EM 1110-2-1100 (Part II)
30 Apr 02
Laboratory measurements indicate that Equation II-1-72 is in agreement with an envelope curve to laboratory
observations (Dean and Dalrymple 1991).
e. Other wave theories.
(1) Introduction.
(a) Extension of the Stokes theory to higher orders has become common with computers, but the
mathematics involved is still tedious. Variations of the Stokes theory have been developed in the last three
decades oriented toward computer implementation. For example, Dean (1965) used the stream function in
place of the velocity potential to develop the stream function theory. Dean (1974) did a limited comparison
of measured horizontal particle velocity in a wave tank with the tenth-order stream function theory and
several other theories. Forty cases were tabulated in dimensionless form to facilitate application of this
theory.
(b) Others (Dalrymple 1974a; Chaplin 1980; Reinecker and Fenton 1981) developed variations of the
stream function theory using different numerical methods. Their studies included currents. For near-breaking
waves, Cokelet (1977) extended the method of Schwartz (1974) for steep waves for the full range of water
depth and wave heights. Using a 110th-order theory for waves up to breaking, Cokelet successfully computed
the wave profile, wave celerity, and various integral properties of waves, including the mean momentum,
momentum flux, kinetic and potential energy, and radiation stress.
(2) Nonlinear shallow-water wave theories.
(a) Stokes' finite amplitude wave theory is applicable when the depth to wavelength ratio d/L is greater
than about 1/8 or kd > 0.78 or Ur < 79. For longer waves a different theory must be used (Peregrine 1976).
As waves move into shallow water, portions of the wave travel faster because of amplitude dispersion or
waves travel faster because they are in deeper water. Waves also feel the effects of frequency dispersion less
in shallow water, e.g., their speed is less and less influenced by water depth.
(b) For the mathematical representation of waves in shallow water, a different perturbation parameter
should be used to account for the combined influence of amplitude and frequency dispersion (Whitham 1974;
Miles 1981; Mei 1991). This can be achieved by constructing two perturbation parameters whose ratio is
equivalent to the Ursell parameters (Peregrine 1972). The set of equations obtained in this manner are termed
the nonlinear shallow-water wave equations. Some common wave theories based on these equations are
briefly described in the following sections.
(3) Korteweg and de Vries and Boussinesq wave theories.
(a) Various shallow-water equations can be derived by assuming the pressure to be hydrostatic so that
vertical water particle accelerations are small and imposing a horizontal velocity on the flow to make it steady
with respect to the moving reference frame. The horizontal velocity might be the velocity at the SWL, at the
bottom, or the velocity averaged over the depth. If equations are written in terms of depth-averaged velocity
u they become:
Mη
M
(d % η)u ' 0
%
Mt
Mx
(II-1-73)
M3u
Mu
Mu
Mη
1
' d2
%u
%g
Mt
Mx
Mx
3 Mx 2Mt
II-1-36
Water Wave Mechanics
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