Wave equation - WikiHQ

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.

Introduction

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 <math>u = u (x, y, z, t)</math> of a time variable <math>t</math> (a variable representing time) and one or more spatial variables <math>x, y, z</math> (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 <math>x</math> 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 <math>(E_x, E_y, E_z)</math> as the representation of an electric vector field wave <math>\vec{E}</math> in the absence of wave sources, each coordinate axis component <math>E_i, i=x,y,z,</math> must satisfy the scalar wave equation. Other scalar wave equation solutions 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

where

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

The equation states that, at any given point, the second derivative of <math>u</math> with respect to time is proportional to the sum of the second derivatives of <math>u</math> 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

<math display="block">u_{tt} = c^2 \Delta u,</math>

where the double subscript denotes the second-order partial derivative with respect to time, <math>\Delta</math> is the Laplace operator and <math>\Box</math> the d'Alembert operator, defined as:

<math display="block"> 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.</math>

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 plane waves with various directions of propagation and wavelengths but all with the same propagation speed <math>c</math>. 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.

Wave equation in one space dimension

thumb| right|French scientist [[Jean-Baptiste le Rond d'Alembert discovered the wave equation in one space dimension.]]

The wave equation in one spatial dimension can be written as follows:

<math display="block">\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}.</math>This equation is typically described as having only one spatial dimension <math>x</math>, because the only other independent variable is the time <math>t</math>.

Derivation

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).

Hooke's law

The 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 <math>m</math> interconnected with massless springs of length . The springs have a spring constant of :

Here the dependent variable <math>u(x)</math> measures the horizontal displacement from equilibrium of the mass situated at , so that <math>u(x)</math> essentially measures the magnitude of a disturbance (i.e. strain) that is traveling in an elastic material. The resulting force exerted on the mass <math>m</math> at the location <math>x+h</math> is:

<math display="block">\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)].

\end{align}</math>

By equating the latter equation with

<math display="block">\begin{align}

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

\end{align}</math>

the equation of motion for the weight at the location is obtained:

<math display="block">\frac{\partial^2}{\partial t^2} u(x + h, t) = \frac{k}{m} [u(x + 2h, t) - u(x + h, t) - u(x + h, t) + u(x, t)].</math>

If the array of weights consists of <math>N</math> weights spaced evenly over the length <math>L=Nh</math> of total mass , and the total spring constant of the array , we can write the above equation as

<math display="block">\frac{\partial^2}{\partial t^2} u(x + h, t) = \frac{KL^2}{M} \frac{[u(x + 2h, t) - 2u(x + h, t) + u(x, t)]}{h^2}.</math>

Taking the limit <math>N \rightarrow \infty, h \rightarrow 0</math> and assuming smoothness, one gets

<math display="block">\frac{\partial^2 u(x, t)}{\partial t^2} = \frac{KL^2}{M} \frac{\partial^2 u(x, t)}{\partial x^2},</math>

which is from the definition of a second derivative. <math>KL^2/M</math> is the square of the propagation speed in this particular case.

thumbnail|1-d standing wave as a superposition of two waves traveling in opposite directions

Stress pulse in a bar

In the case of a stress pulse propagating longitudinally through a bar, the bar acts much like an infinite number of springs in series and can be taken as an extension of the equation derived for Hooke's law. A uniform bar, i.e. of constant cross-section, made from a linear elastic material has a stiffness <math>K</math> given by

where <math>A</math> is the cross-sectional area, and <math>E</math> is the Young's modulus of the material. The wave equation becomes

<math display="block">\frac{\partial^2 u(x, t)}{\partial t^2} = \frac{EAL}{M} \frac{\partial^2 u(x, t)}{\partial x^2}.</math>

<math>AL</math> is equal to the volume of the bar, and therefore

where <math>\rho</math> is the density of the material. The wave equation reduces to

<math display="block">\frac{\partial^2 u(x, t)}{\partial t^2} = \frac{E}{\rho} \frac{\partial^2 u(x, t)}{\partial x^2}.</math>

The speed of a stress wave in a bar is therefore <math>\sqrt{E/\rho}</math>.

General solution

Algebraic approach

For the one-dimensional wave equation a relatively simple general solution may be found. Defining new variables

<math display="block">\begin{align}

\xi &= x - c t, \\

\eta &= x + c t

\end{align}</math>

changes the wave equation into

<math display="block">\frac{\partial^2 u}{\partial \xi \partial \eta}(x, t) = 0,</math>

which leads to the general solution

In other words, the solution is the sum of a right-traveling function <math>F</math> and a left-traveling function <math>G</math>. "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 <math>c</math>. This was derived by Jean le Rond d'Alembert.

