View Full Version : Solving the 3D Scalar Wave Equation

Normandy6644

2005-Feb-28, 12:04 AM

Does anybody know how to solve this equation:

http://img.photobucket.com/albums/v506/Normandy6644/waveeq.jpg

It's the 3D scalar wave equation, and I know the solution is in the form

p=f1(n*r-ct)+f2(n*r+ct)

Basically, I want to somehow separate the operator and then solve it using some kind of change of variables. But I think the problem is with the Laplacian, because I need to show that it's rotationally invariant. Anyone familiar with this or know of any good webpages?

Nicolas

2005-Feb-28, 12:15 AM

You could search for Richard Haberman's "Applied Partial Differential Equations with Fourier series and boundary value problems"

(Pearson Prentice Hall; ISBN 0-13-0645243-1)

Don't immediately buy it, as I'm not sure the answer is inthere :) . But the book does cover a lot about PDE and separation of variables. It discusses the Laplacian, I don't know about Rotational Invariant (that term doesn't ring a loud bell).

ANyway, if you've got a scientific library, you could start your search with this book. I hope it helps!

Normandy6644

2005-Feb-28, 01:00 AM

Thanks! Basically I'm supposed to prove that the Laplacian doesn't change under a rotation of axes, but I'm not sure how to do it.

Severian

2005-Feb-28, 03:28 AM

If you look in the front of Griffiths' E&M book, it'll have the Laplacian in spherical coordinates, and somewhere in the appendices I think he derives it. It looks like your solution is in spherical coordinates, so that's why I mention it. But then the change from cartesian coordinates to spherical isn't really a rotation of the axes, so maybe that's not what you want (but you may be able to show the Laplacian is rotationally invariant by staring at the spherical Laplacian). But if you do want to just rotate the axes and grind out the calculation, then the change of coordinates will just be multiplication by an element of SO(3).

Normandy6644

2005-Feb-28, 03:42 AM

I think I wound up doing it okay. I was able to show that the Laplacian is invariant under a rotation, then argued that you can rotate the original system into the direction of motion of the wave, so it becomes a one dimensional problem. From there the solution is pretty simple.

papageno

2005-Feb-28, 12:52 PM

As far as I remember, the standard approach is to do a Fourier (or similar) transform of the equation (derivatives are transformed into powers).

And shouldn't you solve the time-independent equation first?

Normandy6644

2005-Mar-01, 12:36 AM

As far as I remember, the standard approach is to do a Fourier (or similar) transform of the equation (derivatives are transformed into powers).

And shouldn't you solve the time-independent equation first?

I probably could have done a transform, but it turned out not to be necessary. Thanks though! :D

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