was 6.3 X t010 cm.
when the issuing waves were 250 cm.
long.
.^ The amplitude in the pipe was certainly much greater than in the issuing waves.
^ Since U=n X where U is the velocity of sound, X the wavelength, and n the frequency, it follows that the forward frequency is greater than the backward frequency.
^ This is obvious if we consider that the greater the vibration of the source the greater is the excursion of the air in the issuing waves, and the louder is the sound heard.
.^ Let us take the latter as 0.1 mm.
in the waves  a very extreme value.
.^ The maximum velocity of a particle in the wavetrain is the amplitude of dy/dt.
^ The maximum particle velocity is 21rna (where n is the frequency and a the amplitude), or 27raU/X. This gives maximum u=about 8 cm./sec., which would not seriously change the form of the wave in a few wavelengths.
^ In ordinary soundwaves the effect of the particle velocity in affecting the velocity of transmission must be very small.
.^ Meanwhile the waves are spreading out and the value of u is falling in inverse proportion to the distance from the source, so that very soon its effect must become negligible.
^ In ordinary soundwaves the effect of the particle velocity in affecting the velocity of transmission must be very small.
^ The waves from a source surrounded by a uniform medium at rest spread out as spheres with the source as centre.
.^ In loud sounds, such as a peal of thunder from a near flash, or the report of a gun , the effect may be considerable, and the rumble of the thunder and the prolonged boom of the gun may perhaps be in part due to the breakdown of the wave when the crest of maximum pressure has moved up to the front, though it is probably due in part also to echo from the surfaces of heterogeneous masses of air.
^ When two notes are not quite in unison the resulting sound is found to alternate between a maximum and minimum of loudness recurring periodically.
^ When a wave of sound meets a surface separating two media it is in part reflected, travelling back from the surface into the first medium again with the velocity with which it approached.
.^ But there is no doubt that with very loud explosive sounds the normal velocity is quite considerably exceeded.
^ When two notes are not quite in unison the resulting sound is found to alternate between a maximum and minimum of loudness recurring periodically.
^ But there is no doubt that it is very difficult to detect the summation tone by the ear, and many workers have doubted the possibility, notwithstanding the evidence of such an observer as Helmholtz.
.^ Kirchhoff's formula, Violle and Vautier found for the velocity in open air at o° C. .
^ Thus Regnault in his classical experiments (described below) found that the velocity of the report of a pistol carried through a pipe diminished with the intensity, and his results have been confirmed by J. Violle and T. Vautier (see below).
^ Experiments on the velocity in pipes were carried out by H. Schneebeli ( Pogg.
W. W. Jacques (
Phil.
Mag., 1879, 7, p.
.^ Newton found 979 ft./sec.
at 70, to 90
ft. in the
rear and then fell
off.
A very curious observation is recorded by the Rev. G. Fisher in
an appendix to Captain Parry's
Journal of a Second Voyage the
Arctic Regions. In
describing experiments on the velocity of sound he states that " on
one day and one day only, February 9, 1822, the officer's word of
command ` fire ' was several times heard distinctly both by Captain
Parry and myself about one
beat of the chronometer [nearly half
a second]
after the report of the gun." This is hardly to
be explained by equation (I I), for at the very front of the
disturbance
u =o and the velocity should be normal.
The Energy in a Wave Train
.^ The energy in a train of waves carried forward with the waves is partly strain or potential energy due to change of volume of the air, partly kinetic energy due to the motion of the air as the waves pass.
^ For the superposition of these trains will give a stationary wave between A H A (16) Y which is an equation characteristic of simple harmonic motion.
^ U, for in the stationary wave system the pressure change and the amplitude are both double those in either train, so that the same relation holds.
.^ We shall show that if we sum these up for a whole wave the potential energy is equal to the kinetic energy.
^ During the quarter swing ending with greatest nodal pressure, the kinetic energy is changed to potential energy manifested in the increase of pressure.
^ Intermittent illumination, however, with frequency equal to that of the fork shows at once that the jet is really broken up into drops, one for each vibration, and that these move over in a steady procession .
.^ The kinetic energy per cubic centimetre is 2 pu t, where is the density and u is the velocity of disturbance due to the passage of the wave.
^ We shall investigate the external force needed to make a train of plane waves travel on unchanged in form with velocity U. We shall regard the external force as applied in the form of a pressure X per square centimetre parallel to the line of propagation and varied from point to point as required in order to make the disturbance travel on unchanged in form with the specified velocity U. In addition there will be the internal force due to the change in volume, and consequent change in pressure, from point to point.
^ Hence the work done on the air is (P+zw)v, and the work done per cubic centimetre is (P+Zw)v/V. The term Pv/V added up for a complete wave vanishes, for P/V is constant and Zv=o, since on the whole the compression equals the extension.
.^ If V is the undisturbed volume of a small portion of the air at the undisturbed pressure P, and if it becomes V  v when the pressure increases to P+w, the average pressure during the change may be taken as PH1a), since the pressure excess for a small change is proportional to the change.
^ Let MP =y represent the forward displacement of the particle originally at M, and NQ = y +dy that of the particle originally at N. The layer of air originally of thickness dx now has thickness dx+dy, since N is displaced forwards dy more than M. The volume dx, then, has increased to dx+dy or volume I has increased to I +dy/dx and the increase of volume I is dy/dx.
^ If w is the total pressure excess, and if y is the total displacement at x, then w = E Xchange of volume _original volume =  Edy/dx.
.^ Hence the work done on the air is (P+zw)v, and the work done per cubic centimetre is (P+Zw)v/V. The term Pv/V added up for a complete wave vanishes, for P/V is constant and Zv=o, since on the whole the compression equals the extension.
^ Hence the stream of air does work during half the vibration and this is not abstracted during the other half, and so it goes on increasing the motion until the supply of energy in blowing is equal to the loss by friction and sound.
^ The figure shows that when the curve of displacement slopes down in the direction of propagation there is compression, and the pressure is above the normal, and that when it slopes up there is extension, and the pressure is below the normal.
We have then only to consider the term z iav/V.
.^ But v/V =u/U from equation (2) and w =Eu/U from equation (3) Then 2wv/V = ZEu 2 /U 2 = 2 pu t from equation (6) Then in the whole wave the potential energy equals the kinetic energy and the total energy in a complete wave in a column 1 sq.
^ We shall show that if we sum these up for a whole wave the potential energy is equal to the kinetic energy.
^ During the quarter swing ending with greatest nodal pressure, the kinetic energy is changed to potential energy manifested in the increase of pressure.
cm.
crosssection is
.^ I) We may find here the value of this when we have a train of waves in which the displacement is represented by a sine curve of amplitude a, viz.
^ If a wave travels on without alteration the travelling may be represented by pushing on the displacement curve.
^ Let it be represented by a displacement curve Ahbkc .
y=a sin (x  Ut).
.^ If P is the undisturbed pressure and P+w the pressure at AB, the momentum entering through AB per second isJ01(P+w+pu2)dt.
^ Assuming this energy to be propagated in hemispherical waves, it is easy to find the quantity per second going through I sq.
^ This second plate is capable of rotation about an axis perpendicular to its plane and passing through its centre.
cm.
perpendicular to the line of propagation is
.^ Sound waves, like light waves, exercise a small pressure against any surface upon which they impinge.
^ U3a2/X2 (14) The Pressure of Sound Waves.

