Re: speed of waves in water

["John S. Denker" <jsd@AV8N.COM>]

On 12/03/2003 07:12 PM, Larry Smith wrote:
> From a student: "How come sound waves travel so fast in water but
> water waves (the visual kind) move so slow?"

What's the speed of sound in steel, and how does it
compare to speed of the visble waves on a steel slinky?

1) For starters, figure out what excitation mode
you're talking about.

This includes the polarization.

2) Then figure out what provides the inertia and
what provides the restoring force for the excitation
of interest.

There is no reason to expect that different excitations
should have the same inertia or the same restoring
force.  Ask a different question, get a different
answer.

A) For sound waves in the bulk water, the physics
is pretty simple:  the polarization is longitudinal,
the inertia is just water moving to and fro, and
the restoring force comes from the spring constant
of compressed water.

B) For surface waves at the air/water interface,
things are about as different as they could possibly
be.  The polarization is not longitudinal ... and it's
not purely transverse, either.  It's a weird rolling
motion.  The inertia comes from the moving water,
but you have to understand the excitation mode to
figure out how much water is involved in the motion.
It depends on the wavelength and/or the depth of the
channel.  The restoring force comes from surface tension
and/or gravity acting on the curvature and/or slope
of the wave, and depends strongly on wavelength.

http://ocw.mit.edu/NR/rdonlyres/Civil-and-Environmental-Engineering/1-138JWave-PropagationFall2000/2DBF5F87-AA4E-42D1-AB68-75262A9C42A7/0/Wpchap4.pdf

Surface waves are extremely dispersive, so you must
choose what you mean by "speed of waves":
Phase velocity?  Group velocity?

C) For the slinky, you can analyze the physics of
what's happening by looking closely at the steel
wire.  The "visible" excitations involve bending
the wire.  Bending in turn can be understood in
terms of compressing one side of the wire while
stretching the other side, so it is neither
entirely the same nor entirely different from
the compression/rarefaction associated with
sound waves in the bulk steel.  It is different
by a "mechanical advantage" factor depending
on the ratio of wavelength to wire-thickness,
or the cube thereof, which is typically enormous.
So we expect the "visible" waves to be enormously
slower than the bulk sound waves.