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For our purposes, a simplified version
of Bernoulli's theorem will suffice - the true version would account for
pressure (due to the atmosphere and surface tension). We suppose that at
all locations on a flowline the sum of the potential and the kinetic energy
is constant. If we assume that the amplitude is small, then we may take
Ry2 to be zero.
| KE+PE = 1/ 2 |v|2+gRycoskX |
| = 1/ 2 ( w/ k
)2-( w2/ k
)Rycothkh coskX+gRycoskX
|
To be constant, this quantity must be independent of X. That is, we
must have all the terms in front of coskX totalling to zero.
| 0 = -( w2/
k
)Rycothkh+gRy |
Rearranging terms we have
As desired we have a relationship between the angular velocity w, the wave number k (and hence wavelength
L = 2p/k) and the depth h.
One consequence is that we can find the speed of the wave, c,
|
c = |
L
T
|
= |
w
k
|
= (
(g/k)tanhkh
)1/2 |
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