Wave equation

The wave equation is a second-order linear partial differential equation for the description of waves 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 wave equation is a hyperbolic partial differential equation describing waves, including traveling and standing waves; the latter can be considered as linear superpositions of waves traveling in opposite directions. This article mostly focuses on the scalar wave equation describing waves in scalars by scalar functions \(u = u (x, y, z, t)\) of a time variable \(t\) (a variable representing time) and one or more spatial variables \(x, y, z\) (variables representing a position in a space under discussion). At the same time, there are vector wave equations describing waves in vectors such as waves for an electrical field, magnetic field, and magnetic vector potential and elastic waves. By comparison with vector wave equations, the scalar wave equation can be seen as a special case of the vector wave equations; in the Cartesian coordinate system, the scalar wave equation is the equation to be satisfied by each component (for each coordinate axis, such as the \(x\) component for the x axis) of a vector wave without sources of waves in the considered domain (i.e., space and time). For example, in the Cartesian coordinate system, for \((E_x, E_y, E_z)\) as the representation of an electric vector field wave \(\vec{E}\) in the absence of wave sources, each coordinate axis component \(E_i, i=x,y,z,\) must satisfy the scalar wave equation. Other scalar wave equation solutions u are for physical quantities in scalars such as pressure in a liquid or gas, or the displacement along some specific direction of particles of a vibrating solid away from their resting (equilibrium) positions.

The scalar wave equation is

\(\frac{\partial^2 u}{\partial t^2} = c^2 \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right)\)

where

• \(c\) is a fixed non-negative real 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 \[u_{tt} = c^2 \Delta u,\] or \[\Box u = 0,\] where the double subscript denotes the second-order partial derivative with respect to time, \(\Delta\) is the Laplace operator and \(\Box\) the d'Alembert operator, defined as: \[u_{tt} = \frac{\partial^2 u}{\partial t^2}, \qquad \Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}, \qquad \Box = \frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \Delta.\]

Condensed: the full section is in Wikipedia.

The wave equation in one spatial dimension can be written as follows: \[\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}.\]This equation is typically described as having only one spatial dimension \(x\), because the only other independent variable is the time \(t\).

The wave equation in one space dimension can be derived in a variety of different physical settings. Most famously, it can be derived for the case of a string vibrating in a two-dimensional plane, with each of its elements being pulled in opposite directions by the force of tension.

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).

The vectorial wave equation (from which the scalar wave equation can be directly derived) can be obtained by applying a force equilibrium to an infinitesimal volume element. If the medium has a modulus of elasticity \(E\) that is homogeneous (i.e. independent of \(\mathbf{x}\)) within the volume element, then its stress tensor is given by \(\mathbf{T} = E \nabla \mathbf{u}\), for a vectorial elastic deflection \(\mathbf{u}(\mathbf{x}, t)\). The local equilibrium of:

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

can be written as \(\rho \frac{\partial^2 \mathbf{u}}{\partial t^2} - E \Delta \mathbf{u} = \mathbf{0}.\)

By merging density \(\rho\) and elasticity module \(E,\) the sound velocity \(c = \sqrt{E/\rho}\) results (material law). After insertion, follows the well-known governing wave equation for a homogeneous medium: \[\frac{\partial^2 \mathbf{u}}{\partial t^2} - c^2 \Delta \mathbf{u} = \boldsymbol{0}.\] (Note: Instead of vectorial \(\mathbf{u}(\mathbf{x}, t),\) 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 \(\frac{\partial^2 u}{\partial t^2} - c^2 \frac{\partial^2 u}{\partial x^2} = 0\).)

