Wave equation

 ( Equations Of Physics ) 

The wave equation is a second-order linear partial differential equation for the description of or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics.

This article focuses on waves in classical physics. Quantum physics uses an operator-based wave equation often as a relativistic wave equation.

The scalar wave equation is where

• $c$ is a fixed non-negative real number coefficient representing the propagation speed of the wave
• $u$ is a scalar field representing the displacement or, more generally, the conserved quantity (e.g. pressure or density)
• $x,\; y,$ and $z$ are the three spatial coordinates and $t$ being the time coordinate.

The equation states that, at any given point, the second derivative of $u$ with respect to time is proportional to the sum of the second derivatives of $u$ with respect to space, with the constant of proportionality being the square of the speed of the wave.

Using notations from vector calculus, the wave equation can be written compactly as $$u\_\{tt\}\; =\; c^2\; \backslash Delta\; u,$$ or $$\backslash Box\; u\; =\; 0,$$ where the double subscript denotes the second-order partial derivative with respect to time, $\backslash Delta$ is the Laplace operator and $\backslash Box$ the d'Alembert operator, defined as: $$u\_\{tt\}\; =\; \backslash frac\{\backslash partial^2\; u\}\{\backslash partial\; t^2\},\; \backslash qquad\; \backslash Delta\; =\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; x^2\}\; +\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; y^2\}\; +\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; z^2\},\; \backslash qquad\; \backslash Box\; =\; \backslash frac\{1\}\{c^2\}\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; t^2\}\; -\; \backslash Delta.$$

A solution to this (two-way) wave equation can be quite complicated. Still, it can be analyzed as a linear combination of simple solutions that are sinusoidal with various directions of propagation and wavelengths but all with the same propagation speed $c$. This analysis is possible because the wave equation is linear and homogeneous, so that any multiple of a solution is also a solution, and the sum of any two solutions is again a solution. This property is called the superposition principle in physics.

The wave equation alone does not specify a physical solution; a unique solution is usually obtained by setting a problem with further conditions, such as initial conditions, which prescribe the amplitude and phase of the wave. Another important class of problems occurs in enclosed spaces specified by boundary conditions, for which the solutions represent standing waves, or harmonics, analogous to the harmonics of musical instruments.

Another physical setting for derivation of the wave equation in one space dimension uses Hooke's law. In the theory of elasticity, Hooke's law is an approximation for certain materials, stating that the amount by which a material body is deformed (the strain) is linearly related to the force causing the deformation (the stress).

Here the dependent variable $u(x)$ measures the distance from the equilibrium of the mass situated at , so that $u(x)$ essentially measures the magnitude of a disturbance (i.e. strain) that is traveling in an elastic material. The resulting force exerted on the mass $m$ at the location $x+h$ is: $$\backslash begin\{align\}$$

F_\text{Hooke}  &= F_{x+2h} - F_x = k [u(x + 2h, t) - u(x + h, t)] - k[u(x + h,t) - u(x, t)].
     

By equating the latter equation with $$\backslash begin\{align\}$$

F_\text{Newton} &= m \, a(t) = m \, \frac{\partial^2}{\partial t^2} u(x + h, t),
     

the equation of motion for the weight at the location is obtained: $$\backslash frac\{\backslash partial^2\}\{\backslash partial\; t^2\}\; u(x\; +\; h,\; t)\; =\; \backslash frac\{k\}\{m\}\; u(x.$$ If the array of weights consists of $N$ weights spaced evenly over the length $L=Nh$ of total mass , and the total stiffness of the array , we can write the above equation as $$\backslash frac\{\backslash partial^2\}\{\backslash partial\; t^2\}\; u(x\; +\; h,\; t)\; =\; \backslash frac\{KL^2\}\{M\}\; \backslash frac\{u(x\}\{h^2\}.$$

Taking the limit $N\; \backslash rightarrow\; \backslash infty,\; h\; \backslash rightarrow\; 0$ and assuming smoothness, one gets $$\backslash frac\{\backslash partial^2\; u(x,\; t)\}\{\backslash partial\; t^2\}\; =\; \backslash frac\{KL^2\}\{M\}\; \backslash frac\{\backslash partial^2\; u(x,\; t)\}\{\backslash partial\; x^2\},$$ which is from the definition of a second derivative. $KL^2/M$ is the square of the propagation speed in this particular case.

