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The wave equation is an important second-order linear partial differential equation of waves, such as sound waves, light waves and water waves. It arises in fields such as acoustics, electromagnetics, and fluid dynamics. Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.

A pulse traveling through a string with fixed endpoints as modeled by the wave equation.
Spherical waves coming from a point source.

Contents

Introduction

The wave equation is the prototypical example of a hyperbolic partial differential equation. In its simplest form, the wave equation refers to a scalar function u=(x1, x2,...,xn,t) that satisfies:

{ \partial^2 u \over \partial t^2 } = c^2 \nabla^2 u

where \scriptstyle\nabla^2 is the (spatial) Laplacian and where c is a fixed constant equal to the propagation speed of the wave. This is known as the non-dispersive wave equation. For a sound wave in air at 20°C this constant is about 343 m/s (see speed of sound). For the vibration of a string the speed can vary widely, depending upon the linear density of the string and the tension on it. For a spiral spring (a slinky) it can be as slow as a meter per second. More realistic differential equations for waves allow for the speed of wave propagation to vary with the frequency of the wave, a phenomenon known as dispersion. In such a case, c must be replaced by the phase velocity:

v_\mathrm{p} = \frac{\omega}{k}.

Another common correction in realistic systems is that the speed can also depend on the amplitude of the wave, leading to a nonlinear wave equation:

{ \partial^2 u \over \partial t^2 } = c(u)^2 \nabla^2 u

Also note that a wave may be superimposed onto another movement (for instance sound propagation in a moving medium like a gas flow). In that case the scalar u will contain a Mach factor (which is positive for the wave moving along the flow and negative for the reflected wave).

The elastic wave equation in three dimensions describes the propagation of waves in an isotropic homogeneous elastic medium. Most solid materials are elastic, so this equation describes such phenomena as seismic waves in the Earth and ultrasonic waves used to detect flaws in materials. While linear, this equation has a more complex form than the equations given above, as it must account for both longitudinal and transverse motion:

\rho{ \ddot{\bold{u}}} = \bold{f} + ( \lambda + 2\mu )\nabla(\nabla \cdot \bold{u}) - \mu\nabla \times (\nabla \times \bold{u})

where:

  • \scriptstyle\lambda and \scriptstyle\mu are the so-called Lamé parameters describing the elastic properties of the medium,
  • \scriptstyle\rho is density,
  • \scriptstyle\bold{f} is the source function (driving force),
  • and \scriptstyle\bold{u} is displacement.

Note that in this equation, both force and displacement are vector quantities. Thus, this equation is sometimes known as the vector wave equation.

Variations of the wave equation are also found in quantum mechanics and general relativity.

Scalar wave equation in one space dimension

Derivation of the wave equation

From Hooke's law

wave equation in the one dimensional case can be derived from Hooke's law in the following way: Imagine an array of little weights of mass m interconnected with massless springs of length h . The springs have a stiffness of k:

Array of masses.png

Here u(x) measures the distance from the equilibrium of the mass situated at x. The forces exerted on the mass m at the location x + h are:

F_{\mathit{Newton}}=m \cdot a(t)=m \cdot {{\partial^2 \over \partial t^2}u(x+h,t)}
F_\mathit{Hooke} = F_{x+2h} + F_x = k \left [ {u(x+2h,t) - u(x+h,t)} \right ] + k[u(x,t) - u(x+h,t)]

The equation of motion for the weight at the location x+h is given by equating these two forces:

m{\partial^2u(x+h,t) \over \partial t^2}= k[u(x+2h,t)-u(x+h,t)-u(x+h,t)+u(x,t)]

where the time-dependence of u(x) has been made explicit.

If the array of weights consists of N weights spaced evenly over the length L = N h of total mass M = N m, and the total stiffness of the array K = k/N we can write the above equation as:

{\partial^2u(x+h,t) \over \partial t^2}={KL^2 \over M}{[u(x+2h,t)-2u(x+h,t)+u(x,t)] \over h^2}

Taking the limit N\rightarrow \infty,h\rightarrow 0 (and assuming smoothness) one gets:

 {\partial^2 u(x,t) \over \partial t^2}={KL^2 \over M}{ \partial^2 u(x,t) \over \partial x^2 }

(KL2)/M is the square of the propagation speed in this particular case.