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 \(\nabla\mathbf{c} = \mathbf{0},\) special two-wave equation with the d'Alembert operator results: \[\left(\frac{\partial}{\partial t} - \mathbf{c} \cdot \nabla\right)\left(\frac{\partial}{\partial t} + \mathbf{c} \cdot \nabla \right) \mathbf{u} = \left(\frac{\partial^2}{\partial t^2} + (\mathbf{c} \cdot \nabla) \mathbf{c} \cdot \nabla\right) \mathbf{u} = \left(\frac{\partial^2}{\partial t^2} + (\mathbf{c} \cdot \nabla)^2\right) \mathbf{u} = \mathbf{0}.\] For \(\nabla \mathbf{c} = \mathbf{0},\) this simplifies to \[\left(\frac{\partial^2}{\partial t^2} + c^2\Delta\right) \mathbf{u} = \mathbf{0}.\] Therefore, the vectorial 1st-order one-way wave equation with waves travelling in a pre-defined propagation direction \(\mathbf{c}\) results as \[\frac{\partial \mathbf{u}}{\partial t} - \mathbf{c} \cdot \nabla \mathbf{u} = \mathbf{0}.\]

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

To obtain a solution with constant frequencies, apply the Fourier transform \[\Psi(\mathbf{r}, t) = \int_{-\infty}^\infty \Psi(\mathbf{r}, \omega) e^{-i\omega t} \, d\omega,\] which transforms the wave equation into an elliptic partial differential equation of the form: \[\left(\nabla^2 + \frac{\omega^2}{c^2}\right) \Psi(\mathbf{r}, \omega) = 0.\]

This is the Helmholtz equation and can be solved using separation of variables. In spherical coordinates this leads to a separation of the radial and angular variables, writing the solution as: \[\Psi(\mathbf{r}, \omega) = \sum_{l,m} f_{lm}(r) Y_{lm}(\theta, \phi).\] The angular part of the solution take the form of spherical harmonics and the radial function satisfies: \[\left[\frac{d^2}{dr^2} + \frac{2}{r} \frac{d}{dr} + k^2 - \frac{l(l + 1)}{r^2}\right] f_l(r) = 0.\] independent of \(m\), with \(k^2=\omega^2 / c^2\). Substituting \[f_{l}(r)=\frac{1}{\sqrt{r}}u_{l}(r),\] transforms the equation into \[\left[\frac{d^2}{dr^2} + \frac{1}{r} \frac{d}{dr} + k^2 - \frac{(l + \frac{1}{2})^2}{r^2}\right] u_l(r) = 0,\] which is the Bessel equation.

The wave equation is linear in u and 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. 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: \[u(t, x, y, z) = t M_{ct}[\varphi].\]

It follows that \[u(0, x, y, z) = 0, \quad u_t(0, x, y, z) = \varphi(x, y, z).\]

The mean value is an even function of t, and hence if \[v(t, x, y, z) = \frac{\partial}{\partial t} \big(t M_{ct}[\varphi]\big),\] then \[v(0, x, y, z) = \varphi(x, y, z), \quad v_t(0, x, y, z) = 0.\]

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 only true for odd numbers of space dimension, where for one dimension the integration is performed over the boundary of an interval with respect to the Dirac measure.

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 a double integral over the disc D with center (x, y) and radius ct:

Condensed: the full section is in Wikipedia.

Assume n ≥ 3 is an odd integer, and g ∈ C(R), h ∈ C(R) for m = (n + 1)/2. Let γ_n = 1 × 3 × 5 × ⋯ × (n − 2) and let

\[u(x, t) = \frac{1}{\gamma_n} \left[\partial_t \left(\frac{1}{t} \partial_t \right)^{\frac{n-3}{2}} \left(t^{n-2} \frac{1}{|\partial B_t(x)|} \int_{\partial B_t(x)} g \, dS \right) + \left(\frac{1}{t} \partial_t \right)^{\frac{n-3}{2}} \left(t^{n-2} \frac{1}{|\partial B_t(x)|} \int_{\partial B_t(x)} h \, dS \right) \right]\]

Then

• \(u \in C^2\big(\mathbf{R}^n \times [0, \infty)\big)\),
• \(u_{tt} - \Delta u = 0\) in \(\mathbf{R}^n \times (0, \infty)\),
• \(\lim_{(x,t) \to (x^0,0)} u(x,t) = g(x^0)\),
• \(\lim_{(x,t) \to (x^0,0)} u_t(x,t) = h(x^0)\).

