3D wave equation - spherically symmetric transformations

[Vidatu]

Problem:

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 speed c and the other inward at speed c. 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.

 

[jasonRF]

Vidatu,

is this a question for a class? I'm assuming it is not as it is posted in this forum, but just in case I will be stingy with my hints until I hear otherwise.

for part a, all you have to know how to do is differentiate a product. If you are uncomfortable with this then you need to spend serious time reviewing calculus.

for part b, you should have seen almost all of this when you studied the 1-D wave equation. Review that material and you should find what you need.

good luck

jason

 

[Vidatu]

Its a suggested problem for our class; a learning exercise, not for marks.

I'm pretty sure I've got part a, but b is still eluding me. For the record, we were never taught the wave equation before; it was part of a prerequisite course, but was cut out, and this course wasn't altered to reflect it.

 

[LCKurtz]

What was apparently cut from your previous course was D'Alembert's solution to the wave equation. You can read about it here:

http://mathworld.wolfram.com/dAlembertsSolution.html

That will help you with part b.