$AL$ is equal to the volume of the bar, and therefore $$\backslash frac\{AL\}\{M\}\; =\; \backslash frac\{1\}\{\backslash rho\},$$ where $\backslash rho$ is the density of the material. The wave equation reduces to $$\backslash frac\{\backslash partial^2\; u(x,\; t)\}\{\backslash partial\; t^2\}\; =\; \backslash frac\{E\}\{\backslash rho\}\; \backslash frac\{\backslash partial^2\; u(x,\; t)\}\{\backslash partial\; x^2\}.$$

The speed of a stress wave in a bar is therefore $\backslash sqrt\{E/\backslash rho\}$.

In other words, the solution is the sum 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 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.D'Alembert (1747) "Recherches sur la courbe que forme une corde tenduë mise en vibration" (Researches on the curve that a tense cord forms when set into vibration), Histoire de l'académie royale des sciences et belles lettres de Berlin, vol. 3, p. 214–219.

• See also: D'Alembert (1747) "Suite des recherches sur la courbe que forme une corde tenduë mise en vibration" (Further researches on the curve that a tense cord forms when set into vibration), Histoire de l'académie royale des sciences et belles lettres de Berlin, vol. 3, p. 220–249.
• See also: D'Alembert (1750) "Addition au mémoire sur la courbe que forme une corde tenduë mise en vibration," Histoire de l'académie royale des sciences et belles lettres de Berlin, vol. 6, p. 355–360.

Another way to arrive at this result is to factor the wave equation using two first-order $$\backslash left\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash left\backslash frac\{\backslash partial\}\{\backslash partial\; u\; =\; 0.$$ Then, for our original equation, we can define $$v\; \backslash equiv\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; t\}\; +\; c\backslash frac\{\backslash partial\; u\}\{\backslash partial\; x\},$$ and find that we must have $$\backslash frac\{\backslash partial\; v\}\{\backslash partial\; t\}\; -\; c\backslash frac\{\backslash partial\; v\}\{\backslash partial\; x\}\; =\; 0.$$

This advection equation can be solved by interpreting it as telling us that the directional derivative of $v$ in the $(1,\; -c)$ direction is 0. This means that the value of $v$ is constant on characteristic lines of the form , and thus that $v$ must depend only on , that is, have the form . Then, to solve the first (inhomogenous) equation relating $v$ to , we can note that its homogenous solution must be a function of the form , by logic similar to the above. Guessing a particular solution of the form , we find that

$$\backslash left\backslash frac\{\backslash partial\}\{\backslash partial\; G(x\; +\; ct)\; =\; H(x\; +\; ct).$$

Expanding out the left side, rearranging terms, then using the change of variables simplifies the equation to

$$G\text{'}(s)\; =\; \backslash frac\{H(s)\}\{2c\}.$$

This means we can find a particular solution of the desired form by integration. Thus, we have again shown that obeys .

For an initial-value problem, the arbitrary functions and 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: $$u(x,\; t)\; =\; \backslash frac\{f(x\; -\; ct)\; +\; f(x\; +\; ct)\}\{2\}\; +\; \backslash frac\{1\}\{2c\}\; \backslash int\_\{x-ct\}^\{x+ct\}\; g(s)\; \backslash ,\; ds.$$

In the classical sense, if , and , then . However, the waveforms and 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, and so it will adhere to the superposition principle. This means that the net displacement caused by two or more waves is the sum of the displacements which would have been caused by each wave individually. In addition, the behavior of a wave can be analyzed by breaking up the wave into components, e.g. the Fourier transform breaks up a wave into sinusoidal components.

