Wave equation

The wave equation, also known as the D'Alembert equation after Jean-Baptiste le Rond d'Alembert, determines the propagation of waves such as sound or light. It is one of the hyperbolic differential equations.

If the medium or vacuum only transmits the wave and does not generate waves itself, it is more precisely the homogeneous wave equation, the linear partial differential equation of second order

${\frac {1}{c^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}-\sum _{i=1}^{n}{\frac {\partial ^{2}u}{\partial x_{i}^{2}}}=0$

for a real function $u(t,x_1,\dots, x_n)$ of space-time. Here the dimension of the space. The parameter is the speed of propagation of the wave, i.e. for sound (in a homogeneous and isotropic medium) the speed of sound and for light the speed of light.

The differential operator of the wave equation is called the D'Alembert operator and is $\Box$ notated with the formula symbol

$\Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\sum _{i=1}^{n}{\frac {\partial ^{2}}{\partial x_{i}^{2}}}$ ,

The solutions of the wave equation are called waves. Because the equation is linear, waves superimpose without influencing each other. Since the coefficients of the wave equation do not depend on location or time, waves behave independently of where or when and in which direction you excite them. Shifted, delayed or rotated waves are also solutions of the wave equation.

The inhomogeneous wave equation is understood to be the inhomogeneous linear partial differential equation

$\Box u=v\ .$

It describes the temporal development of waves in a medium that itself generates waves. The inhomogeneity also called the source of the wave .

The wave equation in one spatial dimension

The D'Alembert operator in one spatial dimension

$\Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-{\frac {\partial ^{2}}{\partial x^{2}}}$

decays due to Schwarz's theorem as in the binomial formula $(a^{2}-b^{2})=(a-b)(a+b)$ into the product

$\Box =\left({\frac {1}{c}}{\frac {\partial }{\partial t}}-{\frac {\partial }{\partial x}}\right)\left({\frac {1}{c}}{\frac {\partial }{\partial t}}+{\frac {\partial }{\partial x}}\right)$ .

Therefore, the wave equation in one spatial dimension has the general solution

$u\left(t, x\right) = f(x + ct) + g(x - ct)$

with any twofold differentiable functions f(x) and g(x) . The first summand $f(x+ct)$ is a wave running to the left and the second summand $g(x-ct)$ a wave running to the right with unchanged shape. The straight lines $x\pm ct={\text{konstant}}$ are the characteristics of the wave equation.

$\phi (x)=u(0,x)=f(x)+g(x)$

the initial value and

$\psi (x)={\frac {1}{c}}{\frac {\partial u}{\partial t}}(0,x)=f'(x)-g'(x)$

the initial time derivative of the wave. These functions of space are collectively called initial values of the wave.

The integration of the last equation gives

$f(x)-g(x)=\int _{x_{0}}^{x}\psi (\xi )\,\mathrm {d} \xi \ .$

By dissolving one obtains

$f(x)={\frac {1}{2}}\left(\phi (x)+\int _{x_{0}}^{x}\psi (\xi )\,\mathrm {d} \xi \right)\ ,$

$g(x)={\frac {1}{2}}\left(\phi (x)-\int _{x_{0}}^{x}\psi (\xi )\,\mathrm {d} \xi \right)\ .$

Expressed by their initial values, the solution of the wave equation is therefore

$u(t,x)={\frac {1}{2}}\left(\phi (x+ct)+\phi (x-ct)+\int _{x-ct}^{x+ct}\psi (\xi )\,\mathrm {d} \xi \right)\ .$

This is also known as the D'Alembert solution of the wave equation (d'Alembert, 1740s).

The wave equation in three spatial dimensions

The general solution of the wave equation can be expressed as a linear combination of plane waves

$u({\vec {x}},t)=\int \mathrm {d} \omega \int \mathrm {d} ^{3}{\vec {k}}\,A(\omega ,k)e^{\mathrm {i} ({\vec {k}}\cdot {\vec {x}}-\omega t)}\delta (\omega -c|{\vec {k}}|)$

write. The delta distribution ensures that the dispersion relation ω $\omega =c|{\vec {k}}|$ maintained. Such a plane wave moves in the direction of ${\vec {k}}$ . In the superposition of such solutions, however, it is not obvious how their initial values are related to the later solution.

