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The Green's function for the wave equation is closely related to the
Green's function for
the ideal driven harmonic oscillator.
Since there is only a single degree of freedom (the oscillator displacement
as a function of time) the mathematics is simplified. Further, the
Green's function for the driven
harmonic oscillator
is often covered in undergraduate classical mechanics texts, and everyone
has a good physical intuition on what the motion of the oscillator
will be during and after a driving force is applied.
Once the harmonic oscillator is understood, the Green's function for the
wave equation can be derived by resolving the wave equation into
Fourier components, or more generally into the eigenmodes corresponding
to the spatial boundary condition. Each mode's equation of motion is
identical to a harmonic oscillator. In fact they are often called radiation
oscillators in quantum mechanics. Green's theorem then leads to a
complete solution.