Wave equation

The homogeneous wave equation is a linear partial differential equation of second order

For a real or complex function and a parameter. It is also called d' Alembert equation and one of the hyperbolic differential equations.

With the D'Alembert operator, wherein the Laplacian referred to briefly as the equation

Noted.

The substitution absorbed a factor in the wave equation, it then has the same shape as for (see also natural units).

The solutions of the wave equation are called waves. This overlap without interference and spread regardless of any other existing shafts. Since the coefficients of the wave equation is not the place or time dependent, wavelike behavior regardless of where or when stimulating them. Postponed or delayed waves are therefore also solutions of the wave equation.

Under the inhomogeneous wave equation is defined as the linear inhomogeneous partial differential equation

Solutions of the homogeneous wave equation in one spatial dimension

The homogeneous wave equation in one dimension

Has the general solution

With arbitrary twice differentiable functions and. Here, the first term is the one with speed c to the left and the second term one with the same speed to the right with an unchanged form traveling wave. The functions and are called Riemann invariants.

The functions and can be used as a linear combination of cosine functions

Or of complex exponentials

Write:

The frequency depends by

Together with the wave number. The phase plugged it in the complex amplitude.

Solution with given initial values

So be the general solution of the wave equation and and two initial conditions, it follows:

Integration of the second equation is:

By solving we obtain:

The solution of the wave equation under the above initial conditions is therefore:

The wave equation in three spatial dimensions

Also in multiple dimensions can be the general solution of the wave equation as a linear combination of plane waves

. Write Such a plane wave moving with speed c in the direction of. In the general solution

Is, however, not obvious how their initial values ​​associated with the solution later.

In three spatial dimensions, the solution of the wave equation can be represented by mean values ​​of the initial values ​​. Be the function and its time derivative at the initial time by functions and given

Then, if we choose for simplicity, the linear combination of mean values

The corresponding solution of the wave equation. It referred

The average of the function averaged over a spherical shell around the point with radius particular

As this illustration shows the solution by the initial values ​​, the solution depends continuously on the initial values ​​and depends on the time at the place only on the initial values ​​at the places from, from which you can reach in the runtime speed of light. So that it satisfies Huygens' principle. In one space dimension and in even space dimensions, this principle does not apply. There, the solutions depend on time also depends on initial values ​​to points from which you can reach at a slower speed.

The solution of the inhomogeneous wave equation in three space dimensions

Depends on the place at the moment only on the inhomogeneity on the backward light cone of from. The inhomogeneity and the initial values ​​affect the solution with the speed of light.