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Applied Partial Differential Equations (Richard Heberman) 4ed.

#12.3.6

Consider the three dimensional wave equation

[tex]\partial^{2}u/\partial t^2 = c^2\nabla^2 u[/tex]

Assume the solution is spherically symetric, so that

[tex]\nabla^2 u = (1/\rho^2)(\partial/\partial\rho)(\rho^2\partial u/\partial\rho) [/tex]

(a) Make the transformation [tex]u = (1/\rho)w(\rho,t)[/tex] and verify that

[tex] \partial^2w/\partial t^2 = c^2(\partial^2w/\partial \rho^2)[/tex]

(b) Show that the most general sphereically symmetric solution of the wave equation consists of the sum of two sphereically symmetric waves, one moving outward at speedcand the other inward at speedc. Note the decay of the amplitude.

Attempts

I really have no idea how to do this. Any and all help (hopefully oriented to the level of someone not all that comfortable with PDEs) would be greatly appreciated.

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# 3D wave equation - spherically symmetric transformations

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