In three spatial dimensions, the general solution of the homogeneous wave equation can be represented by mean values of the initial values. Let the function $u(t,{\vec {x}})$ and its time derivative at the initial time t=0 be $\psi$ given by functions ϕ $\phi$ and ψ

$u(0,{\vec {x}})=\phi ({\vec {x}}),\quad {\frac {1}{c}}{\frac {\partial }{\partial t}}u(0,{\vec {x}})=\psi ({\vec {x}})\,,$

then the linear combination of mean values is

$u(t,{\vec {x}})=ct\,M_{t,{\vec {x}}}[\psi ]+{\frac {1}{c}}{\frac {\partial }{\partial t}}(ct\,M_{t,{\vec {x}}}[\phi ])$

is the corresponding solution of the homogeneous wave equation. Here denotes

$M_{t,{\vec {x}}}[\chi ]={\frac {1}{4\,\pi }}\int _{-1}^{1}\mathrm {d} \cos \theta \int _{0}^{2\pi }\mathrm {d} \varphi \,\chi ({\vec {x}}+ct{\vec {n}}(\theta ,\varphi ))\quad {\text{mit}}\quad {\vec {n}}(\theta ,\varphi )={\begin{pmatrix}\sin \theta \cos \varphi \\\sin \theta \sin \varphi \\\cos \theta \end{pmatrix}}$

the mean value of the function χ $\chi\,,$ averaged over a spherical shell around the point ${\vec {x}}$ with radius $c|t|.$ In particular, $M_{0,{\vec {x}}}[\chi ]=\chi ({\vec {x}}).$

As this representation of the solution by the initial values shows, the solution depends continuously on the initial values and depends at time at location ${\vec {x}}$ only on the initial values at locations ${\vec {y}}$ from which one can reach ${\vec {x}}$ in the running time |t| with velocity It thus satisfies Huygens' principle.

For one-dimensional systems and in even spatial dimensions, this principle does not apply. There, the solutions at time also depend on initial values at closer points ${\vec {y}}$ from which one reaches ${\vec {x}}$ with less speed.

The solution of the inhomogeneous wave equation in three spatial dimensions

$u(t,{\vec {x}})=ct\,M_{t,{\vec {x}}}[\psi ]+{\frac {1}{c}}{\frac {\partial }{\partial t}}(ct\,M_{t,{\vec {x}}}[\phi ])+{\frac {1}{4\pi }}\int _{|{\vec {z}}|\leq c|t|}\mathrm {d} ^{3}{\vec {z}}\,{\frac {v(ct-\operatorname {sign} (t)|{\vec {z}}|,{\vec {x}}+{\vec {z}})}{|{\vec {z}}|}}$

depends at the location ${\vec {x}}$ at time t>0 on the inhomogeneity on the backward light cone of x → ${\vec {x}}$ negative times only on the inhomogeneity on the forward light cone. The inhomogeneity and the initial values affect the solution at the speed of light.

Retarded potential

The retarded potential

$u_{\text{retardiert}}(t,{\vec {x}})={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}\mathrm {d} ^{3}{\vec {z}}\,{\frac {v(ct-|{\vec {z}}|,\,{\vec {x}}+{\vec {z}})}{|{\vec {z}}|}}$

is a solution of the inhomogeneous wave equation which requires that the inhomogeneity on all backward light cones decays faster than $1/r^{2}$ It is the wave that is completely generated by the medium without a passing wave.

In electrodynamics, the continuity equation restricts inhomogeneity. Thus, the charge density of a non-vanishing total charge cannot vanish everywhere at any time. In perturbation theory, inhomogeneities occur that do not decay spatially fast enough. Then the associated retarded integral diverges and has a so-called infrared divergence.

The somewhat more elaborate representation of the solution by its initial values at finite time and by integrals over finite sections of the light cone is free from such infrared divergences.

Lorentz invariance of the D'Alembert operator

The D'Alembert operator $\Box$ is invariant under translations and Lorentz transformations Λ $\Lambda$ sense that applied to Lorentz concatenated functions $f\circ \Lambda ^{-1}$ yields the same as the Lorentz concatenated derivative function

$(\Box f)\circ \Lambda ^{-1}=\Box \,(f\circ \Lambda ^{-1})\ .$

Accordingly, the Laplace operator is invariant under translations and rotations.

The homogeneous wave equation is invariant even under conformal transformations, especially under stretching.