Fundamental solution to the radial wave equation

In summary, the radial wave equation is an important equation in physics and engineering, and its fundamental solution is known as the Bessel function. There are various resources available that discuss the derivation of the Bessel function and the use of the radial wave equation, including articles, lecture notes, and textbooks.
  • #1
n!kofeyn
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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.
 
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  • #2
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.
 

Related to Fundamental solution to the radial wave equation

What is the fundamental solution to the radial wave equation?

The fundamental solution to the radial wave equation is a mathematical function that describes the behavior of a wave in a radial system. It represents the amplitude of the wave at any given point in space.

What is the significance of the fundamental solution to the radial wave equation?

The fundamental solution is an important concept in physics and engineering, as it allows us to understand and predict the behavior of waves in radial systems, such as sound waves in a spherical room or electromagnetic waves in a circular antenna.

What is the mathematical formula for the fundamental solution to the radial wave equation?

The formula for the fundamental solution to the radial wave equation varies depending on the specific system and boundary conditions. In general, it involves a combination of trigonometric functions, Bessel functions, and other mathematical functions.

How is the fundamental solution to the radial wave equation derived?

The fundamental solution is derived through the use of mathematical methods, such as separation of variables and Fourier series. It involves solving the wave equation for a specific system and applying boundary conditions to obtain a unique solution.

What are some applications of the fundamental solution to the radial wave equation?

The fundamental solution has various applications in fields such as acoustics, electromagnetics, and quantum mechanics. It is used to study the behavior of waves in circular or spherical systems, and to design and analyze various devices, such as antennas, acoustic resonators, and particle accelerators.

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