Solving wave equation using Green's function Watch
separating by parts, we can write the solution in the form
If we would like to include a source of wave disturbance, we must describe an inhomogeneous equation
where is a linear differential operator. We show has the simple integral representation
This is easily done by direct substitution
Knowing the Green's function we can in principle find .
Let's solve the inhomogeneous equation
We solve this with point source , the general solution is then obtained by substituting in ,
This equation is solved via Fourier transform, , which transforms the differential equation into an algebraic equation. Works the following way,
The contour integral to find is then subtle. The integrand has poles at on the real axis. Probably have questions already... Do you know about Jordan's Lemma? The Cauchy principal value? By the way doing this off the top my head. I'll will think about the general situation later.