Problem interpreting course notes - 3D wave equation

[Darren93]

I've been stuck trying to figure out what's going on in a particular section of my notes for the last couple days. The biggest issue is the lecturer has just not explained where the example has come from and what it represents. I thought I would post the relevant section here and see if anyone could recognise it immediately. He just introduced the 3D wave equation which I understood but then stated the case of spherical symmetry without any explanation. I was wondering why he defined Ψ in the manor he did and had no clue why he involved an angular section. Does anybody know what's going on? The note section is attached

 

[Simon Bridge]

Your lecturer expected you would know what "spherical symmetry" means.
Start with that - how would you write a sine wave that propagates outwards in all directions from a point?

The lecturer introduced the angle bits for the same reason the radius bit was introduced: because it is more convenient to use spherical-polar coordinates. You should have met these already. The notes have misleadingly placed the bit with the angles off to one side - it belongs on the RHS of the first line (before the equals sign) and it is the rest of what $\nabla^2$ looks like in spherical coordinates.

If the wave travels equally in all directions, then does it vary with angle?

 

[Darren93]

Thanks for the reply. I understand that he has swapped to spherical coordinates and placed a spherical symmetric condition on the equation, I just have no idea how he got the result. What is the form of the wave equation in polar coordinates anyway. Plus where did the 1/r section on Ψ come from?

 

[Darren93]

This is my thinking so far if Ψ=f(r)/r

then d²Ψ/dr² =2r^-3f(r) -2r^-2f'(r) +r^-1f''(r)

You could then substitute in dΨ/dr values but I have a - sign on the f'(r) function and can't quite do that. If it was positive it would leave me with 2/r * dΨ/dr +1/r *f''(r) which is somewhat close to the answer but completely wrong. I probably don't need to know the intermediate steps but Its something I've spent so much time on now that I'm not really wanting to move on.

 

[Simon Bridge]

You already have the wave equation in terms of the Laplace operator and a time derivative - so just substitute the spherical coordinate form of the Laplace operator in and you have the wave equation in spherical coordinates.
http://en.wikipedia.org/wiki/Laplace_operator#Three_dimensions
... the 1/r thing is part of the Laplace operator.

 

[Darren93]

Thanks very much, I was trying to solve for the Laplace operator on my own and just making a mess. Seeing it written out like that makes a great deal of sense and I understand now why the solution is what it is. I'm not certain as to how the 1/r gets incorporated into the Ψ expression but at this point it's no major concern. Thanks very much you helped me quite a bit.

 

[Simon Bridge]

There is a 1/r term in the laplacian - but the solution to the wave equation for spherical waves must reduce amplitude with increasing radius in order to conserve energy.

 

[Khashishi]

How would you write solutions for waves in a spherical cavity? You could use x, y, z coordinates, but the answers would be messy as hell.