.^ Sound waves, like light waves, exercise a small pressure against any surface upon which they impinge.
^ This want of proportionality will have a periodicity, that of the impinging waves, and so will produce vibrations just as does the variation of pressure in the case last investigated.
^ When a wave of sound meets a surface separating two media it is in part reflected, travelling back from the surface into the first medium again with the velocity with which it approached.
.^ The existence of this pressure has been demonstrated experimentally by W. Altberg ( Ann.
der . 1903, II,
p.
^ Physik, 1903, II, p.
405).
.^ A small circular disk at one end of a torsion arm formed part of a solid wall, but was free to move through a hole in the wall slightly larger than the disk.
^ When intense sound waves impinged on the wall, the disk moved back through the hole, and by an amount showing a pressure of the order given by the following investigation: Suppose that a train of waves is incident normally on the surface S (fig.
^ The cases interesting in sound are those in which (i) the bar is free at both ends, and (2) it is clamped at one end and free at the other.
When intense sound waves
impinged on the wall, the disk moved back through the hole, and by
an amount showing a pressure of the order given by the following
investigation: Suppose that a train of waves is incident normally
on the surface S (fig. II), and that they are absorbed there
without reflection.
Let ABCD be a column of air 1 sq. cm. crosssection.
.^ The pressure on CD is equal to the A C momentum which it receives per second.
^ This is generally equal to the number of waves issuing from the source per second, and therefore equal to its frequency of vibration.
^ But the tension at P is T, parallel to the tangent, and T sin 4 parallel to PM, and through this  T sin is the momentum passing out at P per second.
On the whole the air S within ABCD neither gains nor
g D loses momentum, so that on the whole it
receives as much through AB as it gives up to CD.
.^ If P is the undisturbed pressure and P+w the pressure at AB, the momentum entering through AB per second isJ01(P+w+pu2)dt.
^ The pressure on CD is equal to the A C momentum which it receives per second.
^ So that no angular momentum enters at B, and therefore on the whole none leaves at A. The transfer of angular momentum through A is of two kinds  first, that due to the passage of rotating matter, and, secondly, that due to the couple with which matter to the right of A acts upon matter to the left of A. The mass of matter moving through A per second is pwa 2 U, where a is the radius of the wire and p is its density.
.^ Buti o Pdt = P is the normal pressure, and as we only wish to find the excess we may leave this out of account.
^ It is evident that the pressure condition will be fulfilled only if the motions in the two tubes are in the same direction at the same time, closing into and opening out from the nodes together.
^ If V is the undisturbed volume of a small portion of the air at the undisturbed pressure P, and if it becomes V  v when the pressure increases to P+w, the average pressure during the change may be taken as PH1a), since the pressure excess for a small change is proportional to the change.
.^ The excess pressure on CD is therefore 4 1 (c:3+ pu 2)dt.
^ But the values of 2 which occur successively during the second at AB exist simultaneously at the beginning of the second over the distance U behind AB. Or if the conditions along this distance U could be maintained constant, and we could travel back along it uniformly in one second, we should meet all the conditions actually arriving at AB and at the same intervals.
^ The pressure on CD will therefore be doubled.
.^ The excess pressure on CD is therefore 4 1 (c:3+ pu 2)dt.
^ If then dl is an element of the path, putting dt = do/U, we have the average excess of pressure p = ° (to } pu 2) dt = Lj j ( +pu2)dI. Here dE is an actual length in the disturbance.
^ If P is the undisturbed pressure and P+w the pressure at AB, the momentum entering through AB per second isJ01(P+w+pu2)dt.
.^ Let MP =y represent the forward displacement of the particle originally at M, and NQ = y +dy that of the particle originally at N. The layer of air originally of thickness dx now has thickness dx+dy, since N is displaced forwards dy more than M. The volume dx, then, has increased to dx+dy or volume I has increased to I +dy/dx and the increase of volume I is dy/dx.
^ A'B'C', where AA ' = Udt, the displacement of the particle originally at M must change from PM to P'M or by PP'.
^ The sum of the disturbance is obtained by adding (24) and (25) y = y l +y 2 = 2a cos Ut s i n 57 x, (26) At any given instant t this is a sine curve of amplitude 2a cos (27r/A)Ut, and of wavelength A, and with nodes at x = o, a A, A, ..., that is, there is no displacement at these nodes whatever the value of t, and between them the displacement is always a sine curve, but of amplitude varying between +2a and  2a.
.^ Missing image Sound7.jpg In a later series of experiments Lord Rayleigh ( Phil.
^ Lord Rayleigh and Sir William Ramsay ( Phil.
^ For a discussion of this type of wave, u = dt =  U¢ cos ( x  Ut ), and ° 4, x Za2 / cos t (x  Ut) dx pu2ax=p = 2p7r 2 U 2 a 2 /X (12) The energy per cubic centimetre on the average is 2 pif2 U2a2 / A2 (13) and the energy passing per second through I sq.
. 1905, 10, p.
^ Mag., 1905, 10, p.
364).
.^ If the train of waves is reflected, the value of p at AB will be the sum of the values for the two trains, and will, on the average, be doubled.
^ When a wave of sound meets a surface separating two media it is in part reflected, travelling back from the surface into the first medium again with the velocity with which it approached.
^ U, for in the stationary wave system the pressure change and the amplitude are both double those in either train, so that the same relation holds.
.^ The pressure on CD will therefore be doubled.
^ The excess pressure on CD is therefore 4 1 (c:3+ pu 2)dt.
.^ But the energy will also be doubled, so that (15) still gives the average excess of pressure.
^ If V is the undisturbed volume of a small portion of the air at the undisturbed pressure P, and if it becomes V  v when the pressure increases to P+w, the average pressure during the change may be taken as PH1a), since the pressure excess for a small change is proportional to the change.
^ If then dl is an element of the path, putting dt = do/U, we have the average excess of pressure p = ° (to } pu 2) dt = Lj j ( +pu2)dI. Here dE is an actual length in the disturbance.
Experimental Determinations of the Velocity of Sound.
An obvi us method of determining the velocity of sound in air
consists in starting some sound, say by firing a gun, and
stationing an observer at some measured distance from the gun.
.^ The times were measured by pendulum clocks.
^ The observer measures by a clock or chronometer the time elapsing between the receipt of the flash, which passes practically instantaneously, and the receipt of the report.
^ An observer with his ear to the tube noted the interval between the arrival of flash and sound.
.^ The distance divided by the time gives the velocity of the sound.
^ An obvi us method of determining the velocity of sound in air consists in starting some sound, say by firing a gun, and stationing an observer at some measured distance from the gun.
^ But when the wind is steady its effect may be eliminated by " reciprocal " observations, that is, by observations of the time of passage of sound in each direction over the measured distance..
The velocity
thus obtained will be affected by the wind.
.^ For instance, William Derham ( Phil.
. 1708) made a series of observations, noting
the time taken by the report of a cannon.
^ Trans., 1708) made a series of observations, noting the time taken by the report of a cannon.
fired on
.^ Blackheath to travel across the Thames to Upminster Church in Essex , 121m.
away.
.^ He found that the time varied between 551seconds when the wind was blowing most strongly with the sound, to 63 seconds when it was most strongly against the sound.
^ When two notes are not quite in unison the resulting sound is found to alternate between a maximum and minimum of loudness recurring periodically.
^ The time between the breaks could be measured in seconds by the clock signals, and in fractions of a second by the tuningfork record.
.^ The value for still air he estimated at 1142 ft.
per second.
.^ He made no correction for temperature or humidity.
^ The temperature of the air traversed and its humidity were observed, and the result was finally corrected to the velocity in dry air at o C. by means of equation (ro).
.^ But when the wind is steady its effect may be eliminated by " reciprocal " observations, that is, by observations of the time of passage of sound in each direction over the measured distance..
^ An obvi us method of determining the velocity of sound in air consists in starting some sound, say by firing a gun, and stationing an observer at some measured distance from the gun.
^ Let T, and T2 be the observed times of passage in the two directions.
.^ Let the velocity of the air from S to R be w, and let U be the velocity of sound in still air.
^ Velocity of Sound in Air and other Gases in Pipes .
^ Let D be the distance, U the velocity of sound in still air, and Tr) the velocity of the wind, supposed for simplicity to blow directly from one station to the other.
.^ Let T, and T2 be the observed times of passage in the two directions.
^ The direct and reflected systems are practically equal, and by suitably timing the vibrations of the hand for each case the rope may be made to vibrate as a whole, as two halves, as threethirds and so on.
^ But when the wind is steady its effect may be eliminated by " reciprocal " observations, that is, by observations of the time of passage of sound in each direction over the measured distance..
We have UFw= D/T
1 and U 
w = D/T . Adding and dividing by 2 U =D T1+T2)'
If T i and T2 are nearly equal, and if T=z(T 1 +T
2), this is very nearly U= D/T.
.^ The reciprocal method was adopted in 1738 by a commission of the French Academy ( Memoires de l'academie des sciences, (1738).
^ Regnault's work, is given in the Memoires de l'academie des sciences, 1868, xxxvii.
.^ Cannons were fired at halfhour intervals, alternately at Montmartre and Montlhery, 17 or 18 m.
^ Cannons were fired at the two stations at intervals of five minutes.
apart. There were also two
intermediate stations at which observations were made.
.^ The times were measured by pendulum clocks.
^ The time between the breaks could be measured in seconds by the clock signals, and in fractions of a second by the tuningfork record.
^ The observer measures by a clock or chronometer the time elapsing between the receipt of the flash, which passes practically instantaneously, and the receipt of the report.
.^ The result obtained at a temperature about 6° C. was, when converted to metres, U=337 metres/second.
The theoretical investigation given above shows that if U is the
velocity in air at 1° C. then the velocity U ° at o° C. in
the same air is independent of the barometric pressure and that Uo
= U /(1 +o o01841), whence U 0 =332 met./sec.
.^ In 1822 a commission of the Bureau des Longitudes made a series of experiments between Montlhery and Villejuif, I r m.
apart.
.^ Cannons were fired at the two stations at intervals of five minutes.
^ Cannons were fired at halfhour intervals, alternately at Montmartre and Montlhery, 17 or 18 m.
^ To eliminate wind as far as possible reciprocal firing was adopted, the interval between the two firings being only a few seconds.
.^ U 0 =331.37 met./sec.; but the number of experiments over the longer distance was greater, and he appears to have put more confidence in the result from them, viz.
^ Chronometers were used for timing, and the result at 15.9° C. was U = 340 9 met./sec., whence U0=330.6 met./sec.
^ Uo=330 6 met./sec., while for a diameter 0.108 it was U 0 =324' 25 met./sec.
.^ (F. J. D. Arago, Connaissance des temps, 1825).
.^ When the measurement of a time interval depends on an.
observer,
his " personal equation " comes in to affect the estimation of the
quantity.
.^ This is the interval between the arrival of an event and his perception that it has arrived, or it may be the interval between arrival and his record of the arrival.
^ An observer with his ear to the tube noted the interval between the arrival of flash and sound.
^ But the interval between 4 and 4`56 414 is quite perceptible, and on the piano, for instance, a separate string must be provided above f.
This personal equation is different for
different observers.
.^ It may differ even by a considerable fraction of a second..
.^ It is different, too, for different senses with the same observer, and different even for the same sense when the external stimuli differ in intensity.
.^ An observer with his ear to the tube noted the interval between the arrival of flash and sound.
^ This personal equation is different for different observers.
^ When the interval between a flash and a report is measured, the personal equations for the two arrivals are, in all probability, different, that for the flash being most likely less than that for the sound.
.^ U/2L, where U is the velocity of sound in the pipe.
^ In the openair experiments the receiver consisted of a large see below.
^ In a long series of experiments carried out by V. Regnault in the years 1862 to 1866 on the velocity of sound in open air, in air in pipes and in various other gases in pipes, he sought to eliminate personal equaticn by dispensing with the human element in the observations, using electric receivers as observers.
.^ A short account of these experiments is given in Phil.
^ An excellent account of these and other jets is given in C. V. Beys' Soap Bubbles, lecture iii.
^ Experiments may be made with plane and curved mirrors to verify these laws, but it is necessary to use short waves, in order to diminish diffraction effects.
Mag., 1868, 35, p.
.^ Regnault's work, is given in the Memoires de l'academie des sciences, 1868, xxxvii.
^ In the memoir cited above Regnault gives an account of determinations of the velocity in air in pipes of great length and of diameters ranging from o 108 metres to i i metres.
^ The example we have given above of the major seventh must serve here.
.^ On page 459 of the Memoire will be found a list of previous careful experiments on the velocity of sound.
^ They found that the velocity of propagation of different musical sounds was the same.
^ Experiments on the velocity of sound in iron have been made on lengths of iron piping by J. B. Biot, and on telegraph wires by Wertheim and Brequet.
.^ In the openair experiments the receiver consisted of a large see below.
^ Thus in the airpump experiment, before exhaustion it travels through the glass of the receiver and the base plate.
We have FIG. II.
cone having a thin
india
rubber membrane stretched over its narrow end.
.^ A small metal disk was attached to the centre of the membrane and connected to earth by a fine wire.
^ To the centre of this membrane is attached a small feather fibre, which, when the reflector is suitably placed, touches lightly the surface of the revolving cylinder.
^ A metal contactpiece adjustable by a screw could be made to just touch a point at the centre of the disk.
.^ A metal contactpiece adjustable by a screw could be made to just touch a point at the centre of the disk.
^ A small metal disk was attached to the centre of the membrane and connected to earth by a fine wire.
^ When the wave travelled to the receiver it pushed back the disk from the contactpiece, and this break, too, was recorded.
.^ When contact was made it completed an electric circuit which passed to a recording station, and there, by means of an electromagnet, actuated a style writing a record on a band of travelling smoked paper.
^ There were also two intermediate stations at which observations were made.
^ On the same band a tuning fork electrically maintained and a seconds clock actuating another style wrote parallel records.
.^ On the same band a tuning fork electrically maintained and a seconds clock actuating another style wrote parallel records.
^ Then the wheel makes 8 N + 1 revolutions in N clock beats, and the fork makes 32 N + 4 vibrations in the same time.
^ If, at the same time, a tuningfork of known number of vibrations per second be made to trace its own line close to the other, a comparison of the two lines gives the number corresponding to the sound under consideration.
The circuit was continued to the gun which served
as a source, and stretched across its muzzle.
.^ When the gun was fired, the circuit was broken, and the break was recorded on the paper.
.^ The circuit was at once remade.
.^ When the wave travelled to the receiver it pushed back the disk from the contactpiece, and this break, too, was recorded.
^ When a wave of sound meets a surface separating two media it is in part reflected, travelling back from the surface into the first medium again with the velocity with which it approached.
^ A metal contactpiece adjustable by a screw could be made to just touch a point at the centre of the disk.
.^ The time between the breaks could be measured in seconds by the clock signals, and in fractions of a second by the tuningfork record.
^ The times were measured by pendulum clocks.
^ If, at the same time, a tuningfork of known number of vibrations per second be made to trace its own line close to the other, a comparison of the two lines gives the number corresponding to the sound under consideration.
.^ The receiving apparatus had what we may term a personal equation, for the break of contact could only take place when the membrane travelled some finite distance, exceedingly small no doubt, from the contactpiece.
^ When the wave travelled to the receiver it pushed back the disk from the contactpiece, and this break, too, was recorded.
^ We may see how this occurs by supposing that the restoring force of the receiving mechanism is represented by Ax fµx 2, where x is the displacement and µx 2 is very small.
But the
apparatus was used in such a way that this could be neglected.
.^ In this experiment the personal equations of the observers were determined and allowed for.
^ In some experiments in which contact was made instead of broken, Regnault determined the personal equation of the apparatus.
^ The receiving apparatus had what we may term a personal equation, for the break of contact could only take place when the membrane travelled some finite distance, exceedingly small no doubt, from the contactpiece.
.^ To eliminate wind as far as possible reciprocal firing was adopted, the interval between the two firings being only a few seconds.
^ But when the wind is steady its effect may be eliminated by " reciprocal " observations, that is, by observations of the time of passage of sound in each direction over the measured distance..
^ We see lightning before we hear the thunder which spreads out from the flash, and the more distant the flash the longer the interval between the two.
.^ The temperature of the air traversed and its humidity were observed, and the result was finally corrected to the velocity in dry air at o C. by means of equation (ro).
^ He made no correction for temperature or humidity.
^ If we take y =1.4 we obtain approximately for the velocity in dry air at 0° C. .
.^ Regnault used two different distances, viz.
.^ U 0 =331.37 met./sec.; but the number of experiments over the longer distance was greater, and he appears to have put more confidence in the result from them, viz.
^ The result obtained at a temperature about 6° C. was, when converted to metres, U=337 metres/second.
^ At present we cannot assign a more exact value than Uo = 331 metres per second.
.^ Uo=330 71 met./sec.
^ The theoretical investigation given above shows that if U is the velocity in air at 1° C. then the velocity U ° at o° C. in the same air is independent of the barometric pressure and that Uo = U /(1 +o o01841), whence U 0 =332 met./sec.
^ Uo =331 To met./sec.
In the
Phil. . 1872, 162, p.
^ Trans., 1872, 162, p.
.^ E. J. Stone at the Cape of Good Hope.
.^ This personal equation is different for different observers.
^ In this experiment the personal equations of the observers were determined and allowed for.
^ In some experiments in which contact was made instead of broken, Regnault determined the personal equation of the apparatus.
Velocity of Sound in Air and other Gases in Pipes
In the memoir cited above Regnault gives an account of
determinations of the velocity in air in pipes of great length and
of diameters ranging from o 108 metres to i i metres.
.^ He used various sources and the method of electric registration .
.^ He found that in all cases the velocity decreased with a diameter.
.^ They found that the velocity of propagation of different musical sounds was the same.
^ Sound takes Time to Travel .
^ The sound travelled to and fro in the pipes several times before the signals died away, and he found that the velocity decreased with the intensity, tending to a limit for very feeble sounds, the limit being the same whatever the source.
This limit for a diameter
1 1
m.
was . 6 met./sec., while for a diameter 0.108 it was
U 0 =324' 25 met./sec.
^ Uo =331 To met./sec.
^ Chronometers were used for timing, and the result at 15.9° C. was U = 340 9 met./sec., whence U0=330.6 met./sec.
^ Uo=330 6 met./sec., while for a diameter 0.108 it was U 0 =324' 25 met./sec.
.^ Regnault also set up a shorter length of pipes of diameter o 108 m.
^ In the memoir cited above Regnault gives an account of determinations of the velocity in air in pipes of great length and of diameters ranging from o 108 metres to i i metres.
in a court at the College de
France, and with this length he could use dry
air, vary the pressure, and fill with other gases. He found that
within wide limits the velocity was independent of the pressure,
thus confirming the theory.
.^ They found that the velocity of propagation of different musical sounds was the same.
^ Comparing the velocities of sound U i and U2 in two different gases with densities and at the same temperature and pressure, and with ratios of specific heats 'yl, 72, theory gives Ui/U2 = 1/ {71 p 2/72 p i }.
^ If yl is known this gives 72.
.^ This formula was very nearly confirmed for hydrogen , carbon dioxide and nitrous oxide .
.^ Violle and Vautier made some later experiments on the propagation of musical sounds in a tunnel 3 metres in diameter ( Ann.
^ J. Violle and T. Vautier ( Ann.
chim. phys., 1890, vol.
19) made observations with a tube o. 7 m. in diameter, and, using
.^ Regnault's apparatus, found, that the velocity could be represented by 33 3(1 +C/P), where P is the mean excess of pressure above the normal.
^ He found that within wide limits the velocity was independent of the pressure, thus confirming the theory.
^ Buti o Pdt = P is the normal pressure, and as we only wish to find the excess we may leave this out of account.
.^ According to von Helmholtz and Kirchhoff the velocity in a tube should be less than that in free air by a quantity depending on the diameter of the tube, the frequency of the note used, and the viscosity of the gas (Rayleigh, Sound, vol.
^ The maintenance of the vibration of the air in the singing tube has been explained by Lord Rayleigh ( Sound, vol.
^ The jet tube, for a reason which will be given when we consider the maintenance of vibrations, must be less than c a9 half the length of the sounding tube.
ii. §§
3478) .
.^ Correcting the velocity obtained in the 0 .
7
m. tube by Kirchhoff's formula, Violle and Vautier found for the
velocity in open air at o° C.
.^ U 0 =331.37 met./sec.; but the number of experiments over the longer distance was greater, and he appears to have put more confidence in the result from them, viz.
^ Uo =331 To met./sec.
^ Uo=330 6 met./sec., while for a diameter 0.108 it was U 0 =324' 25 met./sec.
with a probable error estimated at o Io
metre.
.^ Let the velocity of the air from S to R be w, and let U be the velocity of sound in still air.
