Wavelength of Ripples in a Pond
Darwin
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 sort of related to the waterfall question asked by
another poster (it wasn't me), but should be in principle easier. The
waterfall is known to generate noise, lacking in any particular tone,
so no would claim a simple spectrum. However, the ripples in the pond
appear quite regular. So there may be some pretty simple "filtering
mechanism" to explain the regular pattern.
Timo Nieminen
> If one throws a pebble into a pond, a concentric sequence of circular
> 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?
There are two main types of water waves: capillary waves (aka surface
tension waves, ripples) and gravity waves (like the big waves on the
ocean). For pure capillary waves, you ignore gravity and assume that
surface tension provides all the restoring force affecting the surface.
For gravity waves, you ignore surface tension, and assume it's all
gravity.
You can do the combined case, and for an infinitely deep pond, the result
isn't too hard. Check out wikipedia on "capillary wave"; the dispersion
relation for capillary-gravity waves is given.
The cute thing is that for capillary waves, the group speed is higher than
the phase speed, while for gravity waves, it's the other way around. For a
magic wavelength (wikipedia says 2cm - presumably for an infinitely deep
pond), the group and phase speeds are equal.
For capillary waves, the phase speed is proportional to
1/sqrt(wavelength).
What kind of spectrum do you get? You could approximate the rock as a
delta function excitation. Lamb (Hydrodynamics, art. 269 in 6th ed) gives
the details. Basically, since phase speed -> infinity as wavelength -> 0,
you get infinitesimal wavelengths appearing everywhere first, and at any
point, the wavelength increases. The rate of transport of energy depends
on the product of the speed and the amplitude squared, vA^2. Since all
frequencies get equal energy from the initial impulse, then we can expect
something like A = sqrt(E/v). Since the very short wavelengths have very
high v, they also have very small A. So, the amplitude at any point should
grow with time.
Well, that's for capillary waves, and they'll get large enough so that
gravity matters.
For pure gravity, waves, you'd have the opposite behaviour, where you'd
get infinite wavelengths appearing everywhere first, and then the
wavelength would shrink and the amplitude increase. The maximum
amplitudes look to me to be likely to be close to the wavelengths at
which surface tension meets gravity, perhaps wikipedia's 2cm.
It won't be quite that simple, because the pond will only appear
infinitely deep when the wavelength is much less than the depth, but
perhaps close enough. The approximate solutions that Lamb gives are also
only good when a sufficient distance away from the disturbance, so you
might not see it in a coffee cup. Also, the initial disturbance is not a
sudden delta function impulse.
Also, dependence of loss of energy on wavelength will matter. I expect the
short wavelengths lose energy fastest, and as they start with the smallest
amplitudes, they'll be gone first.
--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html
Uncle Al
Does a small pebble produce the same spectrum as a big rock? Does
velocity at impact matter? Shape of the impactor? Surface
configuration? There is a famous set of underwater high speed
photographs of bowling balls dropped into a swimming pool. A
monolayer sand cap glued onto the center of the impacting hemisphere
makes a world of difference re the boundary layer and propagating
turbulence.
If detergent is added (surface energy), does the same spectrum
result? If you change gee... add salts to change the density...
Water is an elastic membrane plus bulk excitation.
Violent convulsions of a pulsating star's atmosphere, Mach 8
shockwaves at temperatures of 100,000 degrees and higher, are
described by a surprisingly spare set of equations. Horizontal
velocity, surface elevation, normalized density, pressure, and
propagation speed combine to obtain remarkable predictive power.
Ocean waves are similarly modeled with horizontal velocity, surface
elevation, normalized depth, pressure, and propagation speed.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/lajos.htm#a2
Darwin123
> On Wed, 18 Jul 2007, Darwin wrote:
capillary waves. I had assumed that the ripples in the pond were
almost pure capillary waves, based on their small amplitude. But that
may not be true.
> The cute thing is that for capillary waves, the group speed is higher than
> the phase speed, while for gravity waves, it's the other way around. For a
> magic wavelength (wikipedia says 2cm - presumably for an infinitely deep
> pond), the group and phase speeds are equal.
How is this possible? You just said that for each type of wave,
the group velocity is unequal the phase velocity?
I think you mean that the phase velocity at some point is equal
in the capillary and gravity waves. Maybe this is the wavelength we
are seeing? Maybe the gravity waves and the capillary waves
constructively interfere only at this wavelength. This just is a
conjecture based on what you are telling me.
Thank you for your hints.