This produces an ordinary differential equation for the spatial part : $$\backslash frac\{\backslash partial^2\; u\_\backslash omega\; \}\{\backslash partial\; t^2\}\; =\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; t^2\}\; \backslash left(e^\{-i\backslash omega\; t\}\; f(x)\backslash right)\; =\; -\backslash omega^2\; e^\{-i\backslash omega\; t\}\; f(x)\; =\; c^2\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; x^2\}\; \backslash left(e^\{-i\backslash omega\; t\}\; f(x)\backslash right).$$

Therefore, $$\backslash frac\{d^2\}\{dx^2\}f(x)\; =\; -\backslash left(\backslash frac\{\backslash omega\}\{c\}\backslash right)^2\; f(x),$$ which is precisely an eigenvalue equation for , hence the name eigenmode. Known as the Helmholtz equation, it has the well-known plane-wave solutions $$f(x)\; =\; A\; e^\{\backslash pm\; ikx\},$$ with wave number .

The total wave function for this eigenmode is then the linear combination $$u\_\backslash omega(x,\; t)\; =\; e^\{-i\backslash omega\; t\}\; \backslash left(A\; e^\{-ikx\}\; +\; B\; e^\{ikx\}\backslash right)\; =\; A\; e^\{-i\; (kx\; +\; \backslash omega\; t)\}\; +\; B\; e^\{i\; (kx\; -\; \backslash omega\; t)\},$$ where complex numbers , depend in general on any initial and boundary conditions of the problem.

Eigenmodes are useful in constructing a full solution to the wave equation, because each of them evolves in time trivially with the phase factor $e^\{-i\backslash omega\; t\},$ so that a full solution can be decomposed into an eigenmode expansion: $$u(x,\; t)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; s(\backslash omega)\; u\_\backslash omega(x,\; t)\; \backslash ,\; d\backslash omega,$$ or in terms of the plane waves,

&= \int_{-\infty}^\infty s_+(\omega) e^{-ik(x+ct)} \, d\omega + \int_{-\infty}^\infty s_-(\omega) e^{ik (x-ct)} \, d\omega \\
&= F(x - ct) + G(x + ct),
     

1. the tension force $\backslash operatorname\{div\}\; \backslash mathbf\{T\}\; =\; \backslash nabla\backslash cdot(E\; \backslash nabla\; \backslash mathbf\{u\})\; =\; E\; \backslash Delta\backslash mathbf\{u\}$ due to deflection $\backslash mathbf\{u\}$, and
2. the inertial force $\backslash rho\; \backslash partial^2\backslash mathbf\{u\}/\backslash partial\; t^2$ caused by the local acceleration $\backslash partial^2\backslash mathbf\{u\}\; /\; \backslash partial\; t^2$

By merging density $\backslash rho$ and elasticity module $E,$ the sound velocity $c\; =\; \backslash sqrt\{E/\backslash rho\}$ results (material law). After insertion, follows the well-known governing wave equation for a homogeneous medium: Text was copied from this source, which is available under a Creative Commons Attribution 4.0 International License. $$\backslash frac\{\backslash partial^2\; \backslash mathbf\{u\}\}\{\backslash partial\; t^2\}\; -\; c^2\; \backslash Delta\; \backslash mathbf\{u\}\; =\; \backslash boldsymbol\{0\}.$$ (Note: Instead of vectorial $\backslash mathbf\{u\}(\backslash mathbf\{x\},\; t),$ only scalar $u(x,\; t)$ can be used, i.e. waves are travelling only along the $x$ axis, and the scalar wave equation follows as $\backslash frac\{\backslash partial^2\; u\}\{\backslash partial\; t^2\}\; -\; c^2\; \backslash frac\{\backslash partial^2\; u\}\{\backslash partial\; x^2\}\; =\; 0$.)