^ It is obvious from the various experiments that the velocity of sound in dry air at o° C. is not yet known with very great accuracy.
^ In a long series of experiments carried out by V. Regnault in the years 1862 to 1866 on the velocity of sound in open air, in air in pipes and in various other gases in pipes, he sought to eliminate personal equaticn by dispensing with the human element in the observations, using electric receivers as observers.
.^ At present we cannot assign a more exact value than Uo = 331 metres per second.
^ Let us seek a more'exact value for w.
^ Missing image Sound1.jpg Missing image Sound2.jpg The pitch of a sound, the note which we assign to it, depends on the number of waves received by the ear per second.
.^ Violle and Vautier made some later experiments on the propagation of musical sounds in a tunnel 3 metres in diameter ( Ann.
^ Missing image Sound7.jpg In a later series of experiments Lord Rayleigh ( Phil.
^ They found that the velocity of propagation of different musical sounds was the same.
chim.
phys., 1905, vol. 5).
.^ They found that the velocity of propagation of different musical sounds was the same.
^ Helmholtz investigated the velocity of propagation of sound in pipes, taking into account the viscosity of the air (Rayleigh, Sound, ii.
^ Though a musical note has definite pitch or frequency, notes of the same pitch emitted by different instruments have quite different quality or timbre.
.^ Some curious effects were observed in the formation of harmonics in the rear of the primary tone used.
These have yet to find an explanation.
Velocity of Sound in Water
.^ The velocity in water was measured by J. D. Colladon and J. K. F. Sturm ( Ann.
chim. phys., 1827 (2), 36, p. 236) in the
water of
Lake
Geneva.
.^ A bell under water was struck, and at the same instant some gunpowder was flashed in air above the bell.
.^ At a station more than 13 kilometres away a sort of big ear trumpet , closed by a membrane, was placed with the membrane under water, the tube rising above the surface.
^ A " sounding tube," say an inch in diameter, and somewhat more than twice the length of the jet tube, is then lowered over the flame, as in the figure.
^ To the centre of this membrane is attached a small feather fibre, which, when the reflector is suitably placed, touches lightly the surface of the revolving cylinder.
.^ An observer with his ear to the tube noted the interval between the arrival of flash and sound.
^ When two notes are not quite in unison the resulting sound is found to alternate between a maximum and minimum of loudness recurring periodically.
^ We see lightning before we hear the thunder which spreads out from the flash, and the more distant the flash the longer the interval between the two.
.^ The velocity deduced at 8.1° C. was U=1435 met./sec., agreeing very closely with the value calculated from the formula U 2 = E/p.
^ The theoretical investigation given above shows that if U is the velocity in air at 1° C. then the velocity U ° at o° C. in the same air is independent of the barometric pressure and that Uo = U /(1 +o o01841), whence U 0 =332 met./sec.
Experiments on the velocity of sound in
iron have been made on lengths of
iron piping by J. B. Biot, and on
telegraph wires by Wertheim and Brequet.
.^ The experiments were not satisfactory, and it is sufficient to say that the results accorded roughly with the value given by theory.
.^ When a wave of sound meets a surface separating two media it is in part reflected, travelling back from the surface into the first medium again with the velocity with which it approached.
^ It further follows, as in the analogous case of light, that there is a certain angle termed the critical angle, whose sine is found by dividing the less by the greater velocity, such that all rays of sound meeting the surface separating two different bodies will not pass onward, but suffer total reflection back into the first body, if the.
^ Helmholtz investigated the velocity of propagation of sound in pipes, taking into account the viscosity of the air (Rayleigh, Sound, ii.
.^ Echo is a familiar example of this.
.^ The laws of reflection of sound are identical with those of the reflection of light , viz.
^ This change of direction is termed refraction, and takes place, no doubt, according to the same laws as does the refraction of light , viz.
(1) the
planes of incidence and reflection are coincident, and (2) the
angles of incidence and reflection are equal.
.^ Experiments may be made with plane and curved mirrors to verify these laws, but it is necessary to use short waves, in order to diminish diffraction effects.
^ Fourier's theorem asserts that such a curve may be built up by the superposition, or addition of ordinates, of a series of sine curves of wavelengths X, IX, 3A, 4A...
^ If we take one of these spheres a distance from the source very great as compared with a single wavelength, and draw a radius to a point on the sphere, then for some little way round that point the sphere may be regarded as a plane perpendicular to the radius or the line of propagation.
For instance, a
ticking watch may be put at the
focus of a large concave metallic
mirror, which sends a parallel
"
beam " of sound to a second
concave mirror facing the first.
.^ If an eartrumpet is placed at the focus of the second mirror the ticking may be heard easily, though it is quite inaudible by direct waves.
^ For instance, a ticking watch may be put at the focus of a large concave metallic mirror , which sends a parallel " beam " of sound to a second concave mirror facing the first.
^ At a station more than 13 kilometres away a sort of big ear trumpet , closed by a membrane, was placed with the membrane under water, the tube rising above the surface.
.^ Or it may be revealed by placing a sensitive flame of the kind described below with its nozzle at the focus.
^ If an eartrumpet is placed at the focus of the second mirror the ticking may be heard easily, though it is quite inaudible by direct waves.
^ This may be shown by furnishing the pipes with manometric flames placed in the same vertical line.
.^ The flame jumps down at every tick.
^ The membrane vibrates, and alternately checks and increases the gas supply, and the flame jumps up and down with the frequency of the source.
^ Such a flame may jump down, for instance, to each tick of a neighbouring clock.
.^ Examples of reflection of sound in buildings are only too frequent.
.^ In large halls the words of a speaker are echoed or reflected from flat walls or roof or floor; and these reflected sounds follow the direct sounds at such an interval that syllables and words overlap, to the confusion of the speech and the annoyance of the audience .
^ In some cases of echo, when the original sound is a compound musical note, the octave of the fundamental tone is reflected much more strongly than that tone itself.
.^ Some curious examples of echo are given in Herschel's article on " Sound " in the Encyclopaedia Metropolitana, but it appears that he is in error in one case.
^ We shall only consider one interesting case of sound diffraction which may be easily observed.
^ The difference in loss in the two cases measured the energy given up to and sent out by the resonator as sound.
He states that in the
whispering gallery in St Paul's,
London, " the faintest sound is faithfully
conveyed from one side to the other of the
dome but is not heard at any intermediate point."
In some domes, for instance in a dome at the university of
Birmingham, a sound from
one end of a diameter is heard very much more loudly quite close to
the other end of the diameter than elsewhere, but in St Paul's Lord
Rayleigh found that " the abnormal loudness with which a whisper is
heard is not confined to the position diametrically opposite to
that occupied by the whisperer, and therefore, it would appear,
does not depend materially upon the symmetry of the dome.
.^ The whisper seems to creep round the gallery horizontally, not necessarily along the shorter arc, but rather along that arc towards which the whisperer faces.
.^ This is a consequence of the very unequal audibility of a whisper in front and behind the speaker, a phenomenon which may easily be observed in the open air" ( Sound, ii.
^ Helmholtz investigated the velocity of propagation of sound in pipes, taking into account the viscosity of the air (Rayleigh, Sound, ii.
^ We shall only consider one interesting case of sound diffraction which may be easily observed.
§ 287).
Let fig.
.^ A ray making an angle less than 0 with the tangent will, with its reflections, touch a larger circle.
^ P. Let OPA be the radius through P. Let PQ represent a ray of sound making the angle B with the tangent at A. Let ON(= OP cos 0) be the perpendicular on PQ. Then the reflected ray QR and the ray reflected at R, and so on, will all touch the circle drawn with ON as radius.
^ Each section then vibrates, and its amplitude goes through all its values in time given by 21rUT/A =2r, or T =A/U, and the frequency is U/A. We may represent such a train of " stationary waves " by fig.
.^ A ray making an angle less than 0 with the tangent will, with its reflections, touch a larger circle.
^ P. Let OPA be the radius through P. Let PQ represent a ray of sound making the angle B with the tangent at A. Let ON(= OP cos 0) be the perpendicular on PQ. Then the reflected ray QR and the ray reflected at R, and so on, will all touch the circle drawn with ON as radius.
.^ Hence all rays between =0 will be confined in the space between the outer dome and a circle of radius OP cos 0, and the weakening of intensity will be chiefly due to vertical spreading.
^ P. Let OPA be the radius through P. Let PQ represent a ray of sound making the angle B with the tangent at A. Let ON(= OP cos 0) be the perpendicular on PQ. Then the reflected ray QR and the ray reflected at R, and so on, will all touch the circle drawn with ON as radius.
.^ Rayleigh points out that this clinging of the sound to the surface of a concave wall does not depend on the exactness of the spherical form.
^ The essential fact, as pointed out by Lord Rayleigh ( Scientific Papers, i.
^ Since the quality of the note sounded depends on the mixture of harmonics, the quality therefore is to some extent dependent on the point of excitation.
.^ He suggests that the propagation of earthquake disturbances is probably affected by the curvature of the surface of the globe, which may act like a whispering gallery.
^ But transverse disturbances may be propagated as waves in solids.
.^ In some cases of echo, when the original sound is a compound musical note, the octave of the fundamental tone is reflected much more strongly than that tone itself.
^ Sounds may be divided into noises and musical notes.
^ H. N. Dove (18031879) 1879) produced a modification of the siren by which the relations of different musical notes may be more Dove's readily ascertained.
.^ This is explained by Rayleigh ( Sound, ii.
^ Lord Rayleigh ( Sound, ii.
^ Helmholtz investigated the velocity of propagation of sound in pipes, taking into account the viscosity of the air (Rayleigh, Sound, ii.
§ 296) as a consequence of the irregularities
of the reflecting surface. The irregularities send back a scattered
reflection of the different incident trains, and this scattered
reflection becomes more copious the shorter the wavelength.
.^ Hence the octave, though comparatively feeble in the incident train, may predominate in the scattered reflection constituting the echo.
^ The irregularities send back a scattered reflection of the different incident trains, and this scattered reflection becomes more copious the shorter the wavelength.
^ In some cases of echo, when the original sound is a compound musical note, the octave of the fundamental tone is reflected much more strongly than that tone itself.
.^ They found that the velocity of propagation of different musical sounds was the same.
^ Usually air is the medium through which sound travels, but it can travel through solids or liquids.
^ The direction, too, in which the new wave travels is different from the previous one.
.^ The direction, too, in which the new wave travels is different from the previous one.
^ For instance, if a rope is fixed at one end and held in the hand at the other end, a transverse jerk by the hand will travel as a transverse wave along the rope.
^ If the figure thus drawn is spun round its centre in the right direction in its own plane waves appear to travel out from the centre along any radius.
.^ This change of direction is termed refraction, and takes place, no doubt, according to the same laws as does the refraction of light , viz.
^ The receiving apparatus had what we may term a personal equation, for the break of contact could only take place when the membrane travelled some finite distance, exceedingly small no doubt, from the contactpiece.
^ The laws of reflection of sound are identical with those of the reflection of light , viz.
(I) The
new direction or
refracted ray lies always in the
plane of incidence, or plane which contains the incident
ray (i.e. the direction of the wave in the first medium), and the
normal to the surface separating the two media, at the point in
which the incident ray meets it; (2) The
sine of the angle
between the normal and the incident ray bears to the
sine
of the angle between the normal and the refracted ray a ratio which
is constant for the same pair of media.
.^ As with light the ratio involved in the second law is always equal to the ratio of the velocity of the wave in the first medium to the velocity in the second; in other words, the sines of the angles in question are directly proportional to the velocities.
^ The medium then acts for the second train just as if it were undisturbed by the first.
^ This is generally equal to the number of waves issuing from the source per second, and therefore equal to its frequency of vibration.
.^ Hence sound rays, in passing from one medium into another, are bent in towards the normal, or the reverse, according as the velocity of propagation in the former exceeds or falls short of that in the latter.
^ They found that the velocity of propagation of different musical sounds was the same.
^ Helmholtz investigated the velocity of propagation of sound in pipes, taking into account the viscosity of the air (Rayleigh, Sound, ii.
.^ Thus, for instance, sound is refracted towards the perpendicular when passing into air from water, 0 or into carbonic acid gas from air; the converse is the case when the passage takes place the opposite way.
^ Helmholtz investigated the velocity of propagation of sound in pipes, taking into account the viscosity of the air (Rayleigh, Sound, ii.
^ This experiment may be varied by holding the fork over a glass jar into which water is poured to such a depth that the aircolumn within reinforces the note of the fork when suitably placed, and then turning the fork round.
.^ When a wave of sound meets a surface separating two media it is in part reflected, travelling back from the surface into the first medium again with the velocity with which it approached.
^ POM being greater than 1 5z°, will not pass into the water at all, but suffer total reflection.
^ It further follows, as in the analogous case of light, that there is a certain angle termed the critical angle, whose sine is found by dividing the less by the greater velocity, such that all rays of sound meeting the surface separating two different bodies will not pass onward, but suffer total reflection back into the first body, if the.
velocity in that body is less than that in
the other body, and. if the angle of incidence exceeds the limiting
angle.
.^ The velocities in air and water being respectively 1090 and 4700 ft.
^ On the other hand, to produce convergence with water or hydrogen gas, in both which the velocity of sound exceeds its rate in air, the lens ought to be concave.
^ C. being taken as 0.001293, we get for the velocity in dry air at o° C. .
the limiting angle for these media may be easily shown to be
slightly above 152°.
.^ Hence, rays of sound proceeding from a distant source, and therefore nearly parallel to each other,.
and to PO
(fig.
.^ POM being greater than 1 5z°, will not pass into the water at all, but suffer total reflection.
^ It further follows, as in the analogous case of light, that there is a certain angle termed the critical angle, whose sine is found by dividing the less by the greater velocity, such that all rays of sound meeting the surface separating two different bodies will not pass onward, but suffer total reflection back into the first body, if the.
^ When the interval between a flash and a report is measured, the personal equations for the two arrivals are, in all probability, different, that for the flash being most likely less than that for the sound.
.^ Under such circumstances, the report of a gun, however powerful,.
should be
inaudible by an ear placed in the water.
Acoustic Lenses
.^ As light is concentrated into a focus by a.
.^ Let the velocity of the air from S to R be w, and let U be the velocity of sound in still air.
^ The jet tube, for a reason which will be given when we consider the maintenance of vibrations, must be less than c a9 half the length of the sounding tube.
^ Helmholtz investigated the velocity of propagation of sound in pipes, taking into account the viscosity of the air (Rayleigh, Sound, ii.
through a convex lens formed of carbonic
acid gas.
.^ On the other hand, to produce convergence with water or hydrogen gas, in both which the velocity of sound exceeds its rate in air, the lens ought to be concave.
^ Velocity of Sound in Air and other Gases in Pipes .
^ Let the velocity of the air from S to R be w, and let U be the velocity of sound in still air.
.^ The experimental demonstration of these results is easily made by the sonometer or monochord (fig.
^ These results have been confirmed experimentally by K. F. J. Sondhauss ( Pogg.
. 1852, 85. p.
^ Ann., 1852, 85.
378), who used a
collodion lens filled with
carbonic acid.
.^ He found its focal length and hence the refractive index of the gas, C. Hajech ( Ann.
chim. phys., 1858,
(iii). vol.
.^ In a long series of experiments carried out by V. Regnault in the years 1862 to 1866 on the velocity of sound in open air, in air in pipes and in various other gases in pipes, he sought to eliminate personal equaticn by dispensing with the human element in the observations, using electric receivers as observers.
^ An obvi us method of determining the velocity of sound in air consists in starting some sound, say by firing a gun, and stationing an observer at some measured distance from the gun.
^ Thus, for instance, sound is refracted towards the perpendicular when passing into air from water, 0 or into carbonic acid gas from air; the converse is the case when the passage takes place the opposite way.
.^ Osborne Reynolds ( Prot.
Roy. . 1874, 22, p.
^ Soc., 1874, 22, p.
53 1 ) first
pointed out that refraction would result from a variation in the
temperature of the air at different heights. The velocity of sound
in air is independent of the pressure,
T 'e ' but varies
with the temperature, its value at
t° C.
being as we have seen
.^ Now if the temperature is higher overhead than at the surface, the velocity overhead is greater.
^ Since U=n X where U is the velocity of sound, X the wavelength, and n the frequency, it follows that the forward frequency is greater than the backward frequency.
^ But usually the lower layers are warmer than the upper layers, and the velocity below is greater than the velocity above.
.^ If a wave front is in a given position, as a 1 (fig.
^ Then the front tends to swing round and travel upwards as shown in the successive positions b I, 2, 3, and 4, in fig.
^ Since the velocity increases as we go upwards the front tends to swing round and travel downwards, as shown in the successive positions a I, 2, 3 and 4, in fig.
14), at a given instant FIG.
the upper part, moving faster, gains on the lower, and the front
tends to swing round as shown by the successive positions in. a 2,
3 and 4; that is, the sound tends to come down to the surface.
.^ This is well illustrated by the remarkable horizontal.
carriage of sound
on a still clear frosty morning, when the surface layers of air are
decidedly colder than those above.
.^ Let the velocity of the air from S to R be w, and let U be the velocity of sound in still air.
^ At sunset, too, after a warm day, if the air is still, the cooling of the earth by radiation cools the lower layers, and sound carries excellently over a level surface.
^ We may obtain an excellent representation of the motion of the layers of air in a train of sound waves by means of a device due to Crova and known as " Crova's disk."
But usually the lower layers are warmer than the upper
layers, and the velocity below is greater than the velocity above.
.^ If a wave front is in a given position, as a 1 (fig.
^ Then the front tends to swing round and travel upwards as shown in the successive positions b I, 2, 3, and 4, in fig.
^ Consequently a wave front such as b 1 tends to turn upwards, as shown in the successive positions b 2, 3 and 4.
On a hot. summer's day the temperature
of the surface layers may be much higher than that of the higher
layers, and the effect on the horizontal carriage of sound may be
very marked.
.^ It is well known that sound travels far better with the wind than against it.
^ The same kind of thing happens with soundwave fronts when travelling with the wind.
^ Many of the wellknown phenomena of optical diffraction may be imitated with sound waves, especially if the waves be short.
.^ Stokes showed that this effect is one of refraction, due to variation of velocity of the air from the surface upwards Brit.
^ It is obvious from the various experiments that the velocity of sound in dry air at o° C. is not yet known with very great accuracy.
^ The theoretical investigation given above shows that if U is the velocity in air at 1° C. then the velocity U ° at o° C. in the same air is independent of the barometric pressure and that Uo = U /(1 +o o01841), whence U 0 =332 met./sec.
Assoc. . Re f raction P (p
? 1857, P by wind.
^ Rep., Re f raction P (p ?
Missing image
Sound3.jpg
22).
.^ It is, of course, a matter of common observation that the wind increases in velocity from the surface upwards.
^ Since the velocity increases as we go upwards the front tends to swing round and travel downwards, as shown in the successive positions a I, 2, 3 and 4, in fig.
^ Stokes showed that this effect is one of refraction, due to variation of velocity of the air from the surface upwards Brit.
An excellent
illustration of this increase was pointed out by F. Osler in the
shape of old clouds; their upper portions always appear dragged
forward and they lean over, as it were, in the 14.
Missing image
Sound4.jpg
A FIG. 12.
Missing image
Sound5.jpg
FIG. 13.
direction in which the wind is going.
.^ The same kind of thing happens with soundwave fronts when travelling with the wind.
^ When a wave of sound meets a surface separating two media it is in part reflected, travelling back from the surface into the first medium again with the velocity with which it approached.
^ It is well known that sound travels far better with the wind than against it.
.^ The velocity of any part of a wave front relative to the ground will be the normal velocity of sound + the velocity of the wind at that point.
^ But if the wind is against the sound the velocity of a point of the wave front is the normal velocitythe wind velocity at the point, and so decreases as we rise.
^ In ordinary soundwaves the effect of the particle velocity in affecting the velocity of transmission must be very small.
.^ If a wave front is in a given position, as a 1 (fig.
^ Then the front tends to swing round and travel upwards as shown in the successive positions b I, 2, 3, and 4, in fig.
^ Since the velocity increases as we go upwards the front tends to swing round and travel downwards, as shown in the successive positions a I, 2, 3 and 4, in fig.
.^ Suppose that a disturbance is travelling with velocity U unchanged in form along a rod from left to right.
