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Re: speed of a water wave



On Wed, 31 May 2000, Yvon Jean wrote:

I'm presently teaching the refraction of a water wave but I have not
come across a good explanation of why a water wave travels faster in
deep water. Can anyone provide me with a Web link or a few hints.

On Wed, 31 May 2000, Herbert H Gottlieb wrote:

I'm unable to cite a source now, but I recall learning that particles of
water on the surface travel along the path of a vertical ellipse as a
wave passes by. Particles of water below the surface travel in similar,
but not identical, elliptical patterns that depend of the depth of the
water where the particles are located. Where the water is shallow,
particles encounter friction with the bottom and delay the speed of the
wave up to the surface.

I wouldn't say the effect is due to surface friction, since the governing
equations are usually derived assuming a frictionless surface. Rather,
I'd say that the effect is due to the presence of more water undergoing
the same forcing. To explain, first we must acknowledge that the
dependence on depth is only for shallow water waves (i.e., depth is small
compared to wavelength). For such waves, the pressure gradient (due to
the variation in water height) is independent of depth. Consequently,
even the water at the bottom is undergoing an oscillatory motion (just
with no vertical displacement). With a larger depth of water undergoing
the oscillation, less time is needed to "fill" the water troughs and
"evacuate" the water peaks and so the wave "moves" faster.

I know this is a lot of hand-waving and I'm not 100% this is correct.
Perhaps somebody can provide a more rigorous explanation.

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| Robert Cohen Department of Physics |
| East Stroudsburg University |
| bbq@esu.edu East Stroudsburg, PA 18301 |
| http://www.esu.edu/~bbq/ (570) 422-3428 |
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