The above vectorial partial differential equation of the 2nd order delivers two mutually independent solutions. From the quadratic velocity term $c^2\; =\; (+c)^2\; =\; (-c)^2$ can be seen that there are two waves travelling in opposite directions $+c$ and $-c$ are possible, hence results the designation "two-way wave equation". It can be shown for plane longitudinal wave propagation that the synthesis of two one-way wave equations leads to a general two-way wave equation. For $\backslash nabla\backslash mathbf\{c\}\; =\; \backslash mathbf\{0\},$ special two-wave equation with the d'Alembert operator results: $$\backslash left(\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; -\; \backslash mathbf\{c\}\; \backslash cdot\; \backslash nabla\backslash right)\backslash left(\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; +\; \backslash mathbf\{c\}\; \backslash cdot\; \backslash nabla\; \backslash right)\; \backslash mathbf\{u\}\; =$$

\left(\frac{\partial^2}{\partial t^2} + (\mathbf{c} \cdot \nabla) \mathbf{c} \cdot \nabla\right) \mathbf{u} =
\left(\frac{\partial^2}{\partial t^2} + (\mathbf{c} \cdot \nabla)^2\right) \mathbf{u} = \mathbf{0}.
     

\left(\nabla^2 - \frac{1}{c^2} \frac{\partial^2 }{\partial t^2}\right) \Psi(\mathbf{r}, t) = 0,
     

\left(\frac{\partial^2}{\partial r^2} + \frac{2}{r} \frac{\partial}{\partial r} - \frac{1}{c^2} \frac{\partial^2}{\partial t^2}\right) u(r, t) = 0.
     

This equation can be rewritten as $$\backslash frac\{\backslash partial^2(ru)\}\{\backslash partial\; t^2\}\; -\; c^2\; \backslash frac\{\backslash partial^2(ru)\}\{\backslash partial\; r^2\}\; =\; 0,$$ where the quantity satisfies the one-dimensional wave equation. Therefore, there are solutions in the form$$u(r,\; t)\; =\; \backslash frac\{1\}\{r\}\; F(r\; -\; ct)\; +\; \backslash frac\{1\}\{r\}\; G(r\; +\; ct),$$ where and are general solutions to the one-dimensional wave equation and can be interpreted as respectively an outgoing and incoming spherical waves. The outgoing wave can be generated by a point source, and they make possible sharp signals whose form is altered only by a decrease in amplitude as increases (see an illustration of a spherical wave on the top right). Such waves exist only in cases of space with odd dimensions.

For physical examples of solutions to the 3D wave equation that possess angular dependence, see dipole radiation.

From this we can observe that the peak intensity of the spherical-wave oscillation, characterized as the squared wave amplitude $$I\; =\; |u(r,\; t)|^2\; =\; \backslash frac$$

 1+D
     

 u = (\partial_t G) \ast u + G \ast \partial_t u
     

 u(t, x) = \int  (\partial_t G)(t, x-x') u(0, x') dx' +  \int  G(t, x-x') (\partial_t u)(0, x') dx'.
     

 \iint_{t' < t} G(t-t', x-x') s(t', x')dt' dx'.
     

 \hat G (\omega)=  \frac{1}{-\omega_0^2 + \omega_1^2 + \cdots + \omega_D^2},
     

The integral can be solved by analytically continuing the Poisson kernel, giving$$$$

 G(t, x) = \lim _{\epsilon \rightarrow 0^{+}} \frac{C_D}{D-1}
     

 C_D=\pi^{-(D+1) / 2} \Gamma((D+1) / 2)
     

  s(t, x_{1:D}, x_{D+1:D+n}) = s(t, x_{1:D}), \quad u(t, x_{1:D}, x_{D+1:D+n}) = u(t, x_{1:D}).
     

  G_D(t, x_{1:D}) = \int_{\R^n} G_{D+n}(t, x_{1:D}, x_{D+1:D+n}) d^n x_{D+1:D+n}.
     

  G_D(t, r) = \int_{\R^2} G_{D+2}(t, \sqrt{r^2 + y^2 + z^2}) dydz.
     

  G_D(t, r) = 2\pi \int_0^\infty G_{D+2}(t, \sqrt{r^2 + q^2}) qdq = 2\pi \int_r^\infty G_{D+2}(t, q') q'dq',
     

  G_{D+2}(t, r) = -\frac{1}{2\pi r} \partial_r G_D(t, r).
     

