# Expression for circular wave

1. Dec 16, 2009

### scepter

Hi all.
I´m looking for an expression for displacement of the waveform, eta(x,y,t) , of a circular wave that is created when a stone is dropped into a pond. Simplest case, assuming an ideal fluid and neglecting non-linear effects.

Thanks,

/scepter

2. Dec 16, 2009

### Born2bwire

If you are purely interested in cylindrical wave functions, Bessel, Hankel and Airy functions are all appropriate wave functions. I would think a Hankel function is most appropriate, a traveling cylindrical wave. As to the specific application of waves on the surface of a fluid, I am ignorant of the solution to that particular problem.

3. Dec 16, 2009

### scepter

The Hankel functions solves the Laplace equation for the velocity potential in cylindrical coordinates, so it describes a propagating circular wave.
How do I go about finding the expression for the surface displacement from the Hankel functions?
Is there no simpler way? I imagine the displacement must vanish at a rate proportional to 1/r, where r is the distance from the origin.

4. Dec 16, 2009

### uart

Actually it's the energy (density) that falls off at a rate proportional to 1/r, so the amplitude falls off at a rate proportional to $1/ \sqrt{r}$. And that corresponds exactly to the asymptotic behaviour of the Bessel function (for large x).

$$J_m(x) \simeq \sqrt{\frac{2}{\pi x}} \cos(x - \frac{m \pi}{4} - \frac{\pi}{4})$$

Edit. Should be :

$$J_m(x) \simeq \sqrt{\frac{2}{\pi x}} \cos(x - \frac{m \pi}{2} - \frac{\pi}{4})$$

Last edited: Dec 16, 2009
5. Dec 16, 2009

### scepter

Ok, I see.
So to find the displacement at some distance away from the origin, which is what I`m interested in, I could use this asymptotic form of the Bessel function. Do I need to somehow convert it to polar coordinates?
I take it that there is no available simple expression for the displacement that contains the Bessel function? Maybe someone could outline the procedures for arriving at the displacement formula?

6. Dec 16, 2009

### FredGarvin

Since Hankel functions are a combination of Bessel functions J and Y, with this case, there would be a singularity at r=0 for Y and thus the result should be a function of J. The only places I have sen Henkel functions arise is in situations where there is no interest in the r=0 case, i.e. annular ducts, incident flows or a pulsating source.

uart: You may want to check your equation for the asymptotic expansion of Jn. Your second cosine term should be $$\frac{m \pi}{2}$$

Last edited: Dec 16, 2009
7. Dec 16, 2009

### Born2bwire

In electromagnetics, the Bessel function is a standing wave and the Hankel function is a travelling wave. The wave equation that would arise with a line sourc such as the OP's would be a Hankel function unless he required a reflecting boundary.