^ Let the disturbance be supposed to travel unchanged in form from left to right with velocity U. Now suppose that the wire or rod is moved from right to left with velocity U. The disturbance is then fixed in space.
^ Let D be the distance, U the velocity of sound in still air, and Tr) the velocity of the wind, supposed for simplicity to blow directly from one station to the other.
But if the wind
is against the sound the velocity of a point of the wave front is
the normal velocitythe wind velocity at the point, and so
decreases as we rise.
.^ Since the velocity increases as we go upwards the front tends to swing round and travel downwards, as shown in the successive positions a I, 2, 3 and 4, in fig.
^ Consequently a wave front such as b 1 tends to turn upwards, as shown in the successive positions b 2, 3 and 4.
^ If a wave front is in a given position, as a 1 (fig.
.^ Suppose that a disturbance is travelling with velocity U unchanged in form along a rod from left to right.
^ Let a disturbance once set going travel along unchanged in form from A to B with velocity U. Then move AB from right to left with this velocity, and the disturbance remains fixed in space.
^ Let the disturbance be supposed to travel unchanged in form from left to right with velocity U. Now suppose that the wire or rod is moved from right to left with velocity U. The disturbance is then fixed in space.
.^ Sound is then not so well heard along the level, but may still reach an elevated observer.
^ In the first case the waves are more likely to reach and be perceived by an observer level with the source, while in the second case they may go over his head and not be heard at all.
^ But a priori we are hardly justified in assuming that waves can be propagated at all, and certainly not justified in assuming that they go on unchanged by the action of the internal forces alone.
.^ Many of the wellknown phenomena of optical diffraction may be imitated with sound waves, especially if the waves be short.
^ We shall only consider one interesting case of sound diffraction which may be easily observed.
^ We may obtain an excellent representation of the motion of the layers of air in a train of sound waves by means of a device due to Crova and known as " Crova's disk."
.^ The essential fact, as pointed out by Lord Rayleigh ( Scientific Papers, i.
^ Lord Rayleigh ( Scientific Papers, iii.
24) has given various examples, and we refer the
reader to his account.
.^ We shall only consider one interesting case of sound diffraction which may be easily observed.
^ As a preliminary to the investigation of the modes of vibration of certain sources of sound we shall consider the formation of " stationary waves."
^ The cases interesting in sound are those in which (i) the bar is free at both ends, and (2) it is clamped at one end and free at the other.
When we are
walking past a fence formed by equallyspaced vertical rails or
overlapping boards, we may often note that each footstep is
followed by a musical ring.
.^ A sharp clap of the hands may also produce the effect.
^ This change can always be effected if we can apply whatever force may be needed to produce it.
A short impulsive wave travels towards the
fence, and each rail as it is reached by the wave becomes the
centre of a new secondary wave sent out
all
round, or at any rate on the front side of the fence.
FIG. 15.
Let S (fig.
.^ At the instant that the original wave reaches F the wave from E has travelled to a circle of radius very nearly equal to EFnot quite, as S is not quite in the plane of the rails.
^ The more general case of motion of source, medium and receiver may be treated very easily if the motions are all in the line joining source and receiver.
.^ The wave from D has travelled to a circle of radius nearly equal to DF, that from C to a circle of radius nearly CF, and so on.
^ At the instant that the original wave reaches F the wave from E has travelled to a circle of radius very nearly equal to EFnot quite, as S is not quite in the plane of the rails.
^ If the fork has slightly greater frequency, then a white line will not quite reach the next place while the fork is making its swing ip and out, and the waves will travel against the motion of the cylinder.
.^ The wave from D has travelled to a circle of radius nearly equal to DF, that from C to a circle of radius nearly CF, and so on.
^ At the instant that the original wave reaches F the wave from E has travelled to a circle of radius very nearly equal to EFnot quite, as S is not quite in the plane of the rails.
^ The formation of beats may be illustrated by considering the disturbance at any point due to two trains of waves of equal amplitude a and of nearly equal frequencies n, n2.
As these "
secondary waves " return to S their distance apart is nearly equal
to twice the distance between the rails, and the observer then
hears a note of wavelength nearly 2EF. But if an observer is
stationed at S' the waves will be about half as far apart and will
reach him with nearly twice the frequency, so that he hears a note
about an octave higher.
.^ As he travels further round the frequency increases still more.
^ Since the velocity increases as we go upwards the front tends to swing round and travel downwards, as shown in the successive positions a I, 2, 3 and 4, in fig.
.^ The railings in fact do for sound what a diffraction grating does for light.
.^ Sounds may be divided into noises and musical notes.
^ Using the term " note " for the sound produced by a periodic disturbance, there is no doubt that a welltrained ear can resolve a note into pure tones of frequencies equal to those of the fundamental and its harmonics.
^ If, the two pipes are slightly out of tune when sounded separately together they sound a common note which may be higher than that due to either alone.
.^ A mere noise is an irregular disturbance.
.^ If we study the source producing it we find that there is no regularity of vibration.
^ We shall discuss the disturbance which is propagated from the source to the ear , and which there produces sound, and the modes in which various sources vibrate and give rise to the disturbance.
^ There appears to be no doubt that they are produced, and the only question is whether the theory accounts sufficiently for the intensity of the tones actually heard.
.^ A musical note always arises from a source which has some regularity of vibration, and which sends equallyspaced waves into the air.
^ The reason of this is, that the segments of the plate AOD, BOC always vibrate in the same direction, but oppo sitely to the segments AOB, DOC. Hence, when the pasteboard is in its place, there are two waves of same phase starting from the two former segments, and reaching the ear after equal distances of transmission through the air, are again in the same phase, and produce on the ear a conjunct impression.
^ But where it is appropriate, the disturbance sent out into the air contains the same harmonic series as the source.
.^ This is generally equal to the number of waves issuing from the source per second, and therefore equal to its frequency of vibration.
^ A given note has always the same frequency, that is to say, the hearer receives the same number of waves per second whatever the source by which the note is produced.
^ As an example of the application of this siren, suppose that the number of revolutions of the plate, as shown by the indices, amounts to 5400 in a minute, that is, to 90 per second, then the number of vibrations per second of the note heard amounts to 90n, or (if number of holes in each plate = 8) to 720.
.^ Various instruments have been devised which produce any desired note, and which are provided with methods of counting the frequency of vibration.
^ When a cart wheel is ungreased it produces a very high note, probably due to torsional vibrations of the axle .
^ This want of proportionality will have a periodicity, that of the impinging waves, and so will produce vibrations just as does the variation of pressure in the case last investigated.
.^ The results obtained fully confirm the general law that " pitch," or the position of the note in the musical scale, depends solely on its frequency.
^ Kundt also obtained results in general agreement with the formula (Rayleigh, Sound, ii.
^ The result is a note whose pitch rises as the velocity of rotation increases, and becomes steady when that velocity reaches its constant value.
.^ We shall now describe some of the methods of determining frequency.
^ An obvi us method of determining the velocity of sound in air consists in starting some sound, say by firing a gun, and stationing an observer at some measured distance from the gun.
^ The method is easily adapted for the converse determination of speed of revolution when the frequency of a fork is known.
.^ This second plate is capable of rotation about an axis perpendicular to its plane and passing through its centre.
^ Savart's toothed wheel apparatus, named alter Felix Savart (17911841), a French physicist and surgeon, consists of a brass wheel, whose edge is divided into a number of equal projecting teeth distributed uniformly over the circumference, and which is capable of rapid rotation about an axis perpendicular to its plane and passing through its centre, by means of a series of multiplying wheels, the last of which is turned round by the hand.
^ The siren of L. F. W. A. Seebeck (18051849) is the simplest form of apparatus thus designated, and consists of a large circular disk mounted on a central axis, about which it may be made to revolve with moderate rapidity.
.^ The toothed wheel being set in motion, the edge of a card or of a funnel shaped piece of common notepaper is held against the teeth, when a note will be heard arising from the rapidly succeeding displacements of the air in its vicinity.
^ If we divide this by 2 3 or 8, we obtain 440 as the number of vibrations answering to the note A. If, for the single toothed wheel, be substituted a set of four with a common axis, in which the teeth are in the ratios 4: 5: 6: 8, and if the card be rapidly passed along their edges, we shall hear distinctly produced the fundamental chord C, E, G, C 1 and shall thus satisfy ourselves that the intervals C, E; C, G and C, C 1 are, 2 and 2 respectively.
^ When the motion is reversed and the air moves out of the pipe at the embouchure, the sheet is deflected on to the outer side of the sharp edge, and no work is done against it by the air in the pipe.
.^ The result is a note whose pitch rises as the velocity of rotation increases, and becomes steady when that velocity reaches its constant value.
^ The pitch of this note will rise as the rate of rotation increases,_and becomes steady when that rotation is maintained uniform.
^ But as the pitch of the one rises the beats become a jar too frequent to count, and only perhaps to a trained ear recognizable as beats.
.^ Sounds may be divided into noises and musical notes.
^ A determinate musical pitch is not perceived, he says, till about 40 vibrations per second.
^ The relation between the pitch of a note and the frequency of the corresponding vibrations has also been studied by graphic methods.
.^ A3 may be given by a piano.
.^ Thus the wheel made about 8 revolutions per second.
^ Now, suppose that the note produced with Savart's apparatus is in unison with A3, when the experimenter turns round the first wheel at the rate of 60 turns per minute or one per second, and that the circumferences of the various multiplying wheels are such that the rate of revolution of the toothed wheel is thereby increased 44 times, then the latter wheel will perform 44 revolutions in a, second,.
^ As an example of the application of this siren, suppose that the number of revolutions of the plate, as shown by the indices, amounts to 5400 in a minute, that is, to 90 per second, then the number of vibrations per second of the note heard amounts to 90n, or (if number of holes in each plate = 8) to 720.
and hence, if the number of its teeth be 80, the number of
taps imparted to the card every second will amount to 44X80 or
3520. This, therefore, is the number of vibrations corresponding to
the note A3. If we divide this by 2 3 or 8, we obtain 440 as the
number of vibrations answering to the note A. If, for the single
toothed wheel, be substituted a set of four with a common axis, in
which the teeth are in the ratios 4: 5: 6: 8, and if the card be
rapidly passed along their edges, we shall hear distinctly produced
the fundamental chord C, E, G, C 1 and shall thus satisfy ourselves
that the intervals C, E; C, G and C, C 1 are, 2 and 2
respectively.
.^ Neither this instrument nor the next to be described is now used for exact work; they merely serve as illustrations of the law of pitch.
^ In an experiment described by Rayleigh such a wheel provided with four armatures was used to determine the exact frequency of a driving fork known to have a frequency near 32.
.^ The siren of L. F. W. A. Seebeck (18051849) is the simplest form of apparatus thus designated, and consists of a large circular disk mounted on a central axis, about which it may be made to revolve with moderate rapidity.
^ Barrett found that the best form of burner for ordinary gas pressure might be made of glass tubing about $ in.
^ Thus the wheel made about 8 revolutions per second.
.^ This disk is per Siren.
^ For this purpose four vertical mirrors are arranged round the vertical sides of a cube which is rapidly revolved about a vertical axis.
^ A small circular disk at one end of a torsion arm formed part of a solid wall, but was free to move through a hole in the wall slightly larger than the disk.
.^ In the first series of circles, reckoning from the centre the openings are so made as to divide the respective circumferences, on which they are found, in aliquot parts bearing to each other the ratios of the numbers 2, 4, 5, 6, 8, 10, 12, 16, 20,.
^ In the outer series is a circle divided by perforations into four sets, the numbers of aliquot parts being as 3: 4: 5: 6, followed by others which we need not further refer to.
^ In like manner, the first circle in which we have two sets of holes dividing the circumference, the one into say 8 parts, and the other into Io,.
.^ In like manner, the first circle in which we have two sets of holes dividing the circumference, the one into say 8 parts, and the other into Io,.
^ In the outer series is a circle divided by perforations into four sets, the numbers of aliquot parts being as 3: 4: 5: 6, followed by others which we need not further refer to.
^ Hence the note produced with any given circle of holes rises in pitch as the disk revolves more rapidly; and if, the revolution of the disk being kept as steady as possible, the tube be passed rapidly across the circles of the first series, a series of notes is heard, which, if the lowest be denoted by C, form the sequence C, C1, El, G1, C2, &c.
.^ The disk being started, then by means of a tube held at one end between the lips, and applied near to the disk at the other, or more easily with a common bellows , a blast of air is made to fall on the part of the disk which contains any one of the above circles.
^ The phenomena of beats may be easily observed with two organpipes put slightly out of tune by placing the hand near the open end of one of them, with two musical strings on a resonant chest, or with two tuningforks of the same pitch mounted on their resonance boxes, or held over a resonant cavity (such as a glass jar), one of the forks being put out of tune by loading one prong with a small lump of beeswax.
^ Hence the note produced with any given circle of holes rises in pitch as the disk revolves more rapidly; and if, the revolution of the disk being kept as steady as possible, the tube be passed rapidly across the circles of the first series, a series of notes is heard, which, if the lowest be denoted by C, form the sequence C, C1, El, G1, C2, &c.
.^ Thecurrent being alternately transmitted and shut off, as a hole passes on and off the aperture of the tube or bellows, causes a vibratory motion of the air, whose frequency depends on the number of times per second that a perforation passes the mouth of the tube.
^ As an example of the application of this siren, suppose that the number of revolutions of the plate, as shown by the indices, amounts to 5400 in a minute, that is, to 90 per second, then the number of vibrations per second of the note heard amounts to 90n, or (if number of holes in each plate = 8) to 720.
^ The first overtone of the second will beat 64 times per second with the third of the first, and at such height in the scale this frequency will be unpleasant.
.^ Hence the note produced with any given circle of holes rises in pitch as the disk revolves more rapidly; and if, the revolution of the disk being kept as steady as possible, the tube be passed rapidly across the circles of the first series, a series of notes is heard, which, if the lowest be denoted by C, form the sequence C, C1, El, G1, C2, &c.
^ As an example of the application of this siren, suppose that the number of revolutions of the plate, as shown by the indices, amounts to 5400 in a minute, that is, to 90 per second, then the number of vibrations per second of the note heard amounts to 90n, or (if number of holes in each plate = 8) to 720.
^ In like manner, the first circle in which we have two sets of holes dividing the circumference, the one into say 8 parts, and the other into Io,.
.^ In like manner, the first circle in which we have two sets of holes dividing the circumference, the one into say 8 parts, and the other into Io,.
^ When a wave of sound meets a surface separating two media it is in part reflected, travelling back from the surface into the first medium again with the velocity with which it approached.
^ Johann Heinrich Scheibler (17771838) tuned two forks to an exact octave, and then prepared a number of others dividing the octave into such small steps that the beats between each and the next could be counted easily.
or in ratio
.^ Here it is sufficient to say that the frequencies of a note, its major third, its fifth and its octave , are in the ratios of 4: 5: 6: 8.
^ Thus, an interval of onethird of a second elapses between two successive maxima or beats, and there are produced three beats per second.
^ But the interval is still dissonant, and this is to be explained by the fact that the two tones unite to give a third tone of the frequency of the beats, easily heard when the two primary tones are loud.
.^ Similar results are obtainable by means of the remaining perforations.
.^ A still simpler form of siren may be constituted with a good spinning top, a perforated card disk, and a tube for blowing with.
^ The siren of L. F. W. A. Seebeck (18051849) is the simplest form of apparatus thus designated, and consists of a large circular disk mounted on a central axis, about which it may be made to revolve with moderate rapidity.
.^ The siren of C. Cagniard de la Tour is founded on the same principle as the preceding.
.^ It consists of a cylindrical chest of brass, the base of which is pierced at its centre with an opening in which is fixed a brass tube projecting outwards, and Siren of intended for supplying the cavity of the cylinder with Cagniard de compressed air or other gas, or even liquid.
^ The velocity with which waves of longitudinal disturbance travel in air or in any other fluid can be calculated from the resistance to compression and extension and the density of the fluid.
^ When the air in a pipe open at both ends is vibrating in its simplest mode, the air is alternately moving into and out from the centre.
.^ Beneath the lower or fixed plate are four metallic rings furnished with holes corresponding to those in the plates, and which may be pushed round by projecting pins, so as to admit the aircurrent through any one or more of the series of perforations in the fixed plate.
^ A still simpler form of siren may be constituted with a good spinning top, a perforated card disk, and a tube for blowing with.
^ The second series consists of circles each of which is formed of two sets of perforations, in the first circle arranged as 4:5, in the next as 3:4, then as 2:3, 3: 5, 4 : 7.
.^ Immediately above this fixed plate, and almost in contact with it, is another of the same dimensions, and furnished with the same number, n, of openings similarly placed, but passing obliquely through in an opposite direction from those in the fixed plate, the one set being inclined to the left, the other to the right.
^ The reason of this is, that the segments of the plate AOD, BOC always vibrate in the same direction, but oppo sitely to the segments AOB, DOC. Hence, when the pasteboard is in its place, there are two waves of same phase starting from the two former segments, and reaching the ear after equal distances of transmission through the air, are again in the same phase, and produce on the ear a conjunct impression.
^ If, now, the apparatus be so set that the notes from the upper and lower chest are in unison, the upper fixed plate may be placed in four positions, such as to cause the aircurrent to be cut off in the one chest at the exact instant when it is freely passing through the other, and vice versa.
.^ This second plate is capable of rotation about an axis perpendicular to its plane and passing through its centre.
^ Savart's toothed wheel apparatus, named alter Felix Savart (17911841), a French physicist and surgeon, consists of a brass wheel, whose edge is divided into a number of equal projecting teeth distributed uniformly over the circumference, and which is capable of rapid rotation about an axis perpendicular to its plane and passing through its centre, by means of a series of multiplying wheels, the last of which is turned round by the hand.
^ If the bob of the pendulum is iron, and if a coil is placed just below the centre of swing, then, if a current passes through the coil, while and only while the bob is moving towards it, the vibration is maintained.
.^ Now, let the movable plate be at any time in a position such that its holes.
^ Let, now, the plate be made in the usual manner to vibrate so as to exhibit two nodal lines coinciding with two rectangular diameters.
are immediately above those in the fixed plate, and let the bellows
by which air is forced into the cylinder
.^ Let us suppose that a system of stationary waves is formed in the air in a pipe of indefinite length, and let fig.
^ Immediately above this fixed plate, and almost in contact with it, is another of the same dimensions, and furnished with the same number, n, of openings similarly placed, but passing obliquely through in an opposite direction from those in the fixed plate, the one set being inclined to the left, the other to the right.
^ Let D be the distance, U the velocity of sound in still air, and Tr) the velocity of the wind, supposed for simplicity to blow directly from one station to the other.
16), and
will therefore urge the latter to rotation round its centre.
.^ After 1/nth of a revolution, the two sets of perforations will again coincide, the lateral impulse of the air repeated, and hence the rapidity of rotation increased.
^ If both forks are in vibration, and are prefectly in tune, this line may either be increased or diminished permanently in length according to the difference of phase between the two sets of vibrations.
^ Hence arises the same number of successive impulses of the external air immediately in contact with the movable plate, which is thus thrown into a state of vibration at the rate of n for every revolution of the plate.
.^ This will go on continually as long as air is supplied to the cylinder, and the velocity of rotation of the upper plate will be accelerated up to a certain maximum, at which it may be maintained by keeping the force of the current constant.
^ If, now, the apparatus be so set that the notes from the upper and lower chest are in unison, the upper fixed plate may be placed in four positions, such as to cause the aircurrent to be cut off in the one chest at the exact instant when it is freely passing through the other, and vice versa.
^ The result is a note whose pitch rises as the velocity of rotation increases, and becomes steady when that velocity reaches its constant value.