The dimension can be raised to give the $D=3$ case$$G\_3(t,\; r)\; =\; \backslash frac\{\backslash delta(t-r)\}\{4\backslash pi\; r\}$$and similarly for the backward solution. This can be integrated down by one dimension to give the $D=2$ case$$G\_2(t,\; r)\; =\; \backslash int\_\backslash R\; \backslash frac\{\backslash delta(t\; -\; \backslash sqrt\{r^2\; +\; z^2\})\}\{4\backslash pi\; \backslash sqrt\{r^2\; +\; z^2\}\}\; dz$$

In odd dimensions, the forward solution is nonzero only at $t\; =\; r$. As the dimensions increase, the shape of wavefront becomes increasingly complex, involving higher derivatives of the Dirac delta function. For example,where $\backslash tau\; =\; t-\; r$, and the wave speed $c$ is restored.

In even dimensions, the forward solution is nonzero in $r\; \backslash leq\; t$, the entire region behind the wavefront becomes nonzero, called a wake. The wake has equation:$$G\_\{D\}\; (t,\; x\; )\; =\; (-1)^\{1+D\; /\; 2\}\; \backslash frac\{1\}\{(2\; \backslash pi)^\{D\; /\; 2\}\}\; \backslash frac\{1\}\{c^D\}\; \backslash frac\{\backslash theta(t-r\; /\; c)\}\{\backslash left(t^2-r^2\; /\; c^2\backslash right)^\{(D-1)\; /\; 2\}\}$$The wavefront itself also involves increasingly higher derivatives of the Dirac delta function.

This means that a general Huygens' principle – the wave displacement at a point $(t,\; x)$ in spacetime depends only on the state at points on characteristic rays passing $(t,\; x)$ – only holds in odd dimensions. A physical interpretation is that signals transmitted by waves remain undistorted in odd dimensions, but distorted in even dimensions.

Hadamard's conjecture states that this generalized Huygens' principle still holds in all odd dimensions even when the coefficients in the wave equation are no longer constant. It is not strictly correct, but it is correct for certain families of coefficients

Consider the component of the incident wave with an angular frequency of , which has the waveform $$u^\backslash text\{inc\}(x,\; t)\; =\; Ae^\{i(k\_1\; x\; -\; \backslash omega\; t)\},\backslash quad\; A\; \backslash in\; \backslash C.$$ At , the incident reaches the boundary between the two media at . Therefore, the corresponding reflected wave and the transmitted wave will have the waveforms $$u^\backslash text\{refl\}(x,\; t)\; =\; Be^\{i(-k\_1\; x\; -\; \backslash omega\; t)\},\; \backslash quad$$

u^\text{trans}(x, t) = Ce^{i(k_2 x - \omega t)}, \quad
B, C \in \C.
     

u_x^\text{inc}(0, t) + u_x^\text{ref}(0, t) = u_x^\text{trans}(0, t).
     

A - B = \frac{k_2}{k_1} C = \frac{c_1}{c_2} C,
     

\frac{C}{A} = \frac{2c_2}{c_2 + c_1}.
     

The limiting case of corresponds to a "fixed end" that does not move, whereas the limiting case of corresponds to a "free end".

$$\backslash begin\{align\}$$

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

where and are non-negative. The case where is required to vanish at an endpoint (i.e. "fixed end") is the limit of this condition when the respective or 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 $$\backslash frac\{T$$}{c^2 T} = \frac{v}{v} = -\lambda.

The eigenvalue must be determined so that there is a non-trivial solution of the boundary-value problem $$\backslash begin\{align\}$$

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

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

$$\backslash frac\{\backslash partial\; u\}\{\backslash partial\; n\}\; +\; a\; u\; =\; 0,$$

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

$$u(0,\; x)\; =\; f(x),\; \backslash quad\; u\_t(0,\; x)\; =\; g(x),$$

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

$$\backslash nabla\; \backslash cdot\; \backslash nabla\; v\; +\; \backslash lambda\; v\; =\; 0$$

in , and

$$\backslash frac\{\backslash partial\; v\}\{\backslash partial\; n\}\; +\; a\; v\; =\; 0$$

on .