Timo A. Nieminen
> On Jul 19, 3:39 pm, Timo Nieminen <t...@physics.uq.edu.au> wrote:
>> On Wed, 18 Jul 2007, Darwin wrote:
> Thank you for pointing out that there are both gravity and
> capillary waves. I had assumed that the ripples in the pond were
> almost pure capillary waves, based on their small amplitude. But that
> may not be true.
>
>> The cute thing is that for capillary waves, the group speed is higher than
>> the phase speed, while for gravity waves, it's the other way around. For a
>> magic wavelength (wikipedia says 2cm - presumably for an infinitely deep
>> pond), the group and phase speeds are equal.
>
> How is this possible? You just said that for each type of wave,
> the group velocity is unequal the phase velocity?
Real waves on a pond are neither capillary waves or gravity waves, but a
kind of average of the two. At small wavelengths, they look like capillary
waves, and at larger wavelengths they look more like gravity waves. In
between, neither. (All this assumes the amplitude isn't too large either!)
Think of the pure capillary and gravity waves as ideal limits, and real
waves as the intermediate case linking them. Under suitable conditions,
the real wave looks very much like either a pure capillary wave or a pure
gravity wave, so these are very useful ideal limits, but there are
conditions under which the real wave looks like neither.
> I think you mean that the phase velocity at some point is equal
> in the capillary and gravity waves. Maybe this is the wavelength we
> are seeing? Maybe the gravity waves and the capillary waves
> constructively interfere only at this wavelength. This just is a
> conjecture based on what you are telling me.
I am away from my Lamb at the moment, but iirc, this is a valid way of
looking at it. Lamb does write about calculating the gravity part and the
capillary part separately, and adding them together. Mathematically, the
capillary-gravity wave looks like an average of a capillary wave and a
gravity wave (see the wikipedia page - the two terms are just added
together).
Darwin123
> There are two main types of water waves: capillary waves (aka surface
> tension waves, ripples) and gravity waves (like the big waves on the
> ocean). For pure capillary waves, you ignore gravity and assume that
> surface tension provides all the restoring force affecting the surface.
> For gravity waves, you ignore surface tension, and assume it's all
> gravity.
>
> You can do the combined case, and for an infinitely deep pond, the result
> isn't too hard. Check out wikipedia on "capillary wave"; the dispersion
> relation for capillary-gravity waves is given.
>
> The cute thing is that for capillary waves, the group speed is higher than
> the phase speed, while for gravity waves, it's the other way around. For a
> magic wavelength (wikipedia says 2cm - presumably for an infinitely deep
> pond), the group and phase speeds are equal.
>
Thank you for your reference. It answered my question. However,
you said it wrong. For a combined gravity and capillary wave, an
intermediate mixture of the two, the group and phase speeds are equal
at a magic wavelength. At this magic wavelength, there is no
dispersion. So there is the filtering mechanism that I was looking
for. Waves at the other wavelengths just dispersed. Thank you!
>From Wikopedia on "capillary waves."
"Therefore an interesting and common situation occurs when the
dispersion caused by gravity cancels out the dispersion due to the
capillary effect. At wavelength around 2 cm the capillary effect
causes group velocity to equal phase velocity. The dispersion is zero,
and a wave ridge can travel for long distances."
Andy Resnick
> If one throws a pebble into a pond, a concentric sequence of circular
> 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
Timo Nieminen
> For a combined gravity and capillary wave, an
> intermediate mixture of the two, the group and phase speeds are equal
> at a magic wavelength. At this magic wavelength, there is no
> dispersion. So there is the filtering mechanism that I was looking
> for. Waves at the other wavelengths just dispersed. Thank you!
>
> >From Wikopedia on "capillary waves."
> "Therefore an interesting and common situation occurs when the
> dispersion caused by gravity cancels out the dispersion due to the
> capillary effect. At wavelength around 2 cm the capillary effect
> causes group velocity to equal phase velocity. The dispersion is zero,
> and a wave ridge can travel for long distances."
Note that this is not a _filtering_ mechanism. "Dispersion" simply means
that the phase speed depends on the wavelength, rather than being a
constant, and has nothing to do with waves disappearing, or being
absorbed, or such, despite the usage of "disperse" in such a sense in
non-technical English.
For the filtering mechanism, you need to consider the loss of energy due
to viscosity (I guess this is most important at short wavelengths) and
loss of energy due to interaction with the bottom (long wavelengths).
That, and the dependence of amplitude on group speed, is why you should
see a dominant wavelength. This only relates to dispersion insofar as the
group speed matters.