.^ Now, it is evident that each coincidence of the perforations in the two plates is followed by a noncoincidence, during which the aircurrent is shut off, and that consequently, during each revolution of the upper plate, there occur n alternate passages and interceptions of the current.
^ For this purpose the axis is furnished at its upper part with a screw working into a toothed wheel, and driving it round, during each revolution of the plate, through a space equal to the interval between two teeth.
^ Beneath the lower or fixed plate are four metallic rings furnished with holes corresponding to those in the plates, and which may be pushed round by projecting pins, so as to admit the aircurrent through any one or more of the series of perforations in the fixed plate.
.^ Hence arises the same number of successive impulses of the external air immediately in contact with the movable plate, which is thus thrown into a state of vibration at the rate of n for every revolution of the plate.
^ The reason of this is, that the segments of the plate AOD, BOC always vibrate in the same direction, but oppo sitely to the segments AOB, DOC. Hence, when the pasteboard is in its place, there are two waves of same phase starting from the two former segments, and reaching the ear after equal distances of transmission through the air, are again in the same phase, and produce on the ear a conjunct impression.
^ As an example of the application of this siren, suppose that the number of revolutions of the plate, as shown by the indices, amounts to 5400 in a minute, that is, to 90 per second, then the number of vibrations per second of the note heard amounts to 90n, or (if number of holes in each plate = 8) to 720.
.^ The result is a note whose pitch rises as the velocity of rotation increases, and becomes steady when that velocity reaches its constant value.
^ The pitch of this note will rise as the rate of rotation increases,_and becomes steady when that rotation is maintained uniform.
^ It may be noted that the elasticity E is only constant for small volume changes or for small values of dy/dx.
.^ For this purpose the axis is furnished at its upper part with a screw working into a toothed wheel, and driving it round, during each revolution of the plate, through a space equal to the interval between two teeth.
^ Thus the wheel made about 8 revolutions per second.
^ By similar reasoning it may be shown that the number of beats per second is always equal to the difference between the numbers of vibrations in the same time corresponding to the two interfering notes.
.^ An index resembling the hand of a watch partakes of this motion, and points successively to the divisions of a graduated dial .
.^ On the completion of each revolution of this toothed wheel (which, if the number of its teeth be 100, will comprise loo revolutions of the movable plate), a projecting pin fixed to it catches a tooth of another toothed wheel and turns it round, and with it a corresponding index which thus records the number of turns of the first toothed wheel.
^ In it the fixed and movable plates Do Siren.
^ Savart's toothed wheel apparatus, named alter Felix Savart (17911841), a French physicist and surgeon, consists of a brass wheel, whose edge is divided into a number of equal projecting teeth distributed uniformly over the circumference, and which is capable of rapid rotation about an axis perpendicular to its plane and passing through its centre, by means of a series of multiplying wheels, the last of which is turned round by the hand.
As an example of the application of this
siren, suppose that the number of revolutions of the plate, as
shown by the indices, amounts to 5400 in a minute, that is, to 90
per second, then the number of vibrations per second of the note
heard amounts to 90n, or (if number of holes in each plate = 8) to
720.
.^ H. N. Dove (18031879) 1879) produced a modification of the siren by which the relations of different musical notes may be more Dove's readily ascertained.
^ Sounds may be divided into noises and musical notes.
^ It may easily be heard when a double whistle with notes of different pitch is blown strongly, or when two gongs are loudly sounded close to the hearer.
.^ Beneath the lower or fixed plate are four metallic rings furnished with holes corresponding to those in the plates, and which may be pushed round by projecting pins, so as to admit the aircurrent through any one or more of the series of perforations in the fixed plate.
^ In it the fixed and movable plates Do Siren.
^ As an example of the application of this siren, suppose that the number of revolutions of the plate, as shown by the indices, amounts to 5400 in a minute, that is, to 90 per second, then the number of vibrations per second of the note heard amounts to 90n, or (if number of holes in each plate = 8) to 720.
.^ Thus may be obtained, either separately or in various combinations, the four notes whose vibrations are in the ratios of the above numbers, and which therefore form the fundamental chord ( Cegc 1).
^ This, therefore, is the number of vibrations corresponding to the note A3.
^ If we divide this by 2 3 or 8, we obtain 440 as the number of vibrations answering to the note A. If, for the single toothed wheel, be substituted a set of four with a common axis, in which the teeth are in the ratios 4: 5: 6: 8, and if the card be rapidly passed along their edges, we shall hear distinctly produced the fundamental chord C, E, G, C 1 and shall thus satisfy ourselves that the intervals C, E; C, G and C, C 1 are, 2 and 2 respectively.
.^ Helmholtz ( Sensations of Tone, ch.
^ The inventor has given to this instrument the name of the manyvoiced siren.
^ But there is no doubt that it is very difficult to detect the summation tone by the ear, and many workers have doubted the possibility, notwithstanding the evidence of such an observer as Helmholtz.
viii.)
further adapted the siren for more extensive use, by the addition
to
.^ Dove's instrument Helmholtz's of another chest cone), i Double taming its own fixed Siren.
^ In it the fixed and movable plates Do Siren.
^ The siren of L. F. W. A. Seebeck (18051849) is the simplest form of apparatus thus designated, and consists of a large circular disk mounted on a central axis, about which it may be made to revolve with moderate rapidity.
.^ Annexed is a figure of this instru ment (fig.
17).
Graphic Methods
.^ The relation between the pitch of a note and the frequency of the corresponding vibrations has also been studied by graphic methods.
^ But when the pitch of the cavity was exactly that of the fork when vibrating alone, though it resounded most strongly, it did not affect the frequency of the fork.
^ The results obtained fully confirm the general law that " pitch," or the position of the note in the musical scale, depends solely on its frequency.
.^ Thus, if an elastic metal slip or a pig's bristle be attached to one prong of a tuningfork, and if the fork, while in vibration, is moved rapidly over a glass plate coated with lampblack , the attached style touching the plate lightly, a wavy line will be traced on the plate answering to the vibrations to and fro of the FIG. 17.
^ Instead of a mirror, the objective of a microscope is attached to one prong of the first fork and the eyepiece of the microscope is fixed behind the fork.
^ FIG with a bit of cloth powdered with resin , till the rod gives a distinct note; the vibrations are communicated to the plate, which consequently vibrates transversely, and causes the sand to heap itself into one or more concentric rings.
.^ The same result will be ob tained with a stationary fork and a movable glass plate; and, if the time occupied by the plate in moving through a given distance can be ascertained and the number of complete undulations exhibited on the plate for that distance, which is evidently the number of vibrations of the fork in that time, is reckoned, we shall have determined the numerical vibration value of the note yielded by the fork.
^ The reason of this is, that the segments of the plate AOD, BOC always vibrate in the same direction, but oppo sitely to the segments AOB, DOC. Hence, when the pasteboard is in its place, there are two waves of same phase starting from the two former segments, and reaching the ear after equal distances of transmission through the air, are again in the same phase, and produce on the ear a conjunct impression.
^ As an example of the application of this siren, suppose that the number of revolutions of the plate, as shown by the indices, amounts to 5400 in a minute, that is, to 90 per second, then the number of vibrations per second of the note heard amounts to 90n, or (if number of holes in each plate = 8) to 720.
.^ If, at the same time, a tuningfork of known number of vibrations per second be made to trace its own line close to the other, a comparison of the two lines gives the number corresponding to the sound under consideration.
^ If, then, we can determine the number m of revolutions performed by the plate in every second, we shall at once have the number of vibrations per second corresponding to the audible note by multiplying m by n.
^ Or, if the same plate be moved in contact with two tuningforks, we shall, by comparing the number of sinuosities in the one trace with that in the other, be enabled to assign the ratio of the corresponding numbers of vibrations per second.
.^ Thus, if the one note be an octave higher than the other, it will give double the number of waves in the same distance.
^ X 16 (four octaves higher than the first of the preceding) or 512, and 514 vibrations, which are only slightly out of tune.
^ Wien also used the apparatus to find the decrease of intensity with increase of distance, and found that it was somewhat more rapid than the inverse square law would give.
.^ The motion of the plate may be simply produced by dropping it between two vertical grooves, the tuningforks being properly fixed to a frame above.
^ The time between the breaks could be measured in seconds by the clock signals, and in fractions of a second by the tuningfork record.
^ Johann Heinrich Scheibler (17771838) tuned two forks to an exact octave, and then prepared a number of others dividing the octave into such small steps that the beats between each and the next could be counted easily.
.^ Greater accuracy may be attained with a revolving drum chronograph first devised by Thomas Young ( Lett.
^ The phonograph may be regarded as an instrument of this class, in that it records vibrations on a revolving drum or disk.
on Nat. Phil., 1807, i.
.^ The Revol with lampblack, or, better still, a metallic cylinder ving Drum.
.^ The cylinder is mounted on an axis and turned round, while the style attached to the vibrating body is in light contact with it, and traces therefore a wavy circle, which, on taking off the paper and flattening it, becomes a wavy straight line.
^ Thus, if an elastic metal slip or a pig's bristle be attached to one prong of a tuningfork, and if the fork, while in vibration, is moved rapidly over a glass plate coated with lampblack , the attached style touching the plate lightly, a wavy line will be traced on the plate answering to the vibrations to and fro of the FIG. 17.
^ If the figure be photographed as a lantern slide which is mounted so as to turn round, the wave motion is excellently shown on the screen , the compressions and extensions being represented by the crowding in and opening out of the lines.
.^ The superiority of this arrangement arises from the comparative facility with which the number of revolutions of the cylinder in a given time may be ascertained.
.^ R. Koenig, Quelques experiences d'acoustique (1882) describes apparatus and experiments, intended to show, in opposition to Helmholtz, that beats coalesce into tones, and also that the quality of a note is affected by alteration of phase of one of its component overtones relative to the phase of the fundamental.
^ In R. Koenig's arrangement ( Quelques experiences d'acoustique, p.
.^ I) the axis of the cylinder is fashioned as a screw, which works in fixed nuts at the ends, causing a sliding as well as a rotatory motion of the cylinder.
.^ The lines traced out by the vibrating pointer are thus prevented from overlapping when more than one turn is given to the cylinder.
^ Thus, if an elastic metal slip or a pig's bristle be attached to one prong of a tuningfork, and if the fork, while in vibration, is moved rapidly over a glass plate coated with lampblack , the attached style touching the plate lightly, a wavy line will be traced on the plate answering to the vibrations to and fro of the FIG. 17.
^ The jet tube, for a reason which will be given when we consider the maintenance of vibrations, must be less than c a9 half the length of the sounding tube.
.^ In the phonautograph of E. L. Scott ( Comptes rendus, 1861, 53, p.
.^ Paper, parchment , or any other thin membrane stretched over a square, circular, &c., frame, when in the vicinity of a sufficiently powerful vibrating body, will, through the medium of the air, be itself made to vibrate in unison, and, by using sand, as in previous instances, the nodal lines will be depicted to the eye, and seen to vary in form, number and position with the tension of the plate and the pitch of the originating sound.
^ The cases interesting in sound are those in which (i) the bar is free at both ends, and (2) it is clamped at one end and free at the other.
^ At the other end of the scale with increasing frequency there is another limiting frequency somewhere about 20,000 per second, beyond which no sound is heard.
.^ To the centre of this membrane is attached a small feather fibre, which, when the reflector is suitably placed, touches lightly the surface of the revolving cylinder.
^ At a station more than 13 kilometres away a sort of big ear trumpet , closed by a membrane, was placed with the membrane under water, the tube rising above the surface.
^ Any sound (such as that of the human voice) transmitting its rays into the reflector, and communicating vibratory motion to the membrane, will cause the feather to trace a sinuous line on the paper.
.^ Any sound (such as that of the human voice) transmitting its rays into the reflector, and communicating vibratory motion to the membrane, will cause the feather to trace a sinuous line on the paper.
^ The displacement curve of the waves from a tuningfork on its resonance box , or from the human voice sounding oo, are nearly smooth and symmetrical, as in fig.
^ To the centre of this membrane is attached a small feather fibre, which, when the reflector is suitably placed, touches lightly the surface of the revolving cylinder.
.^ If, at the same time, a tuningfork of known number of vibrations per second be made to trace its own line close to the other, a comparison of the two lines gives the number corresponding to the sound under consideration.
^ Or, if the same plate be moved in contact with two tuningforks, we shall, by comparing the number of sinuosities in the one trace with that in the other, be enabled to assign the ratio of the corresponding numbers of vibrations per second.
^ As an example of the application of this siren, suppose that the number of revolutions of the plate, as shown by the indices, amounts to 5400 in a minute, that is, to 90 per second, then the number of vibrations per second of the note heard amounts to 90n, or (if number of holes in each plate = 8) to 720.
.^ The phonograph may be regarded as an instrument of this class, in that it records vibrations on a revolving drum or disk.
^ Greater accuracy may be attained with a revolving drum chronograph first devised by Thomas Young ( Lett.
^ Bells may be regarded as somewhat like circular plates vibrating with radial nodes, and with the edges turned down.
Lissajous Figures
.^ A mode of exhibiting the ratio of the frequencies of two forks was devised by Jules Antoine Lissajous (18221880).
^ Koenig devised a clock in which a fork of frequency 64 takes the place of the pendulum ( Wied.
^ All experiments in frequency show that two notes, forming a definite musical interval, have their frequencies always in the same ratio wherever in the musical scale the two notes are situated.
.^ On one prong of each fork is fixed a small plane mirror.
^ Instead of a mirror, the objective of a microscope is attached to one prong of the first fork and the eyepiece of the microscope is fixed behind the fork.
^ One prong of the fork carries a microscope objective, wart of a vibration microscope, of which the eyepiece is fixed at the back of the clock and the Lissajous figure FIG. 16.
.^ On one prong of each fork is fixed a small plane mirror.
^ The two forks are fixed so that one vibrates in a vertical, and the other in a horizontal, plane, and they are so placed that a converging beam of light received on one mirror is reflected to the other and then brought to a point on a screen.
^ If one prong of each fork be furnished with a small plain mirror, and a beam of light from a luminous point be reflected successively by the two mirrors, so as to form an image on a distinct screen, when one fork alone is put in vibration, the image will move on the screen and be seen as a line of a certain length.
.^ If the first fork alone vibrates, the point on the screen appears lengthened out into a vertical line through the changes in inclination of the first mirror, while if the second fork alone vibrates, the point appears lengthened out into a horizontal line.
^ The flame appears to lengthen, but if the reflection is viewed in a vertical mirror revolving about a vertical axis or in Koenig's cube of mirrors, it is seen that the flame is really intermittent, jumping up and down once with each vibration, sometimes apparently going within the jet tube at its lowest point.
^ The two forks are fixed so that one vibrates in a vertical, and the other in a horizontal, plane, and they are so placed that a converging beam of light received on one mirror is reflected to the other and then brought to a point on a screen.
.^ If both vibrate, the point describes a curve which appears continuous through the persistence of the retinal impression.
^ But if a tuningfork of appropriate frequency be set vibrating with its stalk in contact with the holder of the pipe from which the jet issues, the jet appears to go over in one continuous thread.
^ If the first fork alone vibrates, the point on the screen appears lengthened out into a vertical line through the changes in inclination of the first mirror, while if the second fork alone vibrates, the point appears lengthened out into a horizontal line.
.^ Lissajous also obtained the figures by aid of the vibration microscope , an instrument which he invented.
^ If both forks vibrate, an observer looking through the microscope sees the bright point describing Lissajous figures.
^ One prong of the fork carries a microscope objective, wart of a vibration microscope, of which the eyepiece is fixed at the back of the clock and the Lissajous figure FIG. 16.
.^ Instead of a mirror, the objective of a microscope is attached to one prong of the first fork and the eyepiece of the microscope is fixed behind the fork.
^ On one prong of each fork is fixed a small plane mirror.
^ Instead of a mirror the second fork carries a bright point on one prong, and the microscope is focused on this.
.^ Instead of a mirror, the objective of a microscope is attached to one prong of the first fork and the eyepiece of the microscope is fixed behind the fork.
^ On one prong of each fork is fixed a small plane mirror.
^ Instead of a mirror the second fork carries a bright point on one prong, and the microscope is focused on this.
.^ If both forks vibrate, an observer looking through the microscope sees the bright point describing Lissajous figures.
^ Lissajous also obtained the figures by aid of the vibration microscope , an instrument which he invented.
^ One prong of the fork carries a microscope objective, wart of a vibration microscope, of which the eyepiece is fixed at the back of the clock and the Lissajous figure FIG. 16.
.^ If the two forks have the same frequency, it is easily seen that the figure will be an ellipse (including as limiting cases, depending on relative amplitude and phase, a circle and a straight line).
^ If, at the same time, a tuningfork of known number of vibrations per second be made to trace its own line close to the other, a comparison of the two lines gives the number corresponding to the sound under consideration.
^ If d is measured for two gases in succession for the same frequency N, we have 72 p 2P1 d22 71 p i p s d12' where the suffixes denote the gases to which the quantities relate.
.^ If the forks are not of exactly the same frequency the ellipse will slowly revolve, and from its rate of revolution the ratio of the frequencies may be determined (Rayleigh, Sound, i.
^ N, and it follows that when n is known, the frequency of every fork in the range may be determined.
^ With this apparatus Koenig studied the effect of temperature on a standard fork of 256 frequency, and found that the frequency decreased by o 0286 of a vibration fora rise of I°, the frequency being exactly 256 at 26.2° C. Hence the frequency may be put as 256 { I  0.000113 026.2)1 (From Lord Rayleigh's Theory of Sound, by permission of Macmillan & Co., Ltd.
§ 33).
.^ If one is the octave of the other a figure of 8 may be described, and so on.
^ Thus, if the one note be an octave higher than the other, it will give double the number of waves in the same distance.
Fig. 18 shows curves given by intervals of the octave,
the twelfth and the fifth.
.^ The kaleidophone devised by Charles Wheatstone in 1827 gives these figures in a simple way.
^ For the superposition of these trains will give a stationary wave between A H A (16) Y which is an equation characteristic of simple harmonic motion.
.^ It consists of a straight rod clamped in a vice and carrying a bead at its upper free end.
^ The cases interesting in sound are those in which (i) the bar is free at both ends, and (2) it is clamped at one end and free at the other.
^ When one end is clamped and the other is free the clamped end is always a node.
.^ The bead is illuminated and shows a bright point of light.
^ On the pendulum was fixed an illuminated silver bead which appeared as a bright point of light when seen for an instant.
.^ If the rod is circular in section and perfectly uniform the end will describe a circle, ellipse or straight line; but, as the elasticity is usually not exactly the same in all directions, the figure usually changes and revolves.
^ The ordinate of the curve changes sign as we pass through a node, so that successive sections are moving always in opposite directions and have opposite displacements.
^ The change in virtual length by removal of the flange was thus found, and the open end correction for the unflanged pipe was o 6 R. This correction has also been found by David James Blaikley by direct experiment ( Phil.
.^ Various modifications of the kaleidophone have been made (Rayleigh, Sound, § 38).
.^ Koenig devised a clock in which a fork of frequency 64 takes the place of the pendulum ( Wied.
^ Koenig also used the apparatus to investigate the effect on the frequency of a fork of a resonating cavity placed near it.
^ This apparatus was used to find the temperature coefficient of the frequency of forks, the value obtained  .00011 being the same as that found by Koenig.
. 1880, ix.
^ Ann., 1880, ix.
394).
.^ The motion of the fork is maintained by the clock acting through an escapement, and the dial registers both the number Koenig's of vibrations of the fork and the seconds, minutes and Tuningfork hours.
^ As an example of the application of this siren, suppose that the number of revolutions of the plate, as shown by the indices, amounts to 5400 in a minute, that is, to 90 per second, then the number of vibrations per second of the note heard amounts to 90n, or (if number of holes in each plate = 8) to 720.
^ One prong of the fork carries a microscope objective, wart of a vibration microscope, of which the eyepiece is fixed at the back of the clock and the Lissajous figure FIG. 16.
.^ By comparison with a clock of known rate Glock.
