Waveform means the shape and form of a signal such as a wave moving in a solid, liquid or gaseous medium.
In many cases the medium in which the wave is being propagated does not permit a direct visual image of the form. In these cases, the term 'waveform' refers to the shape of a graph of the varying quantity against time or distance. An instrument called an oscilloscope can be used to pictorially represent the wave as a repeating image on a CRT or LCD screen.
By extension of the above, the term 'waveform' is now also sometimes used to describe the shape of the graph of any varying quantity against time.
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Common periodic waveforms include (t is time):
Other waveforms are often called composite waveforms and can often be described as a combination of a number of sinusoidal waves or other basis functions added together.
The Fourier series describes the decomposition of periodic waveforms, such that any periodic waveform can be formed by the sum of a fundamental component and harmonic components. Finiteenergy nonperiodic waveforms can be analyzed into sinusoids by the Fourier transform.
The waveform is the shape of a wave as it travels. There are many different waveforms. They are usually a shape which is repeated over and over (a "periodic waveform"). A common waveform is the sine wave. It is normally not possible to see a waveform without some device.
The amplitude of a waveform may change a lot. Even though it changes, the waveform has a root mean square (rms) value. For example: in the UK, the AC mains supply is a sine wave and has a voltage of 240 V. This is an rms voltage. The actual voltage varies:
The amplitude of the sine wave keeps changing from 339.4 V to +339.4 V.
Root mean square is important. It lets us work out many useful things, like power and heating in a wire.
This table has information about working out the rms for some waveforms.
Wave type  rms amplitude 

Sine wave  $A\_\{\backslash mbox\{rms\}\}\; =\; \backslash frac\{A\_\{\backslash mbox\{peak\}\}\}\{\backslash sqrt\; 2\}$ 
Square wave  $A\_\{\backslash mbox\{rms\}\}\; =\; A\_\{\backslash mbox\{peak\}\}$ 
Triangular wave  $A\_\{\backslash mbox\{rms\}\}\; =\; \backslash frac\{A\_\{\backslash mbox\{peak\}\}\}\{\backslash sqrt\; 3\}$ 