From the generic scalar transport equation

Starting with the generic scalar transport equation without diffusion,

\frac{\partial\phi}{\partial t}+\frac{\partial(u\phi)}{\partial x}=S_\phi,

we differentiate with respect to t to get

\frac{\partial^2\phi}{\partial t^2}+\frac{\partial^2(u\phi)}{\partial x\partial t}=\frac{\partial S_\phi}{\partial t}.

Assuming that Sφ and u are constant, we may write

\frac{\partial^2\phi}{\partial t^2}+u\frac{\partial}{\partial x}\frac{\partial\phi}{\partial t}=0.

Substituting for the time derivative of φ we get

\frac{\partial^2\phi}{\partial t^2}+u\frac{\partial}{\partial x}\left[S_\phi-\frac{\partial(u\phi)}{\partial x}\right]=\frac{\partial^2\phi}{\partial t^2}-u^2\frac{\partial^2\phi}{\partial x^2}=0,

where u is the speed of propagation of the scalar φ which, in general, is a function of time and position.

General solution

The one dimensional wave equation is unusual for a partial differential equation in that a very simple general solution may be found. Defining new variables[1]:

\xi = x - c t \quad ; \quad \eta = x + c t

changes the wave equation into

\frac{\partial^2 u}{\partial \xi \partial \eta} = 0

which leads to the general solution

u(\xi, \eta) = F(\xi) + G(\eta) \quad \Rightarrow \quad u(x, t) = F(x - c t) + G(x + c t)

In other words, solutions of the 1D wave equation are sums of a right traveling function F and a left traveling function G. "Traveling" means that the shape of these individual arbitrary functions with respect to x stays constant, however the functions are translated left and right with time at the speed c. This was derived by Jean le Rond d'Alembert.

Another way to arrive at this result is to note that the wave equation may be "factored":

\left[\frac{\part}{\part t} - c\frac{\part}{\part x}\right] \left[ \frac{\part}{\part t} + c\frac{\part}{\part x}\right] u = 0
\Big\Downarrow
\frac{\part u}{\part t} - c\frac{\part u}{\part x} = 0 \qquad \mbox{and} \qquad \frac{\part u}{\part t} + c\frac{\part u}{\part x} = 0

These last two equations are advection equations, one left traveling and one right, both with constant speed c.

For an initial value problem, the arbitrary functions F and G can be determined to satisfy initial conditions:

u(x,0)=f(x) \,
u_t(x,0)=g(x) \,

The result is d'Alembert's formula:

u(x,t) = \frac{f(x-ct) + f(x+ct)}{2} + \frac{1}{2c} \int_{x-ct}^{x+ct} g(s) ds

In the classical sense if \scriptstyle f(x) \in C^k and \scriptstyle g(x) \in C^{k-1} then \scriptstyle u(t,x) \in C^k. However, the waveforms F and G may also be generalized functions, such as the delta-function. In that case, the solution may be interpreted as an impulse that travels to the right or the left.

The basic wave equation is a linear differential equation which means that the amplitude of two waves interacting is simply the sum of the waves. This means also that a behavior of a wave can be analyzed by breaking up the wave into components. The Fourier transform breaks up a wave into sinusoidal components and is useful for analyzing the wave equation.

Scalar wave equation in three space dimensions

The solution of the initial-value problem for the wave equation in three space dimensions can be obtained from the solution for a spherical wave. This result can then be used to obtain the solution in two space dimensions.

Spherical waves

The wave equation is unchanged under rotations of the spatial coordinates, and therefore one may expect to find solutions that depend only on the radial distance from a given point. Such solutions must satisfy

 u_{tt} - c^2 \left( u_{rr} + \frac{2}{r} u_r \right) =0. \,

This equation may be rewritten as

 (ru)_{tt} -c^2 (ru)_{rr}=0; \,

the quantity ru satisfies the one-dimensional wave equation. Therefore there are solutions in the form

 u(t,r) = \frac{1}{r} F(r-ct) + \frac{1}{r} G(r+ct), \,

where F and G are arbitrary functions. Each term may be interpreted as a spherical wave that expands or contracts with velocity c. Such waves are generated by a point source, and they make possible sharp signals whose form is altered only by a decrease in amplitude as r increases (see an illustration of a spherical wave on the top right). Such waves exist only in cases of space with odd dimensions.

