Using the d'Alembert operator for wave equations

[adphysics]

1. Is there anywhere where I can find a derivation for the generalized inhomogeneous wave equation? I found a derivation of the 1D wave equation for an infinitesimal region of an elastic string here:

http://www.math.ubc.ca/~feldman/apps/wave.pdf

But I am looking for proof of the generalized version in n dimensions where
[(grad)^2-(1/c^2)d^2/dt^2)]f=0 in free space.

Additionally, I'm having trouble following this derivation of the solution to the time-dependent Green's function here:
http://farside.ph.utexas.edu/teaching/em/lectures/node49.html

I can understand the derivation of the generalized solution to Poisson's equation in the time-independent case, but I'm having trouble understanding lines 487-492 in the link above. Intuitively, I can understand where the solution comes from since an observer at r will only measure the potential resulting from an impulse at t' and r' at time t=t'+(/r-r'//c), so we use the delta-dirac function, and the potential still falls off like 1/r so the spatial component is the same as the time-independent case. Can anyone help with those lines?

 

[BigJDubs]

For a derivation of the generalized inhomogeneous wave equation, you can refer to the book Advanced Engineering Mathematics by Kreyszig. In this book, the derivation of the generalized inhomogenous wave equation is presented using the D'Alembert operator (∂2/∂t2 − c2∆) and the integration by parts technique. Regarding the derivation of the solution to the time-dependent Green's function, the first step is to take the Fourier transform of the Green's function. Then, substitute the Fourier transformed Green's function into the wave equation to obtain a solution in the frequency domain, which is then inverted to get a solution in the time domain. Lines 487-492 show the steps of this inversion process, which essentially involves applying the Fourier inversion theorem.