^ N, and it follows that when n is known, the frequency of every fork in the range may be determined.
^ One prong of the fork carries a microscope objective, wart of a vibration microscope, of which the eyepiece is fixed at the back of the clock and the Lissajous figure FIG. 16.
One prong of the
fork carries a microscope objective,
wart of a vibration microscope, of which the
eyepiece is fixed at the back of the clock and the Lissajous figure
FIG. 16.
Missing image
Sound6.jpg
? made by the clock fork and any other fork may
be observed.
.^ These effects have been explained by Lord Rayleigh ( Sound, i.
^ A fork of frequency 256 was used as the source.
^ With this apparatus Koenig studied the effect of temperature on a standard fork of 256 frequency, and found that the frequency decreased by o 0286 of a vibration fora rise of I°, the frequency being exactly 256 at 26.2° C. Hence the frequency may be put as 256 { I  0.000113 026.2)1 (From Lord Rayleigh's Theory of Sound, by permission of Macmillan & Co., Ltd.
) FIG. 18.
.^ Koenig also used the apparatus to investigate the effect on the frequency of a fork of a resonating cavity placed near it.
^ A fork of frequency 256 was used as the source.
^ With this apparatus Koenig studied the effect of temperature on a standard fork of 256 frequency, and found that the frequency decreased by o 0286 of a vibration fora rise of I°, the frequency being exactly 256 at 26.2° C. Hence the frequency may be put as 256 { I  0.000113 026.2)1 (From Lord Rayleigh's Theory of Sound, by permission of Macmillan & Co., Ltd.
h; e
found that when the pitch of the cavity was below that of the fork
the pitch of the fork was raised, and vice versa.
.^ But when the pitch of the cavity was exactly that of the fork when vibrating alone, though it resounded most strongly, it did not affect the frequency of the fork.
^ The pipe will also resound to forks of frequencies 512, 768, 1024 and so on.
^ But if a tuningfork of appropriate frequency be set vibrating with its stalk in contact with the holder of the pipe from which the jet issues, the jet appears to go over in one continuous thread.
.^ This is explained by Rayleigh ( Sound, ii.
^ Lord Rayleigh ( Sound, ii.
^ These effects have been explained by Lord Rayleigh ( Sound, i.
§ 117).
.^ In the stroboscopic method of H. M'Leod and G. S. Clarke, the full details of which will be found in the original memoir ( Phil.
. 1880, pt.
^ Trans., 1880, pt.
i. p.
.^ I), a cylinder is ruled with equidistant white lines parallel to the axis on a black ground.
^ The boundary between ,the grey cylinder and the black fork will therefore appear wavy with fixed undulations, the distance from crest to crest being the distance between the lines on the cylinder.
^ Let the cylinder be rotated so that each white line moves exactly into the place of the next while the prong moves once in and out.
It is set so that it can be turned at any desired and
determined speed about a horizontal axis, and when going fast
enough it appears grey. Imagine now that a fork with black prongs
is held near the cylinder with its prongs vertical and the plane of
vibration parallel to. the axis, and suppose that we watch the
outer outline of the righthand prong.
.^ Let the cylinder be rotated so that each white line moves exactly into the place of the next while the prong moves once in and out.
^ The most will be cut off in the position of the lines corresponding to the furthest swing out, then less and less till the furthest swing in, then more and more till the furthest swing out, when the appearance will be exactly as at first.
^ If the fork has slightly greater frequency, then a white line will not quite reach the next place while the fork is making its swing ip and out, and the waves will travel against the motion of the cylinder.
.^ Hence when a white line is in a particular position on the cylinder, the prong will always be the same distance along it and cut off the same length from view.
^ If, instead of considering one point in a succession of instants, we consider a succession of points along the line of propagation at the same instant, we evidently have waves of amplitude varying from 2a down to o, and then up to 2a again in distance U/(ni  n2).
^ We shall first investigate the velocity with which a disturbance travels along a string of mass m per unit length when it is stretched with a constant tension T, the same at all points.
.^ The most will be cut off in the position of the lines corresponding to the furthest swing out, then less and less till the furthest swing in, then more and more till the furthest swing out, when the appearance will be exactly as at first.
^ If the first fork alone vibrates, the point on the screen appears lengthened out into a vertical line through the changes in inclination of the first mirror, while if the second fork alone vibrates, the point appears lengthened out into a horizontal line.
^ Let the cylinder be rotated so that each white line moves exactly into the place of the next while the prong moves once in and out.
.^ The boundary between ,the grey cylinder and the black fork will therefore appear wavy with fixed undulations, the distance from crest to crest being the distance between the lines on the cylinder.
^ If both forks are in vibration, and are prefectly in tune, this line may either be increased or diminished permanently in length according to the difference of phase between the two sets of vibrations.
^ If the first fork alone vibrates, the point on the screen appears lengthened out into a vertical line through the changes in inclination of the first mirror, while if the second fork alone vibrates, the point appears lengthened out into a horizontal line.
.^ If the fork has slightly greater frequency, then a white line will not quite reach the next place while the fork is making its swing ip and out, and the waves will travel against the motion of the cylinder.
^ If the fork has slightly less frequency the waves will travel in the opposite direction, and it is easily seen that the frequency of the fork is the number of white lines passing a point in a second t the number of waves passing the point per second.
^ The most will be cut off in the position of the lines corresponding to the furthest swing out, then less and less till the furthest swing in, then more and more till the furthest swing out, when the appearance will be exactly as at first.
If the
fork has slightly less frequency the waves will travel in the
opposite direction, and it is easily seen that the frequency of the
fork is the number of white lines passing a point in a second t the
number of waves passing the point per second.
.^ A fork of frequency 256 was used as the source.
^ This apparatus was used to find the temperature coefficient of the frequency of forks, the value obtained  .00011 being the same as that found by Koenig.
^ With this apparatus Koenig studied the effect of temperature on a standard fork of 256 frequency, and found that the frequency decreased by o 0286 of a vibration fora rise of I°, the frequency being exactly 256 at 26.2° C. Hence the frequency may be put as 256 { I  0.000113 026.2)1 (From Lord Rayleigh's Theory of Sound, by permission of Macmillan & Co., Ltd.
.^ Another important result of the investigation was that the phase of vibration of the fork was not altered by bowing it, the amplitude alone changing.
^ But when the pitch of the cavity was exactly that of the fork when vibrating alone, though it resounded most strongly, it did not affect the frequency of the fork.
^ If both forks are in vibration, and are prefectly in tune, this line may either be increased or diminished permanently in length according to the difference of phase between the two sets of vibrations.
.^ The method is easily adapted for the converse determination of speed of revolution when the frequency of a fork is known.
^ N, and it follows that when n is known, the frequency of every fork in the range may be determined.
^ We shall now describe some of the methods of determining frequency.
.^ Lord Rayleigh ( Sound, ii.
^ The phonic wheel, invented independently by Paul La Cour and Lord Rayleigh (see Sound, i.
^ Missing image Sound7.jpg In a later series of experiments Lord Rayleigh ( Phil.
§
.^ Rayleigh's round its circumference.
.^ If the wheel be driven at such rate that the armatures move one place on in about the period of the current, then on putting on the current the electromagnet controls the rate of the wheel so that the agreement of period is exact, and the wheel settles down to move so that the electric driving forces just supply the work taken out of the wheel.
^ The wheel rotates between Phonic the poles of an electromagnet, which is fed by an intermittent current such as that which is working an electrically maintained tuningfork (see infra).
^ If the wheel has very little work to do it may not be necessary to apply driving power, and uniform rotation may be maintained by the electromagnet.
.^ If the wheel has very little work to do it may not be necessary to apply driving power, and uniform rotation may be maintained by the electromagnet.
^ In a common form of electrically maintained fork, the fork is set horizontal with its prongs in a vertical plane, and a small electromagnet is fixed between them.
^ The wheel rotates between Phonic the poles of an electromagnet, which is fed by an intermittent current such as that which is working an electrically maintained tuningfork (see infra).
In an experiment described by Rayleigh such a
wheel provided with four armatures was used to determine the exact
frequency of a driving fork known to have a frequency near 32. Thus
the wheel made about 8 revolutions per second.
.^ There was one opening in its disk, and through this was viewed the pendulum of a clock beating seconds.
^ A small circular disk at one end of a torsion arm formed part of a solid wall, but was free to move through a hole in the wall slightly larger than the disk.
^ Let us suppose that he notes the positions of two of these next to each other in the beat of the pendulum one way.
.^ The bead is illuminated and shows a bright point of light.
^ On the pendulum was fixed an illuminated silver bead which appeared as a bright point of light when seen for an instant.
^ Suppose now an observer to be looking from a fixed point at the bead through the hole in the phonic wheel, he will see the bead as 8 bright points flashing out in each beat, and in succession at intervals of k second.
.^ Suppose now an observer to be looking from a fixed point at the bead through the hole in the phonic wheel, he will see the bead as 8 bright points flashing out in each beat, and in succession at intervals of k second.
^ If the fork makes exactly 32 vibrations and the wheel 8 revolutions in one pendulum beat, then the positions will be fixed, and every two seconds, the time of a complete pendulum vibration, he will see the two positions looked at flash out in succession at an interval of a second.
^ Thus the interval b'c" with frequencies 495 and 528, giving 33 beats in a second, is very dissonant.
.^ Let us suppose that he notes the positions of two of these next to each other in the beat of the pendulum one way.
^ Let us here suppose that the string AB is displaced into the form AHB (fig.
^ In like manner, the first circle in which we have two sets of holes dividing the circumference, the one into say 8 parts, and the other into Io,.
.^ Then the wheel makes 8 N + 1 revolutions in N clock beats, and the fork makes 32 N + 4 vibrations in the same time.
^ AC = X. If the source makes n vibrations in one second it is said to have " frequency " n.
^ Suppose now an observer to be looking from a fixed point at the bead through the hole in the phonic wheel, he will see the bead as 8 bright points flashing out in each beat, and in succession at intervals of k second.
.^ But if the fork has, say, rather greater frequency, the hole in the wheel comes round at the end of the two seconds before the bead has quite come into position, and the two flashes appear gradually to move back in the opposite way to the pendulum.
^ If the fork makes exactly 32 vibrations and the wheel 8 revolutions in one pendulum beat, then the positions will be fixed, and every two seconds, the time of a complete pendulum vibration, he will see the two positions looked at flash out in succession at an interval of a second.
^ In like manner, the first circle in which we have two sets of holes dividing the circumference, the one into say 8 parts, and the other into Io,.
.^ Suppose that in N beats of the clock the flashes have moved exactly one place back.
^ One prong of the fork carries a microscope objective, wart of a vibration microscope, of which the eyepiece is fixed at the back of the clock and the Lissajous figure FIG. 16.
^ If the clock is going exactly right, this gives a frequency for the fork of 32 + 4/N. If the fork has rather less frequency than 32 then the flashes appear to move forward and the frequency will be 32  4/N. In Rayleigh's experiment the 32 fork was made to drive electrically one of frequency about 128, and somewhat as with the phonic wheel, the frequency was controlled so as to be exactly four times that of the 32 fork.
.^ Then the first flash in the new position is viewed by the 8Nth passage of the opening, and the second flash in the original position of the first is viewed when the pendulum has made exactly N beats and by the (8 N + i)th passage of the hole.
^ There was one opening in its disk, and through this was viewed the pendulum of a clock beating seconds.
^ The most will be cut off in the position of the lines corresponding to the furthest swing out, then less and less till the furthest swing in, then more and more till the furthest swing out, when the appearance will be exactly as at first.
.^ Then the wheel makes 8 N + 1 revolutions in N clock beats, and the fork makes 32 N + 4 vibrations in the same time.
^ If the fork makes exactly 32 vibrations and the wheel 8 revolutions in one pendulum beat, then the positions will be fixed, and every two seconds, the time of a complete pendulum vibration, he will see the two positions looked at flash out in succession at an interval of a second.
^ One prong of the fork carries a microscope objective, wart of a vibration microscope, of which the eyepiece is fixed at the back of the clock and the Lissajous figure FIG. 16.
.^ Then the wheel makes 8 N + 1 revolutions in N clock beats, and the fork makes 32 N + 4 vibrations in the same time.
^ If the clock is going exactly right, this gives a frequency for the fork of 32 + 4/N. If the fork has rather less frequency than 32 then the flashes appear to move forward and the frequency will be 32  4/N. In Rayleigh's experiment the 32 fork was made to drive electrically one of frequency about 128, and somewhat as with the phonic wheel, the frequency was controlled so as to be exactly four times that of the 32 fork.
^ Suppose that in N beats of the clock the flashes have moved exactly one place back.
.^ A standard 128 Efork could then be compared either optically or by beats with the electrically driven fork.
Scheibler's Tonometer
.^ When two tones are sounded together with frequencies not very different, " beats " or swellingsout of the sound are heard of frequency equal to the difference of frequencies of the two tones (see below).
^ The tone of the frequency of the beats was discovered by Georg Andreas Sorge in 1740, and independently a few years later by Giuseppe Tartini , after whom it is named.
^ Thus the interval b'c" with frequencies 495 and 528, giving 33 beats in a second, is very dissonant.
.^ Johann Heinrich Scheibler (17771838) tuned two forks to an exact octave, and then prepared a number of others dividing the octave into such small steps that the beats between each and the next could be counted easily.
^ Any other fork within this octave can then have its frequency determined by finding the two between which it lies.
^ The phenomena of beats may be easily observed with two organpipes put slightly out of tune by placing the hand near the open end of one of them, with two musical strings on a resonant chest, or with two tuningforks of the same pitch mounted on their resonance boxes, or held over a resonant cavity (such as a glass jar), one of the forks being put out of tune by loading one prong with a small lump of beeswax.
.^ Let the forks be numbered o, I, 2,.
.. N. If the frequency of o is
n, that of N is 2n.
Suppose that No. i makes
m 1 beats with No. o, that No.
.^ Suppose, for instance, it makes 3 beats with No.
1, and so on, then the
frequencies are n, n+m i,
n+m l +m 21 ..., n+m1+m2+.. . +
7n N.
Since n+mi+m2+
. + m N =2n, n = mi+m2+
.^ N, and it follows that when n is known, the frequency of every fork in the range may be determined.
^ In an experiment described by Rayleigh such a wheel provided with four armatures was used to determine the exact frequency of a driving fork known to have a frequency near 32.
^ Any other fork within this octave can then have its frequency determined by finding the two between which it lies.
.^ Any other fork within this octave can then have its frequency determined by finding the two between which it lies.
^ Johann Heinrich Scheibler (17771838) tuned two forks to an exact octave, and then prepared a number of others dividing the octave into such small steps that the beats between each and the next could be counted easily.
^ If both forks are in vibration, and are prefectly in tune, this line may either be increased or diminished permanently in length according to the difference of phase between the two sets of vibrations.
.^ Suppose, for instance, it makes 3 beats with No.
10, it might have frequency
either 3 above or below that of No. io. But if it lies above No. io
it will beat less often with No.
.^ II than with No.
9; if below No.
10 less often with No. 9 than with No. 11. Suppose it lies between
No. 10 and No.
.^ II its frequency is that of No.
10+3.
Manometric Flames
.^ This is a device due to Koenig ( Phil.
. 18 73, 45)
and represented diagrammatically in fig.
^ Mag., 18 73, 45) and represented diagrammatically in fig.
^ This rod is supported at 4 and 4 of its length where it enters the two dusttubes, as represented diagrammatically in fig.
19.
f is a flame
FIG. 19.
from a pinhole burner, fed through a cavity
.^ A fork of frequency 256 was used as the source.
^ C, one side of which is closed by a membrane m; on the other side of the membrane is another cavity C', which is put into connexion with a source of sound, as, for instance, a Helmholtz resonator excited by a fork of the same frequency.
^ The phenomena of beats may be easily observed with two organpipes put slightly out of tune by placing the hand near the open end of one of them, with two musical strings on a resonant chest, or with two tuningforks of the same pitch mounted on their resonance boxes, or held over a resonant cavity (such as a glass jar), one of the forks being put out of tune by loading one prong with a small lump of beeswax.
.^ The flame jumps down at every tick.
^ The membrane vibrates, and alternately checks and increases the gas supply, and the flame jumps up and down with the frequency of the source.
^ This is generally equal to the number of waves issuing from the source per second, and therefore equal to its frequency of vibration.
It then appears elongated. To show its
intermittent character its reflection is viewed in a revolving
mirror.
.^ For this purpose four vertical mirrors are arranged round the vertical sides of a cube which is rapidly revolved about a vertical axis.
^ The flame appears to lengthen, but if the reflection is viewed in a vertical mirror revolving about a vertical axis or in Koenig's cube of mirrors, it is seen that the flame is really intermittent, jumping up and down once with each vibration, sometimes apparently going within the jet tube at its lowest point.
.^ The flame then appears toothed as shown.
.^ If several notes are present the flame is jagged by each.
.^ Interesting results are obtained by singing the different vowels into a funnel substituted for the resonator in the figure.
.^ If two such flames are placed one under the other they may be excited by different sources, and the ratio of the frequencies may be approximately determined by counting the number of teeth in each in the same space.
^ The phenomena of beats may be easily observed with two organpipes put slightly out of tune by placing the hand near the open end of one of them, with two musical strings on a resonant chest, or with two tuningforks of the same pitch mounted on their resonance boxes, or held over a resonant cavity (such as a glass jar), one of the forks being put out of tune by loading one prong with a small lump of beeswax.
^ They found that the velocity of propagation of different musical sounds was the same.
.^ It is not necessary here to deal generally with the various musical scales.
^ The results obtained fully confirm the general law that " pitch," or the position of the note in the musical scale, depends solely on its frequency.
.^ We shall treat only of the diatonic scale, which is the basis of European music , and is approximated to as closely as is consistent with convenience of construction in key board instruments, such as the piano, where the eight white notes beginning with C and ending with C an octave higher may be taken as representing the scale with C as the keynote.
^ Now suppose we take G as the keynote and form its diatonic scale.
^ X 16 (four octaves higher than the first of the preceding) or 512, and 514 vibrations, which are only slightly out of tune.
.^ Frequency ratios of diatonic scale .
^ All experiments in frequency show that two notes, forming a definite musical interval, have their frequencies always in the same ratio wherever in the musical scale the two notes are situated.
^ Experiments, which will be described most conveniently when we discuss methods of determining the frequencies of sources, prove conclusively that for a given note the frequency is the same whatever the source of that note, and that the ratio of the frequencies of two notes forming a given musical interval is the same in whatever part of the musical range the two notes are situated.
.^ Here it is sufficient to say that the frequencies of a note, its major third, its fifth and its octave , are in the ratios of 4: 5: 6: 8.
^ E flat as keynote introduces another flat, and so on, each flat not quite coinciding with a sharp but at a very small interval from it.
^ If d is measured for two gases in succession for the same frequency N, we have 72 p 2P1 d22 71 p i p s d12' where the suffixes denote the gases to which the quantities relate.
.^ Frequency ratios of diatonic scale .
^ The frequency ratios in the diatonic scale are all expressible either as fractions, with i, 2, 3 or 5 as numerator and denominator, or as products of such fractions; and it may be shown that for a given note the numerator and denominator are smaller than any other numbers which would give us a note in the immediate neighbourhood.
^ Here it is sufficient to say that the frequencies of a note, its major third, its fifth and its octave , are in the ratios of 4: 5: 6: 8.
.^ Thus the second A = 2 X 2 X 2, and we may regard it as an ascent through two fifths in succession and then a descent through an octave.
^ The third 4= 5 X Z X z or ascent through an interval 4, which has no special name, and a descent through two octaves, and so on.
^ The two sets may be arranged thus c 256 512 768 1024 1280 h 480 960 1440, and we see that the fundamental of the second will beat 32 times per second with the first overtone of the first, giving dissonance.
.^ The third 4= 5 X Z X z or ascent through an interval 4, which has no special name, and a descent through two octaves, and so on.
^ Thus, an interval of onethird of a second elapses between two successive maxima or beats, and there are produced three beats per second.
^ But the interval is still dissonant, and this is to be explained by the fact that the two tones unite to give a third tone of the frequency of the beats, easily heard when the two primary tones are loud.
Notes on scale of