Solution of a general initial-value problem

The wave equation is linear in u and it is left unaltered by translations in space and time. Therefore we can generate a great variety of solutions by translating and summing spherical waves. Let φ(ξ,η,ζ) be an arbitrary function of three independent variables, and let the spherical wave form F be a delta-function: that is, let F be a weak limit of continuous functions whose integral is unity, but whose support (the region where the function is non-zero) shrinks to the origin. Let a family of spherical waves have center at (ξ,η,ζ), and let r be the radial distance from that point. Thus

 r^2 = (x-\xi)^2 + (y-\eta)^2 + (z-\zeta)^2. \,

If u is a superposition of such waves with weighting function φ, then

 u(t,x,y,z) = \frac{1}{4\pi c} \iiint \varphi(\xi,\eta,\zeta) \frac{\delta(r-ct)}{r} d\xi\,d\eta\,d\zeta; \,

the denominator 4πc is a convenience.

From the definition of the delta-function, u may also be written as

 u(t,x,y,z) = \frac{t}{4\pi} \iint_S \varphi(x +ct\alpha, y +ct\beta, z+ct\gamma) d\omega, \,

where α, β, and γ are coordinates on the unit sphere S, and ω is the area element on S. This result has the interpretation that u(t,x) is t times the mean value of φ on a sphere of radius ct centered at x:

 u(t,x,y,z) = t M_{ct}[\phi]. \,

It follows that

 u(0,x,y,z) = 0, \quad u_t(0,x,y,z) = \phi(x,y,z). \,

The mean value is an even function of t, and hence if

 v(t,x,y,z) = \frac{\part}{\part t} \left( t M_{ct}[\psi] \right), \,

then

 v(0,x,y,z) = \psi(x,y,z), \quad v_t(0,x,y,z) = 0. \,

These formulas provide the solution for the initial-value problem for the wave equation. They show that the solution at a given point P, given (t,x,y,z) depends only on the data on the sphere of radius ct that is intersected by the light cone drawn backwards from P. It does not depend upon data on the interior of this sphere. Thus the interior of the sphere is a lacuna for the solution. This phenomenon is called Huygens' principle. It is true for odd numbers of space dimension, where for one dimension the integration is performed over the boundary of an interval w.r.t. the Dirac measure. It is not satisfied in even space dimensions. The phenomenon of lacunas has been extensively investigated in Atiyah, Bott and Gårding (1970, 1973).

Scalar wave equation in two space dimensions

In two space dimensions, the wave equation is

 u_{tt} = c^2 \left( u_{xx} + u_{yy} \right). \,

We can use the three-dimensional theory to solve this problem if we regard u as a function in three dimensions that is independent of the third dimension. If

 u(0,x,y)=0, \quad u_t(0,x,y) = \phi(x,y), \,

then the three-dimensional solution formula becomes

 u(t,x,y) = tM_{ct}[\phi] = \frac{t}{4\pi} \iint_S \phi(x + ct\alpha,\, y + ct\beta) d\omega,\,

where α and β are the first two coordinates on the unit sphere, and dω is the area element on the sphere. This integral may be rewritten as an integral over the disc D with center (x,y) and radius ct:

 u(t,x,y) = \frac{1}{2\pi c} \iint_D \frac{\phi(x+\xi, y +\eta)}{\sqrt{(ct)^2 - \xi^2 - \eta^2}} d\xi\,d\eta. \,

It is apparent that the solution at (t,x,y) depends not only on the data on the light cone where

 (x -\xi)^2 + (y - \eta)^2 = c^2 t^2, \,

but also on data that are interior to that cone.

