Ψ(*x*,η) ' &*Q*

at z ' η

(II-1-101)

where *η *is the sea surface elevation. The dynamic free-surface boundary condition is an expression of

specifying the pressure at the free surface that is constant and equal to the atmospheric pressure. In terms of

the stream function Ψ this condition may be stated as

2

2

MΨ

MΨ

1

% *g*η ' *R*

at *z *' η

(II-1-102)

%

M*x*

M*z*

2

in which *R *is the Bernoulli constant.

(i) The boundary-value problem for wave motion as formulated above is complete. The time-

dependency may be removed from the problem formulation by simply adapting a coordinate system that

moves with the same velocity as the wave phase speed (Fenton 1988; Fenton and McKee 1990; Sobey et al.

1987). This is equivalent to introducing an underlying current relative to which the wave motion is measured.

The current (also called *Stokes' drift velocity *or *Eulerian current*) causes a Doppler shift of the apparent wave

period measured relative to a stationary observer or gauge. The underlying current velocity must therefore

also be known in order to solve the wave problem in the steady (moving) reference frame.

(j) Fenton's solution method uses the Fourier cosine series in *kx *to the governing equations. It is clearly

an approximation, but very accurate, since results of this theory appear not to be restricted to any water

depths. 0 = kH/2 is the expansion parameter replacing *ka *in the Stokes wave theory. The dependent variable

is the stream function Ψ represented by a Fourier cosine series in kx, expressed up to the Nth order as

1

j *B*j

sinh *jk*(*z *% *d*)

2

cos *jkx*

Ψ(*x*,*z*) ' &*u*(*z*%*d*) %

(II-1-103)

cosh *jkd*

where the *B*j are dimensionless Fourier coefficients. The truncation limit of the series *N *determines the order

of the theory. The nonlinear free-surface boundary conditions are satisfied at each of *M+1 *equi-spaced points

on the surface. Wave height, wave period, water depth, and either the mean Eulerian velocity or the Stokes

drift velocity must be specified to obtain a solution.

(k) The solution is obtained by numerically computing the N Fourier coefficients that satisfy a system

of simultaneous equations. The numerical solution solves a set of *2M+6 *algebraic equations to find unknown

Fourier coefficients. The problem is uniquely specified when *M = N *and overspecified when *M > N*. Earlier

wave theories based on stream function consider the overspecified case and used a least-squares method to

find the coefficients. Fenton was the first to consider the uniquely specified case and used the collocation

method to produce the most accurate and computationally efficient solution valid for any water depth.

(l) An initial estimate is required to determine the *M+N+6 *variables. The linear theory provides this

initial estimate for deep water. In relatively shallow water, additional Fourier components are introduced.

An alternative method is used in the shallow-water case by increasing the wave height in a number of steps.

Smaller heights are used as starting solutions for subsequent higher wave heights. This approach eliminates

the triple-crested waves reported by others (Huang and Hudspeth 1984; Dalrymple and Solana 1986).

(m) Sobey et al. (1987) compared several numerical methods for steady water wave problems, including

Fenton's. Their comparison indicated that accurate results may be obtained with Fourier series of 10 to 20

terms, even for waves close to breaking. Comparisons with other numerical methods and experimental data

(Fenton and McKee 1990; Sobey 1990) showed that results from Fenton's theory and experiments agree

II-1-52

Water Wave Mechanics

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