Wave Equation for Circular Waves

[greswd]

What's the solution to the wave equation for circular waves on a two-dimensional membrane?

The waves have a constant wavelength throughout. For spherical waves, you have to multiply the amplitude by 1/r. I tried 1/√r for circular waves but it didn't work.

 

[Orodruin]

I suggest you make an ansatz regarding the phase function and insert it into thewave equation. This will give you an ordinary differential equation to solve for the amplitude.

 

[greswd]

This seems like an easily conceivable problem, has it been solved before? I googled but I couldn't find anything that deals with circular waves.

 

[Orodruin]

Of course it has been solved, I just told you one way you can do it relatively easily.

 

[jtbell]

I googled but I couldn't find anything that deals with circular waves.

A Google search for "vibrating circular membrane" gave me lecture notes such as these:

https://www.math.hmc.edu/~ajb/PCMI/lecture14.pdf
http://ramanujan.math.trinity.edu/rdaileda/teach/s12/m3357/lectures/lecture_3_29.pdf

The radial part of the solution involves Bessel functions. I think undergraduate physics courses usually just set up the differential equation in the form of Bessel's equation and then state that the solutions are called Bessel functions. In graduate school I took a course in "intermediate differential equations" which IIRC actually derived a power-series solution for Bessel's equation. I don't remember which textbook we used. Googling for "Bessel functions" and "Bessel's equation" should find something.

The math methods book by Boas which is often referred to on PF, discusses the series solution to Bessel's equation in chapter 12.

 

[greswd]

These involve drumskins. My problem has no boundary conditions, just circular waves from a point source rippling across the surface of the water.

 

[ZapperZ]

These involve drumskins. My problem has no boundary conditions, just circular waves from a point source rippling across the surface of the water.

But the waveform is the same other than the boundary conditions. It may have something of the form of a Bessel function. But you need to set up the differential equation first so that you know what needs to be solved. Asking for a "solution" without setting that up first is rather meaningless.

Mary Boas's text on the Special Functions chapter covers this.

Zz.

 

[Jilang]

Perhaps you need some factor of pi in there? The three dimensional case will involve a 4 pi and the two dimensional case a 2 pi.

 

[vanhees71]

The radial part of the solution involves Bessel functions. I think undergraduate physics courses usually just set up the differential equation in the form of Bessel's equation and then state that the solutions are called Bessel functions. In graduate school I took a course in "intermediate differential equations" which IIRC actually derived a power-series solution for Bessel's equation. I don't remember which textbook we used. Googling for "Bessel functions" and "Bessel's equation" should find something.

The math methods book by Boas which is often referred to on PF, discusses the series solution to Bessel's equation in chapter 12.

One of my alltime favorites if it comes to classical (i.e., non-quantum) physics is the 6-volume "Lectures on Theoretical Physics" by A. Sommerfeld, and vol. 6 ("Partial differential Equations") is the best of them. Although this lecture series is written in the 1940ies-1950ies it's still quite up to date, and the mathematics is just taught in a beautiful way. I really love it for the treatmend of the standard special functions, including of course the Bessel functions. Particularly the clever use of the generalized Fourier tranformation in the complex plane to define them, is a masterpiece of its own!