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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
 w2 = gktanhkh
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