Ripples in ponds and the wave equation

[Steven E. Landsburg]

When you drop a pebble in a pond and see a series of ripples, are
those ripples a) part of the solution to the 2-dimensional wave
equation on the surface of the water, or b) the result of vertical
oscillations of the water emanating from points below where the pebble
was dropped?

The remainder of this post is a more elaborate version of the above
question.

In three dimensions, any spherically symmetric solution to the wave
equation is of the form psi(r,t) = f(r-t)/r + g(r+t)/r. In two
dimensions, this is false.

In particular, consider initial conditions of the form

psi(r,0) = 0
d psi(r,0)/dt = q(r)

Then in three dimensions, the value of psi at a point (r,t) is
determined completely by the values of q at points a such that
(a,0) is on the past "light" cone of (r,t).

By contrast, in two dimensions, the value of psi at a point (r,t)
dependes on the values of q at points a such that (a,0) is either
*on or inside* the past "light" cone of (r,t).

(These facts follow from the Asgeirsson Mean Value Theorem, which
I've quoted at http://www.landsburg.com/appsmr.txt . The 3-dimensional
result generalizes to any odd dimension and the 2-dimensional result
to any even dimension, but here I'll be concerned only with the 2
and 3 dimensional cases.)

Many years ago, I had a conversation with an assistant professor
here at Rochester, in which he observed that sounds---that is,
disturbances in the three-dimensional air---propagate according
to the 3-d wave equation, which is why, when you hum for one
second, I hear the humming (after a slight delay) for exactly one
second. By contrast, a pebble tossed into a pond creates a
disturbance in the two-dimensional surface, which propagates
according to the 2-d wave equation, which is why the initial wave
crest is followed by a series of ripples. If sound waves were
followed by analogous ripples, the world would be a very noisy
place.

That assistant professor has left academics and I can find no trace
of him on the World Wide Web. However, in preparing this post, I've
found two web sites and one paper (by Sigurdur Helgason) that
make essentially the same assertion; I wouldn't be surprised if
the Helgason paper inspired my assistant professor and both web sites.

I've recently been engaged in an extended conversation with a
physicist who doubts that the ripple story is correct. His guess
---based, I am sure, on far better intuition than mine---is this:

1) Although the 2-d wave solution to the wave equation allows for
"ripples" (by which I mean effects of anything outside the past
light cone), these ripples are likely to be small and insignificant
at distances far from the source on the scale of a wavelength.

2) The ripples we actually see on ponds are therefore not part of
the 2-d solution, but instead the result of nonlinear vertical
oscillations of the water at the center of the disturbance initiated
by the dropped pebble.

In partial support of this view, one of my correspondent's colleagues
makes the following observation:

"The reason you can hear the conversations of very distant fisherman
so clearly when you're out fishing on a calm day, is that the
sounds are carried by surface vibrations of the water (the intensity
of which only diminishes as the inverse distance in 2 dimensions rather
than the inverse square of the distance in 3 dimensions), which disagrees
with your description of how sounds would get smeared out in 2D.
(The conversations are not smeared out.)"

So my big question is:

Question 1: Why does a dropped pebble cause ripples? Is it (primarily)
because these ripples are part of the solution to the 2-d wave equation,
or (primarily) because of vertical oscillations in the water, or something
else?

I can imagine two ways to tackle this question. Method A is to actually
solve the 2-d wave equation (at least numerically) for an appropriate
disturbance and see what it looks like. Method B is to drop pebbles into
ponds of various depths; it seems to me that if vertical oscillations
were the culprit, then the ripples would look very different in a shallow
pond than in a deep pond.

Question 2: Has anyone carried out either of these methods? What are
the results?

Question 3: Would either of these methods in fact be definitive or is
there something I'm overlooking?

Question 4: Is there some other way to decide this question?

--

Steven E. Landsburg
http://www.landsburg.com/about2.html

[re...@cornell.edu]

On Oct 22, 5:00 pm, lands...@troi.cc.rochester.edu (Steven E.

Landsburg) wrote:
> When you drop a pebble in a pond and see a series of ripples, are
> those ripples a) part of the solution to the 2-dimensional wave
> equation on the surface of the water, or b)  the result of vertical
> oscillations of the water emanating from points below where the pebble
> was dropped?

> ...
It is certainly b), because the wave equations are not
very applicable to water waves. Usually what one
does is start from the Euler equations for fluid flow, and make
additional
assumptions in an attempt to create a useful model. I think it is safe
to say that there is no universally accepted model.
For example you might imagine a vertical axis where the pebble was
dropped,
and look for radial and vertical velocity components u_r and u_z
defined
for r > 0 and for 0 < z < h(r,t) where h(r,t) is the variable depth.
The Euler equations and mass conservation will hold provided that
u_r = df/dr and
u_z = df/dz for some harmonic function f(r,z,t).
That is the (relatively) easy part.
The boundary conditions are that
u_z is 0 at the bottom z=0,
u_r = 0 at r=0,
u_z = dh/dt+dh/dr u_r at the variable surface, and probably
u_r, u_z go to 0 at large r.
You'll need some initial condition also.
This is a hard problem.