G

A

B

c

d

, e

f

g

Frequency ratios with


s

1 8  5

2

4

2

s


Frequency ratios of diatonic scale









with G .1



i

s

2

s

y

2

Frequency ratios with


is

1 a

2

1

g

fl

3

.^ Now suppose we take G as the keynote and form its diatonic scale.
^ If we take the new note B flat as keynote, another note, E flat, is required.
^ If we write down the eight notes from G to g in the key of C, their frequency ratios to C, the frequency ratios required by the diatonic scale for G, we get the frequency ratios required in the last line:  We see that all but two notes coincidewith notes on the scale of C. But instead of A = we have n, and instead of f= 4 we have 4 b.
.^ Frequency ratios of diatonic scale .
^ If we write down the eight notes from G to g in the key of C, their frequency ratios to C, the frequency ratios required by the diatonic scale for G, we get the frequency ratios required in the last line:  We see that all but two notes coincidewith notes on the scale of C. But instead of A = we have n, and instead of f= 4 we have 4 b.
^ Now suppose we take G as the keynote and form its diatonic scale.
.^ The interval between and 44 = 44   8 01 is termed a " comma ," and is so small that the same note on an instrument may serve for both.
^ An observer with his ear to the tube noted the interval between the arrival of flash and sound.
^ E flat as keynote introduces another flat, and so on, each flat not quite coinciding with a sharp but at a very small interval from it.
.^ This note is f sharp, and the interval t is termed a sharp.
^ But the interval between 4 and 4`56 414 is quite perceptible, and on the piano, for instance, a separate string must be provided above f.
^ When two notes are not quite in unison the resulting sound is found to alternate between a maximum and minimum of loudness recurring periodically.
.^ If we take the new note B flat as keynote, another note, E flat, is required.
^ If we start with F as keynote, besides a small difference at we have as the fourth from it 3 X 4 = y, making with B = I R 5 an interval and requiring a new note, B flat.
^ Taking the successive keynotes D, A, E, B, it is found that besides small and negligible differences, each introduces a new sharp, and so we get the five sharps, C, D, F, G, A, represented nearly by the black keys.
.^ If we take the new note B flat as keynote, another note, E flat, is required.
^ If we start with F as keynote, besides a small difference at we have as the fourth from it 3 X 4 = y, making with B = I R 5 an interval and requiring a new note, B flat.
^ The interval between and 44 = 44   8 01 is termed a " comma ," and is so small that the same note on an instrument may serve for both.
.^ 'This does not coincide with A sharp which is the octave below the seventh from B or '85 X 1 8 5 X z = M. R. It makes with it an interval = 1 w 6  j 2 rather less than a comma; so that the same string in the piano may serve for both.
^ E flat as keynote introduces another flat, and so on, each flat not quite coinciding with a sharp but at a very small interval from it.
^ The effective length of the string is then AD. Keeping the same tension, it may be shown that nl is constant by finding n for various lengths.
.^ If we take the new note B flat as keynote, another note, E flat, is required.
^ E flat as keynote introduces another flat, and so on, each flat not quite coinciding with a sharp but at a very small interval from it.
^ If we start with F as keynote, besides a small difference at we have as the fourth from it 3 X 4 = y, making with B = I R 5 an interval and requiring a new note, B flat.
.^ If we take the new note B flat as keynote, another note, E flat, is required.
^ E flat as keynote introduces another flat, and so on, each flat not quite coinciding with a sharp but at a very small interval from it.
^ This note is f sharp, and the interval t is termed a sharp.
.^ It is evident that for exact diatonic scales for even a limited number of keynotes, keyboard instruments would have to be provided with a great number of separate strings or pipes, and the corresponding keys would be required.
^ Now suppose we take G as the keynote and form its diatonic scale.
^ If we take the new note B flat as keynote, another note, E flat, is required.
.^ The construction would be complicated and the playing exceedingly difficult.
.^ The same string or pipe and the same key have therefore to serve for what should be slightly different notes.
^ If we start with F as keynote, besides a small difference at we have as the fourth from it 3 X 4 = y, making with B = I R 5 an interval and requiring a new note, B flat.
^ If, the two pipes are slightly out of tune when sounded separately together they sound a common note which may be higher than that due to either alone.
.^ A compromise has to be made, and the note has to be tuned so as to make the compromise as little unsatisfactory as possible.
.^ At present twelve notes are used in the octave, and these are arranged at equal intervals 2= 7.
^ This note is f sharp, and the interval t is termed a sharp.
^ This is termed the equal temperament scale, and it is obviously only an approach to the diatonic scale.
Helmholtz's Notation
.^ In works on sound it is usual to adopt Helmholtz's notation, in which the octave from bass to middle C is written c d e f g a b c'.
^ The next octave above has two accents, and each succeeding octave another accent .
^ The next mode has a node in the middle and two others each 0.132 from the end.
.^ In works on sound it is usual to adopt Helmholtz's notation, in which the octave from bass to middle C is written c d e f g a b c'.
^ The octave below bass C is written C D E F GA B c.
.^ The next octave above has two accents, and each succeeding octave another accent .
^ The next octave below is C] D i E 1 F] G] A] B 1 C, and each preceding octave has another accent as suffix.
^ The next higher octave has the suffix 2, the next higher the suffix 3, and so on.
.^ The standard frequency for laboratory work is c =128, so that middle L' = 256 and treble c"= 512 The standard for musical instruments has varied (see Pitch, Musical ).
^ The results obtained fully confirm the general law that " pitch," or the position of the note in the musical scale, depends solely on its frequency.
^ Neither this instrument nor the next to be described is now used for exact work; they merely serve as illustrations of the law of pitch.
.^ Here it is sufficient to say that the French standard is a' =435 with c" practically 522, and that in England the pitch is somewhat higher.
^ Here it is sufficient to say that the frequencies of a note, its major third, its fifth and its octave , are in the ratios of 4: 5: 6: 8.
.^ The French notation is as under: C D E F G A B c Ut 1 Re f M] Fa] Sol i La i Si, Utz.
.^ The next higher octave has the suffix 2, the next higher the suffix 3, and so on.
^ The next octave below is C] D i E 1 F] G] A] B 1 C, and each preceding octave has another accent as suffix.
French forks are marked with double the true
frequency, so that Uta is marked 512.
Limiting Frequencies for Musical Sounds
.^ Until the vibrations of a source have a frequency in the neighbourhood of 30 per second the ear can hear the separate impulses, if strong enough, but does not hear a note.
^ AC = X. If the source makes n vibrations in one second it is said to have " frequency " n.
^ As an example of the application of this siren, suppose that the number of revolutions of the plate, as shown by the indices, amounts to 5400 in a minute, that is, to 90 per second, then the number of vibrations per second of the note heard amounts to 90n, or (if number of holes in each plate = 8) to 720.
.^ It is not easy to determine the exact point at which the impulses fuse into a continuous tone, for higher tones are usually present with the deepest of which the frequency is being counted, and these may be mistaken for it.
^ A practised ear easily discerns the coexistence of these various tones when a pianoforte or violin string is thrown into vibration.
^ It is evident that we may have tones of frequency hn 1 kn 2 hn i  kn 2 hnl+kn2, where h and k are any integers.
.^ Helmholtz ( Sensations of Tone, ch.
^ We may illustrate the first method by taking a case discussed by Helmholtz ( Sensations of Tone, app.
^ For a full discussion see his Sensations of Tone, ch.
ix.)
used a string loaded at the middle point so that the higher tones
were several octaves above the fundamental, and so not likely to be
mistaken for it; he found that with
.^ As an example of the application of this siren, suppose that the number of revolutions of the plate, as shown by the indices, amounts to 5400 in a minute, that is, to 90 per second, then the number of vibrations per second of the note heard amounts to 90n, or (if number of holes in each plate = 8) to 720.
^ This is generally equal to the number of waves issuing from the source per second, and therefore equal to its frequency of vibration.
^ There will be a tone frequency 480  256 =224, and this will be very dissonant with 256.
.^ A determinate musical pitch is not perceived, he says, till about 40 vibrations per second.
^ This is generally equal to the number of waves issuing from the source per second, and therefore equal to its frequency of vibration.
^ Thus the wheel made about 8 revolutions per second.
.^ At the other end of the scale with increasing frequency there is another limiting frequency somewhere about 20,000 per second, beyond which no sound is heard.
^ For an iron wire Y is about 10 12 /4, so that for a frequency of 500 in a wire fixed at both ends a length about 5 metres is required.
^ If, at the same time, a tuningfork of known number of vibrations per second be made to trace its own line close to the other, a comparison of the two lines gives the number corresponding to the sound under consideration.
.^ But this limit varies greatly with different individuals and with age for the same individual.
^ Keeping AD constant and varying W it may be shown that n oo / W. Lastly, by using different strings, it may be shown that, with the same T and 1, (i/m).
.^ Persons who when young could hear the squeaks of bats may be quite deaf to them when older.
Koenig
constructed a series of bars forming a harmonicon, the frequency of
each
bar being calculable, and he
found the limit to be between 16,000 and 24,000.
The Number of Vibrations needed to give the Perception of
Pitch
.^ Experiments have been made on this subject by various workers, the most extensive by W. Kohlrausch ( Wied.
Ann.,
1880, x. 1). .^ He allowed a limited number of teeth on the arc of a circle to strike against a card.
With sixteen teeth the pitch was
well defined; with nine teeth it was fairly determinate; and even
with two teeth it could be assigned with no great error. His
remarkable result that two waves give some sense of pitch, in fact
a tone with wavelength equal to the interval between the waves, has
been confirmed by other observers.
Alteration of Pitch with Motion of Source or Hearer: Doppler's
Principle
.^ A very noticeable illustration of the alteration of pitch by motion occurs when a whistling locomotive moves rapidly past an observer.
^ An observer in the plane of the motion can easily hear a change in the pitch as the pitchpipe moves to and from him.
^ Alteration of Pitch with Motion of Source or Hearer: Doppler's Principle .
.^ As it passes, the pitch of the whistle falls quite appreciably.
The explanation is simple.
.^ The engine follows up any wave that it has sent forward, and so crowds up the succeeding waves into a less distance than if it remained at rest.
^ It draws off from any wave it has sent backward and so spreads the succeeding waves over a longer distance than if it had remained at rest.
^ Since U=n X where U is the velocity of sound, X the wavelength, and n the frequency, it follows that the forward frequency is greater than the backward frequency.
.^ The engine follows up any wave that it has sent forward, and so crowds up the succeeding waves into a less distance than if it remained at rest.
^ It draws off from any wave it has sent backward and so spreads the succeeding waves over a longer distance than if it had remained at rest.
^ The waves from a source surrounded by a uniform medium at rest spread out as spheres with the source as centre.
.^ Hence the forward waves are shorter and the backward waves are longer.
^ Since U=n X where U is the velocity of sound, X the wavelength, and n the frequency, it follows that the forward frequency is greater than the backward frequency.
^ It draws off from any wave it has sent backward and so spreads the succeeding waves over a longer distance than if it had remained at rest.
.^ Since U=n X where U is the velocity of sound, X the wavelength, and n the frequency, it follows that the forward frequency is greater than the backward frequency.
^ The jet tube, for a reason which will be given when we consider the maintenance of vibrations, must be less than c a9 half the length of the sounding tube.
^ In ordinary soundwaves the effect of the particle velocity in affecting the velocity of transmission must be very small.
Note
Interval with
Frequency .

I

D
second
a

E
major
third
4

F
fourth

G
fifth

A
major
sixth

B
seventh
18

octave
2

ive fre
Successive
quency ratios.



,


8




Successive


major

minor

major

major

minor

major

major

tervals. .


tone

tone

semi
tone

tone

tone

tone

semi
tone

.^ The more general case of motion of source, medium and receiver may be treated very easily if the motions are all in the line joining source and receiver.
^ We may see how this occurs by supposing that the restoring force of the receiving mechanism is represented by Ax fµx 2, where x is the displacement and µx 2 is very small.
^ We shall only consider one interesting case of sound diffraction which may be easily observed.
Let S (fig.
.^ This is generally equal to the number of waves issuing from the source per second, and therefore equal to its frequency of vibration.
^ AC = X. If the source makes n vibrations in one second it is said to have " frequency " n.
^ A given note has always the same frequency, that is to say, the hearer receives the same number of waves per second whatever the source by which the note is produced.
.^ Let the velocity of the air from S to R be w, and let U be the velocity of sound in still air.
^ Let the particle which was at M originally be at m at the given instant, being displaced to the left or backwards.
^ Helmholtz investigated the velocity of propagation of sound in pipes, taking into account the viscosity of the air (Rayleigh, Sound, ii.
FIG. 20.
.^ If all were still, the n waves emitted by S in one second would spread over a length U. But through the wind velocity the first wave is carried to a distance U + w from S, while through the motion of the source the last wave is a distance u from S. Then the n waves occupy a space U + w  u.
^ If the velocities of source and receiver are equal then the frequency is not affected by their motion or by the wind.
^ If he were at rest, it would be the waves in length U + w, for the wave passing him at the beginning of a second would be so far distant at the end of the second.
.^ If he were at rest, it would be the waves in length U + w, for the wave passing him at the beginning of a second would be so far distant at the end of the second.
^ If all were still, the n waves emitted by S in one second would spread over a length U. But through the wind velocity the first wave is carried to a distance U + w from S, while through the motion of the source the last wave is a distance u from S. Then the n waves occupy a space U + w  u.
^ He used a tube of variable length and determined the length resounding to a given fork, (1) when the closed end was the first node, (2) when it was the second node.
.^ This is generally equal to the number of waves issuing from the source per second, and therefore equal to its frequency of vibration.
^ The velocity thus obtained will be affected by the wind.
^ If the velocities of source and receiver are equal then the frequency is not affected by their motion or by the wind.
.^ The velocity thus obtained will be affected by the wind.
^ If the velocities of source and receiver are equal then the frequency is not affected by their motion or by the wind.
^ But if their velocities are different, the frequency of the waves received is affected both by these velocities and by that of the wind.
.^ The change in pitch through motion of the source may be illustrated by putting a pitchpipe in one end of a few feet of rubber tubing and blowing through the other end while the tubing is whirled round the head.
^ A very noticeable illustration of the alteration of pitch by motion occurs when a whistling locomotive moves rapidly past an observer.
^ If we take one of these spheres a distance from the source very great as compared with a single wavelength, and draw a radius to a point on the sphere, then for some little way round that point the sphere may be regarded as a plane perpendicular to the radius or the line of propagation.
.^ An observer in the plane of the motion can easily hear a change in the pitch as the pitchpipe moves to and from him.
^ A very noticeable illustration of the alteration of pitch by motion occurs when a whistling locomotive moves rapidly past an observer.
^ When the motion is reversed and the air moves out of the pipe at the embouchure, the sheet is deflected on to the outer side of the sharp edge, and no work is done against it by the air in the pipe.
.^ Though a musical note has definite pitch or frequency, notes of the same pitch emitted by different instruments have quite different quality or timbre.
^ Musical Quality or Timbre.

