Fundamental solution to the radial wave equation

[n!kofeyn]

i was wondering if anyone could point me towards any resources (including books, papers, and/or notes) that discusses and explicitly derives the fundamental solution to the radial wave equation. i have evans' PDE book, but it isn't contained there, and I've been having trouble searching for it. I'm really more concerned about the derivation and not the solution and how to use it.

i'm also a little confused on what the radial wave equation is and its importance, so if you want to elaborate on that as well, it would be welcomed. thanks for the help.

 

[gtoguy97]

The radial wave equation is a special case of the Helmholtz equation, which is an important partial differential equation in physics and engineering. The fundamental solution to this equation is known as the Bessel function, which is a special function of mathematical physics. There are several good resources available on the web that explain the derivation of the Bessel function and the radial wave equation. One useful resource is a short article written by J. D. Jackson entitled "Bessel Functions and the Radial Wave Equation", which is available at https://www.mth.kcl.ac.uk/~james/pp/bessel.pdf. This article provides a concise overview of the radial wave equation, its solution in terms of the Bessel function, and the derivation of the Bessel function from the radial wave equation. Another useful resource is a series of lecture notes by A. T. Filippov entitled "Partial Differential Equations" which is available at http://www.math.spbu.ru/user/filippov/pdffiles/pdelectures.pdf. This set of lecture notes provides a detailed overview of the radial wave equation and its solution in terms of the Bessel function, as well as a detailed derivation of the Bessel function from the radial wave equation.Finally, there are many textbooks available that discuss the radial wave equation and its solution in terms of the Bessel function, such as the book "Partial Differential Equations: Theory and Completely Solved Problems" by S. L. Sobolev and A. V. Filippov.