i am trying to find resources on the derivation of the fundamental solution to the radial wave equation. any suggestions of or links to books, papers, and/or notes would be much appreciated. i have tried searching google, arxiv, etc. for information, but i haven't come up with anything useful yet. there's a lot of material pertaining to physics concerning using the solution, but i need the derivation.

i've been learning about solutions to the wave equation for euclidean dimensions 1-3 (d'Alembert's formula, spherical means, method of descent), and the above is my next task. if you have any general insight about the radial wave equation, this would be helpful as well. thanks!

(background: 3rd year graduate student in mathematics)

edit: if no one has any specific references to the derivation, then what about context? does the spherically symmetric wave equation mean the same thing as the radial wave equation? i know that seems like a possibly ignorant question, but to someone unfamiliar with or new to this, i think it's good to be cautious and not say well they sound the same.

  • $\begingroup$ Do you really need 'resources' on it, or would someone here giving you the derivation work too? $\endgroup$ – user5810 Oct 7 '10 at 20:26
  • $\begingroup$ someone giving the derivation here would work. this isn't for a course or anything, but i just need to learn it and work through it myself. i am just really not for sure of the idea/approach needed or motivation, other than i guess the equation governs waves whose amplitude only depends on the radial distance from the origin. i don't know if there are different methods depending on the dimension, but the 3-dimensional case or as general solution as possible would be best. thank you for the response. $\endgroup$ – nikofeyn Oct 7 '10 at 20:33
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    $\begingroup$ In you already know about spherical means, then don't you already have a derivation in front of you? (E.g. the first 5 pages of Chris Sogge's Lecture on Nonlinear Wave Equations.) $\endgroup$ – Willie Wong Oct 7 '10 at 21:59
  • $\begingroup$ ... or S.Selberg's lecture notes math.ntnu.no/~sselberg/HopkinsLectures.pdf $\endgroup$ – Willie Wong Oct 7 '10 at 22:03
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    $\begingroup$ As far as I know, "radial wave equation" means the same thing as "spherically symmetric wave equation". $\endgroup$ – Deane Yang Oct 8 '10 at 2:14

This is discussed both in many introductory textbooks on electrodynamics and quantum mechanics. An elementary discussion in the context of quantum mechanics can e.g. be found in Griffiths' "Introduction to Quantum Mechanics", while a discussion in the context of electrodynamics can be found in Jackson's standard tome "Classical Electrodynamics".

  • $\begingroup$ thank you for the response. i have griffiths' book, but i am really needing a more mathematical treatment, as right now the physical applications aren't the goal. for instance, my professor mentioned the use of some sort of kernel (heat?, dirichlet?). now these are just things he very briefly mentioned so what he meant i don't know, but i need as general of an approach as possible. $\endgroup$ – nikofeyn Oct 7 '10 at 21:09
  • $\begingroup$ I must confess, I am a bit confused. Asking for "as general of an approach as possible" while focussing on the radial wave equation is a bit of an oxymoron. $\endgroup$ – Willie Wong Oct 7 '10 at 22:17
  • $\begingroup$ i'm not familiar with the specifics, as evidenced by this post, but i think it was clear that i meant as general of an approach as possible to solving the radial wave equation. $\endgroup$ – nikofeyn Oct 8 '10 at 0:29
  • $\begingroup$ It does not become clear from the discussions here what exactly the problem is. If you have a Laplace equation in n-dimensions and you have a spherical symmetric problem you choose spherical coordinates and by separation of variables the equation will factorize. This will result in particular in what's called the radial factor of the wave equation. This is a completely general procedure, independent of any physics context. You also see from this outline that while radial doesn't mean spherically symmetric, they are linked. Maybe more background in your question would help. $\endgroup$ – Laie Oct 8 '10 at 15:58

If you need something more mathematical, try looking into Section 5.2 of the paper Related Partial Differential Equations and Their Applications by L. R. Bragg and J. W. Dettman in SIAM Journal on Applied Mathematics, Vol. 16, No. 3 (May, 1968), pp. 459-467.

  • $\begingroup$ i'll take a look at this, although the notation seems like it might take some work. thank you for the link. $\endgroup$ – nikofeyn Oct 8 '10 at 0:47

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