What Is the Expression for Circular Wave Displacement in a Pond?

[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

 

[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.

 

[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.

 

[uart]

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.

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 :

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

 

[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?

 

[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 $$\frac{m \pi}{2}$$

 

[Born2bwire]

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 $$\frac{m \pi}{2}$$

In electromagnetics, the Bessel function is a standing wave and the Hankel function is a traveling 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.