Another way to arrive at this result is to factor the wave equation using two first-order differential operators:

<math display="block">\left[\frac{\partial}{\partial t} - c\frac{\partial}{\partial x}\right] \left[\frac{\partial}{\partial t} + c\frac{\partial}{\partial x}\right] u = 0.</math>

Then, for our original equation, we can define

<math display="block">v \equiv \frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x},</math>

and find that we must have

<math display="block">\frac{\partial v}{\partial t} - c\frac{\partial v}{\partial x} = 0.</math>

This advection equation can be solved by interpreting it as telling us that the directional derivative of <math>v</math> in the <math>(1, -c)</math> direction is 0. This means that the value of <math>v</math> is constant on characteristic lines of the form , and thus that <math>v</math> must depend only on , that is, have the form . Then, to solve the first (inhomogenous) equation relating <math>v</math> 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

<math display="block"> \left[\frac{\partial}{\partial t} + c\frac{\partial}{\partial x}\right] G(x + ct) = H(x + ct).</math>

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

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:

The result is d'Alembert's formula:

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.

Plane-wave eigenmodes

Another way to solve the one-dimensional wave equation is to first analyze its frequency eigenmodes. A so-called eigenmode is a solution that oscillates in time with a well-defined constant angular frequency , so that the temporal part of the wave function takes the form , and the amplitude is a function of the spatial variable , giving a separation of variables for the wave function:

<math display="block">u_\omega(x, t) = e^{-i\omega t} f(x).</math>

This produces an ordinary differential equation for the spatial part :

<math display="block">\frac{\partial^2 u_\omega }{\partial t^2} = \frac{\partial^2}{\partial t^2} \left(e^{-i\omega t} f(x)\right) = -\omega^2 e^{-i\omega t} f(x) = c^2 \frac{\partial^2}{\partial x^2} \left(e^{-i\omega t} f(x)\right).</math>

Therefore,

<math display="block">\frac{d^2}{dx^2}f(x) = -\left(\frac{\omega}{c}\right)^2 f(x),</math>

which is precisely an eigenvalue equation for , hence the name eigenmode. Known as the Helmholtz equation, it has the well-known plane-wave solutions

with wave number .

The total wave function for this eigenmode is then the linear combination

<math display="block">u_\omega(x, t) = e^{-i\omega t} \left(A e^{-ikx} + B e^{ikx}\right) = A e^{-i (kx + \omega t)} + B e^{i (kx - \omega t)},</math>

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 <math>e^{-i\omega t},</math> so that a full solution can be decomposed into an eigenmode expansion:

<math display="block">u(x, t) = \int_{-\infty}^\infty s(\omega) u_\omega(x, t) \, d\omega,</math>

or in terms of the plane waves,

<math display="block">\begin{align}

u(x, t) &= \int_{-\infty}^\infty s_+(\omega) e^{-i(kx+\omega t)} \, d\omega + \int_{-\infty}^\infty s_-(\omega) e^{i(kx-\omega t)} \, d\omega \\

&= \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),

\end{align}</math>

which is exactly in the same form as in the algebraic approach. Functions are known as the Fourier component and are determined by initial and boundary conditions. This is a so-called frequency-domain method, alternative to direct time-domain propagations, such as FDTD method, of the wave packet , which is complete for representing waves in absence of time dilations. Completeness of the Fourier expansion for representing waves in the presence of time dilations has been challenged by chirp wave solutions allowing for time variation of . The chirp wave solutions seem particularly implied by very large but previously inexplicable radar residuals in the flyby anomaly and differ from the sinusoidal solutions in being receivable at any distance only at proportionally shifted frequencies and time dilations, corresponding to past chirp states of the source.

Vectorial wave equation in three space dimensions

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 <math>E</math> that is homogeneous (i.e. independent of <math>\mathbf{x}</math>) within the volume element, then its stress tensor is given by <math>\mathbf{T} = E \nabla \mathbf{u}</math>, for a vectorial elastic deflection <math>\mathbf{u}(\mathbf{x}, t)</math>. The local equilibrium of:

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

can be written as <math>\rho \frac{\partial^2 \mathbf{u{\partial t^2} - E \Delta \mathbf{u} = \mathbf{0}.</math>

By merging density <math>\rho</math> and elasticity module <math>E,</math> the sound velocity <math>c = \sqrt{E/\rho}</math> results (material law). After insertion, follows the well-known governing wave equation for a homogeneous medium:

<math display="block">\frac{\partial^2 \mathbf{u{\partial t^2} - c^2 \Delta \mathbf{u} = \boldsymbol{0}.</math>

(Note: Instead of vectorial <math>\mathbf{u}(\mathbf{x}, t),</math> only scalar <math>u(x, t)</math> can be used, i.e. waves are travelling only along the <math>x</math> axis, and the scalar wave equation follows as <math>\frac{\partial^2 u}{\partial t^2} - c^2 \frac{\partial^2 u}{\partial x^2} = 0</math>.)