.^ Though a musical note has definite pitch or frequency, notes of the same pitch emitted by different instruments have quite different quality or timbre.
^ They found that the velocity of propagation of different musical sounds was the same.
^ Now we can see that two notes of the same pitch, but of different quality, or different form of displacement curve, will, when thus analysed, break up into a series having the same harmonic wavelengths; but they may differ as regards the members of the series present and their amplitudes and epochs.
.^ Its periodicity implies that after a certain distance the displacement curve exactly repeats itself.
^ The three characteristics of a longitudinal periodic disturbance are its amplitude, the length after which it repeats itself, and its form, which may be represented by the shape of the displacement curve.
^ The form of the curve is evidently as represented in fig.
.^ Now the amplitude evidently corresponds to the loudness, and the length of period corresponds to the pitch or frequency.
^ The three characteristics of a longitudinal periodic disturbance are its amplitude, the length after which it repeats itself, and its form, which may be represented by the shape of the displacement curve.
^ The question now arises whether the sensation produced by a periodic disturbance can be analysed in correspondence with this geometrical analysis.
.^ Hence we must put down the quality or timbre as depending on the form.
.^ The form of the curve is evidently as represented in fig.
^ For instance, if we have a wave with displacement curve of form ABC (fig.
^ The simplest form of wave, so far as our sensation goes  that is, the one giving rise to a pure tone  is, we have every reason to suppose, one in which the displacement is represented by a harmonic curve or a curve of sines, y=a sin m(x  e).
If we put this in the form
y=a sin (x  e), we see that y=o, for
x =e,
e +aX, e+2A, e+;X, and so on, that
y is +
from
x=e to
x=e+iX,  from e+zX to e+?A, and so
on, and that it alternates between the values+a and
 a.
The form of the curve is evidently as represented in fig. 21, and
it may easily be drawn to exact scale from a table of sines.
K FIG. 21.
.^ In this curve ABCD are nodes.
.^ OA=e is termed the epoch, being the distance from 0 of the first ascending node .
AC is the shortest distance after which the
curve begins to repeat itself; this length X is termed the
wavelength.
.^ The maximum height of the curve HM =a is the amplitude.
.^ If we transfer 0 to A, e=o, and the curve may be represented by y=a sin A x.
^ I) We may find here the value of this when we have a train of waves in which the displacement is represented by a sine curve of amplitude a, viz.
^ The three characteristics of a longitudinal periodic disturbance are its amplitude, the length after which it repeats itself, and its form, which may be represented by the shape of the displacement curve.
.^ The maximum height of the curve HM =a is the amplitude.
^ The excess pressure on CD is therefore 4 1 (c:3+ pu 2)dt.
^ I) We may find here the value of this when we have a train of waves in which the displacement is represented by a sine curve of amplitude a, viz.
.^ The chief experimental basis for supposing that a train of longitudinal waves with displacement curve of this kind arouses the sensation of a pure tone is that the more nearly a source is made to vibrate with a single simple harmonic motion, and therefore, presumably, the more nearly it sends out such a harmonic train, the more nearly does the note heard approximate to a single pure tone.
^ It sends out n waves in each second.
^ The vibration in some way arouses the sensation of the corresponding tone.
.^ Any periodic curve may be resolved into sine or harmonic curves by Fourier's theorem.
^ Fourier's theorem asserts that such a curve may be built up by the superposition, or addition of ordinates, of a series of sine curves of wavelengths X, IX, 3A, 4A...
^ Using the term " note " for the sound produced by a periodic disturbance, there is no doubt that a welltrained ear can resolve a note into pure tones of frequencies equal to those of the fundamental and its harmonics.
.^ Suppose that any periodic sound disturbance, consisting of plane waves, is being propagated in the direction ABCD (fig.
^ When intense sound waves impinged on the wall, the disk moved back through the hole, and by an amount showing a pressure of the order given by the following investigation: Suppose that a train of waves is incident normally on the surface S (fig.
^ Let a train of waves be passing from left to right in the direction ABCD (fig.
22).
.^ Let it be represented by a displacement curve Ahbkc .
^ If this is done for every point we obtain a continuous curve Apbqcrd , which represents the displacement at every point at the given instant, though by a length at right angles to the actual displacement and on an arbitrary scale.
^ If a wave travels on without alteration the travelling may be represented by pushing on the displacement curve.
.^ Its periodicity implies that after a certain distance the displacement curve exactly repeats itself.
^ The three characteristics of a longitudinal periodic disturbance are its amplitude, the length after which it repeats itself, and its form, which may be represented by the shape of the displacement curve.
^ The tangent to the displacement curve is always parallel to the axis, that is, for a small distance the successive particles are always equally displaced, and therefore always occupy the same volume.
Let AC be the H L FIG.
22.
shortest distance after which the repetition occurs, so that
.^ Cldme is merely Ahbkc moved on a distance AC. Then AC =A is the wavelength or period of the curve.
^ AC is the shortest distance after which the curve begins to repeat itself; this length X is termed the wavelength.
^ Fourier's theorem asserts that such a curve may be built up by the superposition, or addition of ordinates, of a series of sine curves of wavelengths X, IX, 3A, 4A...
.^ Let ABCD be drawn at such level that the areas above and below it are equal; then ABCD is the axis of the curve.
^ Let two trains of equal waves moving in opposite directions along such a string of indefinite length form the stationary system of fig.
^ The distribution of velocity then is represented by the dotted curve and is forward when the curve is above the axis and Dackward when it is below.
.^ Since the curve represents a longitudinal disturbance in air it is always continuous, at a finite distance from the axis, and with only one ordinate for each abscissa .
^ Tdx/ds, and when the disturbance is sufficiently small the curve of displacement is so nearly parallel to the axis that /ds = I, and this component is T. The component of T perpendicular to the axis is Tdy/ds=Tdy/dx.
^ We can represent waves of longitudinal displacement by a curve , and this enables us to draw very important conclusions in a very simple way.
.^ Fourier's theorem asserts that such a curve may be built up by the superposition, or addition of ordinates, of a series of sine curves of wavelengths X, IX, 3A, 4A...
^ But we may also have a shorter wavelength such that the length AK occupies the tube.
^ With a still shorter wavelength we may have the length.
if the amplitudes
.^ The wellknown rule of a mile for every five seconds between flash and peal gives a fair estimate of the distance of the lightning.
.^ We may therefore put y=a sin T (x  e) +b sin ( x  f)+c sin 6 (x  g ) +&c.
.^ Cldme is merely Ahbkc moved on a distance AC. Then AC =A is the wavelength or period of the curve.
^ The shortest distance in which this repetition occurs is called the wavelength.
^ But we may also have a shorter wavelength such that the length AK occupies the tube.
.^ If the series were complete we should have terms which separately would correspond to the fundamental, its octave, its twelfth, its double octave, and so on.
^ The whole series of fundamental and overtones gives the complete set of harmonics of frequencies proportional to 1, 2, 3, 4, ..., and wavelengths proportional to 1, 2, 3, 4 ..
.^ Further, the same harmonics with the same amplitude will always be present.
^ The whole series forms the series of odd harmonics.
^ Now we can see that two notes of the same pitch, but of different quality, or different form of displacement curve, will, when thus analysed, break up into a series having the same harmonic wavelengths; but they may differ as regards the members of the series present and their amplitudes and epochs.
.^ We may regard quality, then, as determined by the members of the harmonic series present and their amplitudes and epochs.
^ Now we can see that two notes of the same pitch, but of different quality, or different form of displacement curve, will, when thus analysed, break up into a series having the same harmonic wavelengths; but they may differ as regards the members of the series present and their amplitudes and epochs.
^ Further, the same harmonics with the same amplitude will always be present.
.^ It may, however, be stated here that certain experiments of Helmholtz appear to show that the epoch of the harmonics has not much effect on the quality.
^ R. Koenig, Quelques experiences d'acoustique (1882) describes apparatus and experiments, intended to show, in opposition to Helmholtz, that beats coalesce into tones, and also that the quality of a note is affected by alteration of phase of one of its component overtones relative to the phase of the fundamental.
^ We may regard quality, then, as determined by the members of the harmonic series present and their amplitudes and epochs.
.^ Fourier's theorem can also be usefully applied to the disturbance of a source of sound under certain conditions.
^ As a preliminary to the investigation of the modes of vibration of certain sources of sound we shall consider the formation of " stationary waves."
^ We shall discuss the disturbance which is propagated from the source to the ear , and which there produces sound, and the modes in which various sources vibrate and give rise to the disturbance.
.^ The nature of these conditions will be best realized by considering the case of a stretched string.
.^ It is easy to deduce the modes of vibration from stationary waves as in the previous cases.
^ It is shown below how the vibrations of a string may be deduced from stationary waves.
^ As a preliminary to the investigation of the modes of vibration of certain sources of sound we shall consider the formation of " stationary waves."
.^ Let us here suppose that the string AB is displaced into the form AHB (fig.
^ If then we resolve Ahbkc into harmonics by Fourier's theorem, we may follow the motion of the separate harmonics, and their superposition will give the form of the string at any instant.
^ For instance, if we have a wave with displacement curve of form ABC (fig.
23) and is
then let go. Let H FIG. 23.
us imagine it to form half a wavelength of the extended train
.^ Zgahbkc , on an indefinitely extended stretched string, the values of y at equal distances from A (or from B) being equal and opposite.
^ Let two trains of equal waves moving in opposite directions along such a string of indefinite length form the stationary system of fig.
^ If L 1 is the internodal distance and U 1 the velocity in a gas, L and U being the corresponding values for air, we have U 1 /U =L1/L. .
.^ We shall first investigate the velocity with which a disturbance travels along a string of mass m per unit length when it is stretched with a constant tension T, the same at all points.
^ Then, as we shall prove later, the vibrations of the string may be represented by the travelling of two trains in opposite directions each with velocity /tension=mass per unit length each half the height of the train represented in fig.
^ Since the nodes are always at rest we may represent the vibration of a given string by the length between any two nodes.
.^ For the superposition of these trains will give a stationary wave between A H A (16) Y which is an equation characteristic of simple harmonic motion.
^ U, for in the stationary wave system the pressure change and the amplitude are both double those in either train, so that the same relation holds.
^ It follows from this that any periodic disturbance in air can be resolved into a definite series of simple harmonic disturbances of wavelengths equal to the original wavelength and its successive submultiples, and each of these would separately give the sensation of a pure tone.
.^ The maximum velocity of a particle in the wavetrain is the amplitude of dy/dt.
^ In the time dt which the wave takes to travel over MN the particle displacement at N changes by QR, and QR=  udt, so that QR/MN =  u/U. But QR/MN = dy/dx.
^ U, for in the stationary wave system the pressure change and the amplitude are both double those in either train, so that the same relation holds.
= . (19) (18) [[[Intensity Or Loudness]] and B. Now we may
resolve these trains by Fourier's theorem into harmonics of
wavelengths X, 2X, 3A, &c., where X=2AB and the conditions as
to the values of
y can be shown to require that the
harmonics shall all have nodes, coinciding with the nodes of the
fundamental curve.
^ [[[Intensity Or Loudness]] and B. Now we may resolve these trains by Fourier's theorem into harmonics of wavelengths X, 2X, 3A, &c., where X=2AB and the conditions as to the values of y can be shown to require that the harmonics shall all have nodes, coinciding with the nodes of the fundamental curve.
^ With a still shorter wavelength we may have the length.
^ (I) represent a wavelength of each train when they are coincident.
.^ Since the velocity is the same for all disturbances they all travel at the same speed, and the two trains will always remain of the same form.
^ They found that the velocity of propagation of different musical sounds was the same.
^ When a wave of sound meets a surface separating two media it is in part reflected, travelling back from the surface into the first medium again with the velocity with which it approached.
.^ If then we resolve Ahbkc into harmonics by Fourier's theorem, we may follow the motion of the separate harmonics, and their superposition will give the form of the string at any instant.
^ Any periodic curve may be resolved into sine or harmonic curves by Fourier's theorem.
^ Fourier's theorem asserts that such a curve may be built up by the superposition, or addition of ordinates, of a series of sine curves of wavelengths X, IX, 3A, 4A...
.^ Further, the same harmonics with the same amplitude will always be present.
^ We may regard quality, then, as determined by the members of the harmonic series present and their amplitudes and epochs.
^ Further, the greater the dissipation of energy the less is the prominence of the amplitude of vibration for exact coincidence over the amplitude when the periods are not quite the same, though it is still the greatest for coincidence.
.^ We see, then, that the conditions for the application of Fourier's theorem are equivalent to saying that all disturbances will travel along the system with the same velocity.
^ The velocity of a disturbance along such a bar, and its modes of vibration, depend therefore on the elastic properties of the material and the dimensions of the bar .
^ Suppose that a disturbance is travelling with velocity U unchanged in form along a rod from left to right.
.^ In many vibrating systems this does not hold, and then Fourier's theorem is no longer an appropriate resolution.
^ We see, then, that the conditions for the application of Fourier's theorem are equivalent to saying that all disturbances will travel along the system with the same velocity.
^ There are shears of the order dry/dx and the simple Young's modulus system can no longer be taken to represent the actual condition (see Rayleigh, Sound, i.
.^ But where it is appropriate, the disturbance sent out into the air contains the same harmonic series as the source.
^ The combination tones thus produced in the source should have a physical existence in the air, and the amplitudes of those represented in (35) should be of the same order.
^ It follows from this that any periodic disturbance in air can be resolved into a definite series of simple harmonic disturbances of wavelengths equal to the original wavelength and its successive submultiples, and each of these would separately give the sensation of a pure tone.
.^ The question now arises whether the sensation produced by a periodic disturbance can be analysed in correspondence with this geometrical analysis.
^ Now the amplitude evidently corresponds to the loudness, and the length of period corresponds to the pitch or frequency.
^ There appears to be no doubt that they are produced, and the only question is whether the theory accounts sufficiently for the intensity of the tones actually heard.
.^ Using the term " note " for the sound produced by a periodic disturbance, there is no doubt that a welltrained ear can resolve a note into pure tones of frequencies equal to those of the fundamental and its harmonics.
^ It follows from this that any periodic disturbance in air can be resolved into a definite series of simple harmonic disturbances of wavelengths equal to the original wavelength and its successive submultiples, and each of these would separately give the sensation of a pure tone.
^ The combination tones thus produced in the source should have a physical existence in the air, and the amplitudes of those represented in (35) should be of the same order.
.^ If, for instance, a note is struck and held down on a piano, a little practice enables us to hear both the octave and the twelfth with the fundamental, especially if we have previously directed our attention to these tones by sounding them.
^ If the series were complete we should have terms which separately would correspond to the fundamental, its octave, its twelfth, its double octave, and so on.
^ Using the term " note " for the sound produced by a periodic disturbance, there is no doubt that a welltrained ear can resolve a note into pure tones of frequencies equal to those of the fundamental and its harmonics.
.^ But the harmonics are most readily heard if we fortify the ear by an air cavity with a natural period equal to that of the harmonic to be sought.
^ Using the term " note " for the sound produced by a periodic disturbance, there is no doubt that a welltrained ear can resolve a note into pure tones of frequencies equal to those of the fundamental and its harmonics.
^ It follows from this that any periodic disturbance in air can be resolved into a definite series of simple harmonic disturbances of wavelengths equal to the original wavelength and its successive submultiples, and each of these would separately give the sensation of a pure tone.
.^ The form used by Helmholtz is a glove of thin brass (fig.
.^ In like manner, the first circle in which we have two sets of holes dividing the circumference, the one into say 8 parts, and the other into Io,.
^ For instance, if a rope is fixed at one end and held in the hand at the other end, a transverse jerk by the hand will travel as a transverse wave along the rope.
^ S fitting cork or card piston being fixed on one end of the sounder, which is inserted within the dusttube.
.^ But a cardboard tube closed at one end, with the open end near the ear, will often suffice, and it may be tuned by more or less covering up the open end.
^ The phenomena of beats may be easily observed with two organpipes put slightly out of tune by placing the hand near the open end of one of them, with two musical strings on a resonant chest, or with two tuningforks of the same pitch mounted on their resonance boxes, or held over a resonant cavity (such as a glass jar), one of the forks being put out of tune by loading one prong with a small lump of beeswax.
^ At a station more than 13 kilometres away a sort of big ear trumpet , closed by a membrane, was placed with the membrane under water, the tube rising above the surface.
.^ If the harmonic corresponding to the resonator is present its tone swells out loudly.
^ When two tones are sounded together with frequencies not very different, " beats " or swellingsout of the sound are heard of frequency equal to the difference of frequencies of the two tones (see below).
.^ This resonance is a particular example of the general principle that a vibrating system will be set in vibration by any periodic Forced VI force applied to it, and ultimately in the period of the force, its own natural vibrations gradually dying down.
^ V i brat i ons thus excited are termed forced vibrations, and their amplitude is greater the more nearly the period of the applied force approaches that of the system when vibrating freely.
^ The disturbance made at the commencement of the blowing will no doubt set the air in the pipe vibrating in its own natural period, just as any irregular air disturbance will set a suspended body swinging in its natural period, but we are to consider how the vibration is maintained when once set going.
.
.^ V i brat i ons thus excited are termed forced vibrations, and their amplitude is greater the more nearly the period of the applied force approaches that of the system when vibrating freely.
^ Let an external force F act on the system, and for simplicity suppose its period is so great compared with that of the mechanism that we may take it as practically in equilibrium with the restoring force.
^ Hence the attracting force does more work in the return journey than is done against it in the outgoing, and the balance is available to increase the vibration.
.^ The mathematical investigation of forced vibrations (Rayleigh,Sound, i.
^ Helmholtz investigated the velocity of propagation of sound in pipes, taking into account the viscosity of the air (Rayleigh, Sound, ii.
^ As a preliminary to the investigation of the modes of vibration of certain sources of sound we shall consider the formation of " stationary waves."
§
.^ There is always loss of energy by dissi pation in the vibrating machinery and by radiation into the medium, and the amplitude only increases until this loss is balanced by the gain from the work done by the applied force.
^ If it is always on it only acts as if the value of gravity were increased, and does not help to maintain or check the vibration, but merely to shorten the period.
^ But there is always leakage of energy either through friction or through waveemission, so that the vibration only increases up to the point at which the leakage of energy balances the energy put in by the applied force.
.^ But there is always leakage of energy either through friction or through waveemission, so that the vibration only increases up to the point at which the leakage of energy balances the energy put in by the applied force.
^ The reason of this is, that the segments of the plate AOD, BOC always vibrate in the same direction, but oppo sitely to the segments AOB, DOC. Hence, when the pasteboard is in its place, there are two waves of same phase starting from the two former segments, and reaching the ear after equal distances of transmission through the air, are again in the same phase, and produce on the ear a conjunct impression.
^ There is always loss of energy by dissi pation in the vibrating machinery and by radiation into the medium, and the amplitude only increases until this loss is balanced by the gain from the work done by the applied force.
.^ Further, the greater the dissipation of energy the less is the prominence of the amplitude of vibration for exact coincidence over the amplitude when the periods are not quite the same, though it is still the greatest for coincidence.
^ V i brat i ons thus excited are termed forced vibrations, and their amplitude is greater the more nearly the period of the applied force approaches that of the system when vibrating freely.
^ Further, the same harmonics with the same amplitude will always be present.
.^ The principle of forced vibration may be illustrated by a simple case.
^ We may illustrate the first method by taking a case discussed by Helmholtz ( Sensations of Tone, app.
^ We may illustrate the successive modes of vibration by using as pipe a tall cylindrical jar, and as exciter a vibrating tuningfork held over the mouth.
.^ Suppose that a mass M is controlled by some sort of spring, so that moving freely it executes harmonic vibrations given by µx, where µx is the restoring force to the centre of vibration.
^ We may see how this occurs by supposing that the restoring force of the receiving mechanism is represented by Ax fµx 2, where x is the displacement and µx 2 is very small.
^ V i brat i ons thus excited are termed forced vibrations, and their amplitude is greater the more nearly the period of the applied force approaches that of the system when vibrating freely.
.^ Putting A /M =n 2 the equation becomes x+n 2 x=o, whence x =A sin nt, and the period is 27r/n.
^ Let us assume that the body makes vibrations in the new period 27rp, and let us put x = B sin pt; substituting in (22) we have p 2 B +n 2 B +P/M =0, whence P r B  M p2 _ n2 and the " forced " oscillation due to P sin pt is x = P .
.^ Now suppose that in addition to the internal force represented by ,ux, an external harmonic force of period 27r/p is applied.
^ We may see how this occurs by supposing that the restoring force of the receiving mechanism is represented by Ax fµx 2, where x is the displacement and µx 2 is very small.
^ V i brat i ons thus excited are termed forced vibrations, and their amplitude is greater the more nearly the period of the applied force approaches that of the system when vibrating freely.
Repre  senting it by P sin
pt, the equation of motion is
now
2 M sin
pt=o. (22) Let us assume that the
body makes vibrations in the new period 27rp, and let us put x = B
sin pt; substituting in (22) we have p 2 B +n 2 B +P/M =0, whence P
r B  M
p2 _ n2 and the " forced " oscillation due to P
sin
pt is
x = P
.