What are the equations for circular waves from a pebble dropped in a pond?

[rlduncan]

What are the equation(s) for circular waves such as pebble dropped in a pond.

 

[ZapperZ]

OK, I'll bite. Bessel function of the First Kind?

Zz.

 

[rlduncan]

Circular Wave

Yes, a Bessel function. How about a circular wave in which the wavelength is constant. I have looked at graphs of Bessel functions in a plane and they appear to decrease in amplitude and wavelength.

 

[Tide]

If you look at the Bessel functions carefully you will see that the wavelength approaches a constant value as you go from the near field to the far field regions. You also infer that behavior from the asymptotic behavior of the governing differential equation (wave equation) for the Bessel functions.

 

[rlduncan]

I am studing the equation y=sinkx/x^2 and find that the second order differential equation found for this equation is Bessel like, but not identical to Bessel functions. The first derivative term, for example, is off by a constant.

 

[ZapperZ]

I am studing the equation y=sinkx/x^2 and find that the second order differential equation found for this equation is Bessel like, but not identical to Bessel functions. The first derivative term, for example, is off by a constant.

Are you sure it is not

$$y=\frac{sin^2(kx)}{(kx)^2}$$

which is the Fraunhofer diffraction equation? If it is, then I don't see the connection with asking for circular water waves.

Zz.

 

[Bladibla]

Are you sure it is not

$$y=\frac{sin^2(kx)}{(kx)^2}$$

which is the Fraunhofer diffraction equation? If it is, then I don't see the connection with asking for circular water waves.

Zz.

First time I've seen you use Latex!

 

[rlduncan]

If you graph just sinx/x^2 where k=1 then you get a damped sine curve in which the wavelength is constant. I assume for ripples on a pond the wavelengths are constant and I was trying to make a connection if any.

 

[ZapperZ]

If you graph just sinx/x^2 where k=1 then you get a damped sine curve in which the wavelength is constant. I assume for ripples on a pond the wavelengths are constant and I was trying to make a connection if any.

I'm not sure how you can assume that when you haven't set up the diff. equation to solve for such a problem. Note that for a drum-membrane problem, you do have bessel functions as the solution to the diff. equation with the proper boundary conditions.

Zz.