Re: Wavelength of Ripples in a Pond
- From: Andy Resnick <andy.resnick@xxxxxxxxxxx>
- Date: Mon, 23 Jul 2007 18:57:26 +0000 (UTC)
Darwin wrote:
If one throws a pebble into a pond, a concentric sequence of circular<snip>
waves are formed. The spacing of the concentric waves looks even, so
that as a first guess the ripple spectrum has a narrow bandwidth.
However, I don't understand why this would be true since there was no
intrinsic frequency in the rock. Based on this observation, I have
three questions.
1) Are the ripples produced by a pebble mostly one wavelength?
1a) If so, why are they the same wavelength?
In other words, what filtering mechanism if any is
responsible for the concentric circles.
2) How would one calculate the spectrum of ripples in a pond, produced
by throwing a small stone in the water?
This is a deceptively complicated problem, mostly because it is
intrinsically nonlinear: the boundary conditions involving the pressure,
curvature, velocity, and surface tension have to be applied to the
surface, and the surface is known only when the problem is solved. This
would be a great problem to work through in class.....
Both Segel "Mathematics Applied to Continuum Mechanics" and Lamb
"Hydrodynamics" have extensive sections on this specific problem.
What is done to simplify is the linearize the equations: small amplitude
waves, the pressure is related to the wave amplitude, constant density,
the air has no effect, the excitation is a pure delta function, etc.
etc. What appears is a balance between surface tension (Bond number)
and gravity (Froude number)
Two things fall out: one, that there is a minimum wave speed (for water,
this is 23 cm/s), and the other, that the wavelength does change with
propogation, albeit slowly. I suspect that your assumption that the
wavelength does not depend on the mass of the rock (or size or whatever)
has more to do with the degree to which the disturbance approximates a
delta function and the amount of energy deposited into the fluid within
a wavelength or so from the surface.
--
Andrew Resnick, Ph.D.
Department of Physiology and Biophysics
Case Western Reserve University
.
- References:
- Wavelength of Ripples in a Pond
- From: Darwin
- Wavelength of Ripples in a Pond
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