Assume n ≥ 2 is an even integer and g ∈ C(R), h ∈ C(R), for m = (n + 2)/2. Let γ_n = 2 × 4 × ⋯ × n and let

\[u(x,t) = \frac{1}{\gamma_n} \left [\partial_t \left (\frac{1}{t} \partial_t \right )^{\frac{n-2}{2}} \left (t^n \frac{1}{|B_t(x)|}\int_{B_t(x)} \frac{g}{(t^2 - |y - x|^2)^{\frac{1}{2}}} dy \right ) + \left (\frac{1}{t} \partial_t \right )^{\frac{n-2}{2}} \left (t^n \frac{1}{|B_t(x)|}\int_{B_t(x)} \frac{h}{(t^2 - |y-x|^2)^{\frac{1}{2}}} dy \right ) \right ]\]

then

• u ∈ C(R × [0, ∞))
• u_tt − Δu = 0 in R × (0, ∞)
• \(\lim_{(x,t)\to (x^0,0)} u(x,t) = g(x^0)\)
• \(\lim_{(x,t)\to (x^0,0)} u_t(x,t) = h(x^0)\)

Consider the inhomogeneous wave equation in \(1+D\) dimensions\[(\partial_{tt} - c^2\nabla^2) u = s(t, x)\]By rescaling time, we can set wave speed \(c = 1\).

Since the wave equation \((\partial_{tt} - \nabla^2) u = s(t, x)\) has order 2 in time, there are two impulse responses: an acceleration impulse and a velocity impulse. The effect of inflicting an acceleration impulse is to suddenly change the wave velocity \(\partial_t u\). The effect of inflicting a velocity impulse is to suddenly change the wave displacement \(u\).

For acceleration impulse, \(s(t,x) = \delta^{D+1}(t,x)\) where \(\delta\) is the Dirac delta function. The solution to this case is called the Green's function \(G\) for the wave equation.

For velocity impulse, \(s(t, x) = \partial_t \delta^{D+1}(t,x)\), so if we solve the Green function \(G\), the solution for this case is just \(\partial_t G\).

The main use of Green's functions is to solve initial value problems by Duhamel's principle, both for the homogeneous and the inhomogeneous case.

Given the Green function \(G\), and initial conditions \(u(0,x), \partial_t u(0,x)\), the solution to the homogeneous wave equation is\[u = (\partial_t G) \ast u + G \ast \partial_t u\]where the asterisk is convolution in space. More explicitly, \[u(t, x) = \int (\partial_t G)(t, x-x') u(0, x') dx' + \int G(t, x-x') (\partial_t u)(0, x') dx'.\]For the inhomogeneous case, the solution has one additional term by convolution over spacetime:\[\iint_{t' < t} G(t-t', x-x') s(t', x')dt' dx'.\]

By a Fourier transform,\[\hat G (\omega)= \frac{1}{-\omega_0^2 + \omega_1^2 + \cdots + \omega_D^2}, \quad G(t, x) = \frac{1}{(2\pi)^{D+1}} \int \hat G(\omega) e^{+i \omega_0 t + i \vec \omega \cdot \vec x}d\omega_0 d\vec\omega.\]The \(\omega_0\) term can be integrated by the residue theorem. It would require us to perturb the integral slightly either by \(+i\epsilon\) or by \(-i\epsilon\), because it is an improper integral. One perturbation gives the forward solution, and the other the backward solution. The forward solution gives\[G(t,x) = \frac{1}{(2\pi)^D} \int \frac{\sin (\|\vec \omega\| t)}{\|\vec \omega\|} e^{i \vec \omega \cdot \vec x}d\vec \omega, \quad \partial_t G(t, x) = \frac{1}{(2\pi)^D} \int \cos(\|\vec \omega\| t) e^{i \vec \omega \cdot \vec x}d\vec \omega.\]The integral can be solved by analytically continuing the Poisson kernel, giving\[G(t, x) = \lim _{\epsilon \rightarrow 0^{+}} \frac{C_D}{D-1} \operatorname{Im}\left[\|x\|^2-(t-i \epsilon)^2\right]^{-(D-1) / 2}\]where \[C_D=\pi^{-(D+1) / 2} \Gamma((D+1) / 2)\] is half the surface area of a \((D + 1)\)-dimensional hypersphere.