Problems with boundaries

One space dimension

A flexible string that is stretched between two points x=0 and x=L satisfies the wave equation for t>0 and 0 < x < L. On the boundary points, u may satisfy a variety of boundary conditions. A general form that is appropriate for applications is

 -u_x(t,0) + a u(t,0) = 0, \,
 u_x(t,L) + b u(t,L) = 0,\,

where a and b are non-negative. The case where u is required to vanish at an endpoint is the limit of this condition when the respective a or b approaches infinity. The method of separation of variables consists in looking for solutions of this problem in the special form

 u(t,x) = T(t) v(x).\,

A consequence is that

 \frac{T''}{c^2T} = \frac{v''}{v} = -\lambda. \,

The eigenvalue λ must be determined so that there is a non-trivial solution of the boundary-value problem

 v'' + \lambda v=0, \,
 -v'(0) + a v(0) = 0, \quad v'(L) + b v(L)=0.\,

This is a special case of the general problem of Sturm-Liouville theory. If a and b are positive, the eigenvalues are all positive, and the solutions are trigonometric functions. A solution that satisfies square-integrable initial conditions for u and ut can be obtained from expansion of these functions in the appropriate trigonometric series.

Several space dimensions

A solution of the wave equation in two dimensions with a zero-displacement boundary condition along the entire outer edge.

The one-dimensional initial-boundary value theory may be extended to an arbitrary number of space dimensions. Consider a domain D in m-dimensional x space, with boundary B. Then the wave equation is to be satisfied if x is in D and t > 0. On the boundary of D, the solution u shall satisfy

 \frac{\part u}{\part n} + a u =0, \,

where n is the unit outward normal to B, and a is a non-negative function defined on B. The case where u vanishes on B is a limiting case for a approaching infinity. The initial conditions are

 u(0,x) = f(x), \quad u_t=g(x), \,

where f and g are defined in D. This problem may be solved by expanding f and g in the eigenfunctions of the Laplacian in D, which satisfy the boundary conditions. Thus the eigenfunction v satisfies

 \nabla \cdot \nabla v + \lambda v = 0, \,

in D, and

 \frac{\part v}{\part n} + a v =0, \,

on B.

In the case of two space dimensions, the eigenfunctions may be interpreted as the modes of vibration of a drumhead stretched over the boundary B. If B is a circle, then these eigenfunctions have an angular component that is a trigonometric function of the polar angle θ, multiplied by a Bessel function (of integer order) of the radial component. Further details are in Helmholtz equation.

If the boundary is a sphere in three space dimensions, the angular components of the eigenfunctions are spherical harmonics, and the radial components are Bessel functions of half-integer order.

Inhomogenous wave equation in one dimension

The inhomogenous wave equation in one dimension is the following:

c^2 u_{x x}(x,t) - u_{t t}(x,t) = s(x,t) \,

with initial conditions given by

u(x,0)=f(x) \,
u_t(x,0)=g(x) \,

The function s(x,t) is often called the source function because in practice it describes the effects of the sources of waves on the medium carrying them. Physical examples of source functions include the force driving a wave on a string, or the charge or current density in the Lorenz gauge of electromagnetism.

One method to solve the initial value problem (with the initial values as posed above) is to take advantage of the property of the wave equation that its solutions obey causality. That is, for any point (xi,ti), the value of \scriptstyle u(x_i,t_i) depends only on the values of \scriptstyle f(x_i + c t_i) and \scriptstyle f(x_i - c t_i) and the values of the function \scriptstyle g(x) between \scriptstyle (x_i - c t_i) and \scriptstyle (x_i + c t_i). This can be seen in d'Alembert's formula, stated above, where these quantities are the only ones that show up in it. Physically, if the maximum propagation speed is \scriptstyle c, then no part of the wave that can't propagate to a given point by a given time can affect the amplitude at the same point and time.

In terms of finding a solution, this causality property means that for any given point on the line being considered, the only area that needs to be considered is the area encompassing all the points that could causally affect the point being considered. Denote the area that casually affects point \scriptstyle (x_i,t_i) as \scriptstyle R_C. Suppose we integrate the nonhomogenous wave equation over this region.

\iint \limits_{R_C} \left ( c^2 u_{x x}(x,t) - u_{t t}(x,t) \right ) dx dt = \iint \limits_{R_C} s(x,t) dx dt.

To simplify this greatly, we can use Green's theorem to simplify the left side to get the following:

\int_{ L_0 + L_1 + L_2 } \left ( - c^2 u_x(x,t) dt - u_t(x,t) dx \right ) = \iint \limits_{R_C} s(x,t) dx dt.

