NONLINEAR WAVES
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TABLE 3.2 Results for Stokes Second Order Wave Theory
Parameter
Formula
Surface wave profile
r, $cos(kz
+
x [2 + cosh (2fcd)] cos [2(fcz — wt)]
Horizontal particle velocity
Vertical particle velocity
Horizontal particle
accélération
Vertical particle
w = -^-^^coS(kX-^
accélération
Hydrostatic pressure
P*- P9Z îxLh2kd{COshï2k(Z +
0
Dynamic pressure
» - “F-•*> *
Wave celerity
c = ( £ tanh fcd)1
Ursell (1953) investigated the accuracy of Stokes second order theory by
comparing the amplitude of the second order term to the amplitude of the first
order term. He has generalized this comparison and expressed it in terms of the
Ursell parameter, defined by
%
A
Ur(3-24)
where rj0 is the maximum élévation above the still water level. When the Ursell
parameter is very small, linear wave theory is valid. However, one should note
that, although the Ursell parameter is a useful guide, it is not the sole measure of
determining the relative importance of the nonlinear terms. In shallow water, for
instance, the relative amplitude H/d becomes the more important parameter.