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.
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.
| KE+PE = 1/ 2 |v|2+gRycoskX
| = 1/ 2 ( w/ k
)2-( w2/ k
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,
| 0 = -( w2/
c = ||
<<previous | next>>