The left side is now the sum of three line integrals along the bounds of the causality region. These turn out to be fairly easy to compute

\int^{x_i + c t_i}_{x_i - c t_i} - u_t(x,0) dx = - \int^{x_i + c t_i}_{x_i - c t_i} g(x) dx.

In the above, the term to be integrated with respect to time disappears because the time interval involved is zero, thus dt = 0.

For the other two sides of the region, it is worth noting that \scriptstyle x \pm c t is a constant, namingly \scriptstyle x_i \pm c t_i, where the sign is chosen appropriately. Using this, we can get the relation \scriptstyle dx \pm c dt = 0, again choosing the right sign:

\int_{L_1} \left ( - c^2 u_x(x,t) dt - u_t(x,t) dx \right ) \,
= \int_{L_1} \left ( c u_x(x,t) dx + c u_t(x,t) dt \right)\,
= c \int_{L_1} d u(x,t) = c u(x_i,t_i) - c f(x_i + c t_i).\,

And similarly for the final boundary segment:

\int_{L_2} \left ( - c^2 u_x(x,t) dt - u_t(x,t) dx \right )
= - \int_{L_2} \left ( c u_x(x,t) dx + c u_t(x,t) dt \right )
= - c \int_{L_2} d u(x,t) = - \left ( c f(x_i - c t_i) - c u(x_i,t_i) \right )
= c u(x_i,t_i) - c f(x_i - c t_i).\,

Adding the three results together and putting them back in the original integral:

- \int^{x_i + c t_i}_{x_i - c t_i} g(x) dx + c u(x_i,t_i) - c f(x_i + c t_i) + c u(x_i,t_i) - c f(x_i - c t_i) = \iint \limits_{R_C} s(x,t) dx dt
2 c u(x_i,t_i) - \int^{x_i + c t_i}_{x_i - c t_i} g(x) dx - c f(x_i + c t_i) - c f(x_i - c t_i) = \iint \limits_{R_C} s(x,t) dx dt
2 c u(x_i,t_i) = \int^{x_i + c t_i}_{x_i - c t_i} g(x) dx + c f(x_i + c t_i) + c f(x_i - c t_i) + \iint \limits_{R_C} s(x,t) dx dt
u(x_i,t_i) = \frac{f(x_i + c t_i) + f(x_i - c t_i)}{2} + \frac{1}{2 c}\int^{x_i + c t_i}_{x_i - c t_i} g(x) dx + \frac{1}{2 c}\int^{t_i}_0 \int^{x_i + c \left ( t_i - t \right )}_{x_i - c \left ( t_i - t \right )} s(x,t) dx dt. \,

In the last equation of the sequence, the bounds of the integral over the source function have been made explicit. Looking at this solution, which is valid for all choices (xi,ti) compatible with the wave equation, it is clear that the first two terms are simply d'Alembert's formula, as stated above as the solution of the homogenous wave equation in one dimension. The difference is in the third term, the integral over the source.

Other coordinate systems

In three dimensions, the wave equation, when written in elliptic cylindrical coordinates, may be solved by separation of variables, leading to the Mathieu differential equation.

See also

References

  • M. F. Atiyah, R. Bott, L. Garding, "Lacunas for hyperbolic differential operators with constant coefficients I", Acta Math., 124 (1970), 109–189.
  • M.F. Atiyah, R. Bott, and L. Garding, "Lacunas for hyperbolic differential operators with constant coefficients II", Acta Math., 131 (1973), 145–206.
  • R. Courant, D. Hilbert, Methods of Mathematical Physics, vol II. Interscience (Wiley) New York, 1962.
  • "Linear Wave Equations", EqWorld: The World of Mathematical Equations.
  • "Nonlinear Wave Equations", EqWorld: The World of Mathematical Equations.
  • William C. Lane, "MISN-0-201 The Wave Equation and Its Solutions", Project PHYSNET.
  • Relativistic wave equations with fractional derivatives and pseudodifferential operators, by Petr Zavada, Journal of Applied Mathematics, vol. 2, no. 4, pp. 163–197, 2002. doi:10.1155/S1110757X02110102 (available online or as the arXiv preprint)

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