The above vectorial partial differential equation of the 2nd order delivers two mutually independent solutions. From the quadratic velocity term <math>c^2 = (+c)^2 = (-c)^2</math> can be seen that there are two waves travelling in opposite directions <math>+c</math> and <math>-c</math> 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 <math>\nabla\mathbf{c} = \mathbf{0},</math> special two-wave equation with the d'Alembert operator results:

<math display="block">\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}.</math>

For <math>\nabla \mathbf{c} = \mathbf{0},</math> this simplifies to

<math display="block">\left(\frac{\partial^2}{\partial t^2} + c^2\Delta\right) \mathbf{u} = \mathbf{0}.</math>

Therefore, the vectorial 1st-order one-way wave equation with waves travelling in a pre-defined propagation direction <math>\mathbf{c}</math> results as

<math display="block">\frac{\partial \mathbf{u{\partial t} - \mathbf{c} \cdot \nabla \mathbf{u} = \mathbf{0}.</math>

Scalar wave equation in three space dimensions

thumb|right|Swiss mathematician and physicist [[Leonhard Euler (b. 1707) discovered the wave equation in three space dimensions.

<math display="block">\Psi(\mathbf{r}, \omega) = \sum_{l,m} f_{lm}(r) Y_{lm}(\theta, \phi).</math>

The angular part of the solution take the form of spherical harmonics and the radial function satisfies:

<math display="block"> \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.</math>

independent of <math>m</math>, with <math>k^2=\omega^2 / c^2</math>. Substituting

transforms the equation into

<math display="block"> \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,</math>

which is the Bessel equation.

Example

Consider the case . Then there is no angular dependence and the amplitude depends only on the radial distance, i.e., . In this case, the wave equation reduces to<math display="block">

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

</math>

\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.

</math>

This equation can be rewritten as

<math display="block">\frac{\partial^2(ru)}{\partial t^2} - c^2 \frac{\partial^2(ru)}{\partial r^2} = 0,</math>

where the quantity satisfies the one-dimensional wave equation. Therefore, there are solutions in the form<math display="block">u(r, t) = \frac{1}{r} F(r - ct) + \frac{1}{r} G(r + ct),</math>

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.

Monochromatic spherical wave

thumb|Cut-away of spherical wavefronts, with a wavelength of 10 units, propagating from a point source

Although the word "monochromatic" is not exactly accurate, since it refers to light or electromagnetic radiation with well-defined frequency, the spirit is to discover the eigenmode of the wave equation in three dimensions. Following the derivation in the previous section on plane-wave eigenmodes, if we again restrict our solutions to spherical waves that oscillate in time with well-defined constant angular frequency , then the transformed function has simply plane-wave solutions:<math display="block">r u(r, t) = Ae^{i(\omega t \pm kr)},</math>

<math display="block">u(r, t) = \frac{A}{r} e^{i(\omega t \pm kr)}.</math>

From this we can observe that the peak intensity of the spherical-wave oscillation, characterized as the squared wave amplitude

drops at the rate proportional to , an example of the inverse-square law.

Solution of a general initial-value problem

The wave equation is linear in 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 be an arbitrary function of three independent variables, and let the spherical wave form be a delta function. Let a family of spherical waves have center at , and let be the radial distance from that point. Thus

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

<math display="block">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;</math>

the denominator is a convenience.

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

<math display="block">u(t, x, y, z) = \frac{t}{4\pi} \iint_S \varphi(x + ct\alpha, y + ct\beta, z + ct\gamma) \, d\omega,</math>

where , , and are coordinates on the unit sphere , and is the area element on . This result has the interpretation that is times the mean value of on a sphere of radius centered at :

<math display="block">u(t, x, y, z) = t M_{ct}[\varphi].</math>

It follows that

<math display="block">u(0, x, y, z) = 0, \quad u_t(0, x, y, z) = \varphi(x, y, z).</math>

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

<math display="block">v(t, x, y, z) = \frac{\partial}{\partial t} \big(t M_{ct}[\varphi]\big),</math>

then

<math display="block">v(0, x, y, z) = \varphi(x, y, z), \quad v_t(0, x, y, z) = 0.</math>

These formulas provide the solution for the initial-value problem for the wave equation. They show that the solution at a given point , given depends only on the data on the sphere of radius that is intersected by the light cone drawn backwards from . 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.