In the case of two space dimensions, the eigenfunctions may be interpreted as the modes of vibration of a drumhead stretched over the boundary . If is a circle, then these eigenfunctions have an angular component that is a trigonometric function of the polar angle , 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 of half-integer order.

The function 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 a special property of the wave equation in an odd number of space dimensions, namely that its solutions respect causality. That is, for any point , the value of depends only on the values of and and the values of the function between and . 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 , then no part of the wave that cannot 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 causally affects point as . Suppose we integrate the inhomogeneous wave equation over this region: $$$$

\iint_{R_C} \big(c^2 u_{xx}(x, t) - u_{tt}(x, t)\big) \, dx \, dt = \iint_{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} \big({-}c^2 u_x(x, t) \, dt - u_t(x, t) \, dx\big) = \iint_{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 .

For the other two sides of the region, it is worth noting that is a constant, namely , where the sign is chosen appropriately. Using this, we can get the relation , again choosing the right sign: $$\backslash begin\{align\}$$

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

And similarly for the final boundary segment: $$\backslash begin\{align\}$$

\int_{L_2} \big({-}c^2 u_x(x, t) \, dt - u_t(x, t) \, dx\big) &= -\int_{L_2} \big(c u_x(x, t) \, dx + c u_t(x, t) \, dt \big) \\
&= -c \int_{L_2} \, du(x, t) \\
&= 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 gives $$\backslash begin\{align\}$$

\iint_{R_C} s(x, t) \, dx \, dt &= - \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) \\
&= 2 c u(x_i, t_i) - c f(x_i + c t_i) - c f(x_i - c t_i) - \int^{x_i + c t_i}_{x_i - c t_i} g(x) \, dx.
     

Solving for , we arrive at $$$$

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(t_i - t)}_{x_i - c(t_i - t)} 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 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 homogeneous wave equation in one dimension. The difference is in the third term, the integral over the source.

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

and are the so-called Lamé parameters describing the elastic properties of the medium,

is the density,

is the source function (driving force),

is the displacement vector.

By using , the elastic wave equation can be rewritten into the more common form of the Navier–Cauchy equation.

Note that in the elastic wave equation, both force and displacement are vector quantities. Thus, this equation is sometimes known as the vector wave equation. As an aid to understanding, the reader will observe that if and are set to zero, this becomes (effectively) Maxwell's equation for the propagation of the electric field , which has only transverse waves.

$$\backslash omega\; =\; \backslash omega(\backslash mathbf\{k\}),$$

where is the angular frequency, and is the wavevector describing plane-wave solutions. For light waves, the dispersion relation is , but in general, the constant speed gets replaced by a variable phase velocity:

$$v\_\backslash text\{p\}\; =\; \backslash frac\{\backslash omega(k)\}\{k\}.$$

• Acoustic attenuation
• Acoustic wave equation
• Bateman transform
• Electromagnetic wave equation
• Helmholtz equation
• Inhomogeneous electromagnetic wave equation
• Laplace operator
• Mathematics of oscillation
• Maxwell's equations
• Schrödinger equation
• Standing wave
• Vibrations of a circular membrane
• Wheeler–Feynman absorber theory

• Flint, H.T. (1929) "Wave Mechanics" Methuen & Co. Ltd. London.
• Richard Courant, David Hilbert, Methods of Mathematical Physics, vol II. Interscience (Wiley) New York, 1962.
• Lawrence C. Evans (2026). 9780821849743, American Mathematical Soc.. ISBN 9780821849743
• " 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.

• Nonlinear Wave Equations by Stephen Wolfram and Rob Knapp, Nonlinear Wave Equation Explorer by Wolfram Demonstrations Project.
• Mathematical aspects of wave equations are discussed on the Dispersive PDE Wiki .
• Graham W Griffiths and William E. Schiesser (2009). Linear and nonlinear waves. Scholarpedia, 4(7):4308. doi:10.4249/scholarpedia.4308