Laboratory measurements indicate that Equation II-1-72 is in agreement with an envelope curve to laboratory

observations (Dean and Dalrymple 1991).

(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 *U*r < 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

Mη

M

(*d *% η)*u *' 0

%

M*t*

M*x*

(II-1-73)

M3u

M*u*

M*u*

Mη

1

' * d*2

%*u*

%*g*

M*t*

M*x*

M*x*

3 M*x * 2M*t*

II-1-36

Water Wave Mechanics

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