Scalar wave equation in two space dimensions

In two space dimensions, the wave equation is

<math display="block"> u_{tt} = c^2 \left( u_{xx} + u_{yy} \right). </math>

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

then the three-dimensional solution formula becomes

<math display="block"> u(t,x,y) = tM_{ct}[\phi] = \frac{t}{4\pi} \iint_S \phi(x + ct\alpha,\, y + ct\beta) \, d\omega,</math>

where and are the first two coordinates on the unit sphere, and is the area element on the sphere. This integral may be rewritten as a double integral over the disc with center and radius

It is apparent that the solution at depends not only on the data on the light cone where

but also on data that are interior to that cone.

Scalar wave equation in general dimension and Kirchhoff's formulae

We want to find solutions to for with and .

Odd dimensions

Assume is an odd integer, and , for . Let and let

<math display="block">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]</math>

Then

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

Even dimensions

Assume is an even integer and , , for . Let and let

<math display="block">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 ] </math>

then

in
<math>\lim_{(x,t)\to (x^0,0)} u(x,t) = g(x^0)</math>
<math>\lim_{(x,t)\to (x^0,0)} u_t(x,t) = h(x^0)</math>

Green's function

Consider the inhomogeneous wave equation in <math>

1+D

</math> dimensions<math display="block">

(\partial_{tt} - c^2\nabla^2) u = s(t, x)

</math>By rescaling time, we can set wave speed <math> c = 1</math>.

Since the wave equation <math>

(\partial_{tt} - \nabla^2) u = s(t, x)

</math> 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 <math>\partial_t u</math>. The effect of inflicting a velocity impulse is to suddenly change the wave displacement <math>u</math>.

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

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

Duhamel's principle

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 <math>G</math>, and initial conditions <math>u(0,x), \partial_t u(0,x)</math>, the solution to the homogeneous wave equation is The forward solution gives<math display="block">

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.

</math>The integral can be solved by analytically continuing the Poisson kernel, giving

Solutions in particular dimensions

We can relate the Green's function in <math>D</math> dimensions to the Green's function in <math>D+n</math> dimensions (lowering the dimension is possible in any case, raising is possible in spherical symmetry).

Lowering dimensions

Given a function <math>s(t, x)</math> and a solution <math>u(t, x)</math> of a differential equation in <math>(1+D)</math> dimensions, we can trivially extend it to <math>(1+D+n)</math> dimensions by setting the additional <math>n</math> dimensions to be constant:

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}).

</math>Since the Green's function is constructed from <math>s</math> and <math>u</math>, the Green's function in <math>(1+D+n)</math> dimensions integrates to the Green's function in <math>(1+D)</math> dimensions:

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}.

</math>

Raising dimensions

The Green's function in <math>D</math> dimensions can be related to the Green's function in <math>D+2</math> dimensions. By spherical symmetry,

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

</math>

Integrating in polar coordinates,

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',

</math>

where in the last equality we made the change of variables <math>q' = \sqrt{r^2 + q^2}</math>. Thus, we obtain the recurrence relation<math display="block">

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

</math>

Solutions in D = 1, 2, 3

When <math>D=1</math>, the integrand in the Fourier transform is the sinc function<math display="block">\begin{aligned}

G_1(t, x) &= \frac{1}{2\pi} \int_\R \frac{\sin(|\omega| t)}{|\omega|} e^{i\omega x}d\omega \\

&= \frac{1}{2\pi} \int \operatorname{sinc}(\omega) e^{i \omega \frac xt} d\omega \\

&= \frac{\sgn(t-x) + \sgn(t+x)}{4} \\

&= \begin{cases}

\frac 12 \theta(t-|x|) \quad t > 0 \\

-\frac 12 \theta(-t-|x|) \quad t < 0

\end{cases}

\end{aligned}</math>

where <math>\sgn</math> is the sign function and <math>\theta</math> is the unit step function.

The dimension can be raised to give the <math>D=3</math> case<math display="block">G_3(t, r) = \frac{\delta(t-r)}{4\pi r}</math>and similarly for the backward solution. This can be integrated down by one dimension to give the <math>D=2</math> case<math display="block">G_2(t, r) = \int_\R \frac{\delta(t - \sqrt{r^2 + z^2})}{4\pi \sqrt{r^2 + z^2 dz

= \frac{\theta(t - r)}{2\pi \sqrt{t^2 - r^2

</math>

Wavefronts and wakes

In <math>D=1</math> case, the Green's function solution is the sum of two wavefronts <math>\frac{\sgn(t-x)}{4} +

\frac{\sgn(t+x)}{4}</math> moving in opposite directions.

In odd dimensions, the forward solution is nonzero only at <math> t = r</math>. As the dimensions increase, the shape of wavefront becomes increasingly complex, involving higher derivatives of the Dirac delta function. For example,

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