Wave Equation - Inhomogeneous Wave Equation in One Dimension

Inhomogeneous Wave Equation in One Dimension

The inhomogeneous wave equation in one dimension is the following:

with initial conditions given by

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 the property of the wave equation that its solutions obey 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 can't 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 casually affects point as . Suppose we integrate the inhomogeneous wave equation over this region.

To simplify this greatly, we can use Green's theorem to simplify the left side to get the following:

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

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, namingly, where the sign is chosen appropriately. Using this, we can get the relation, again choosing the right sign:

And similarly for the final boundary segment:

Adding the three results together and putting them back in the original integral:

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.