applicable to deepwater applications. An accurate steady wave theory may be developed by numerically

solving the full nonlinear equations with results that are applicable for short waves (deep water) and for long

waves (shallow water). This is the Fourier approximation method. The method is termed *Fenton's theory*

here. Any periodic function can be approximated by Fourier series, provided the coefficients of the series

can be found. In principal, the coefficients are found numerically. Using this approach, Chappelear (1961)

developed a Fourier series solution by adopting the velocity potential as the primary field variable. Dean

(1965, 1974) developed the stream function theory. The solutions by both Chappelear and Dean successively

correct an initial estimate to minimize errors in the nonlinear free-surface boundary conditions.

(d) A simple Fourier approximation wave theory was introduced by Rienecker and Fenton (1981) and

was subsequently improved by Fenton (1985, 1988; Fenton and McKee 1990). It is an improved numerical

theory that has a range of applicability broader than the Stokes and cnoidal theories. Details of the theory

are given by Reinecker and Fenton (1981) and Fenton (1985, 1988; Fenton and McKee 1990). Sobey et al.

(1987) recasted Fenton's work into a standardized format including currents in the formulation up to fifth

order. The theory has been implemented to calculate wave kinematics and the loading of offshore structures

(Demirbilek 1985). For coastal applications, a PC-based computer code of Fenton's theory is available in

the Automated Coastal Engineering System (ACES) (Leenknecht, Szuwalski, and Sherlock 1992). A brief

description of Fenton's theory is given here; details are provided in ACES.

(e) Fenton's Fourier approximation wave theory satisfies field equations and boundary conditions to

a specified level of accuracy. The hydrodynamic equations governing the problem are identical to those used

in Stokes' theory (Table II-1-2). Various approximations introduced in earlier developments are indicated

in the table. Like other theories, Fenton's theory adopts the same field equation and boundary conditions.

There are three major differences between Fenton's theory and the others. First, Fenton's theory is valid for

deep- and shallow-water depths, and any of the two quantities' wave height, period or energy flux can be

specified to obtain a solution. Second, the Fourier coefficients are computed numerically with efficient

algorithms. Third, the expansion parameter for the Fourier coefficients is 0 = kH/2 rather than 0 = ka, which

is used in Stokes theories. The coefficients are found numerically from simultaneous algebraic equations by

satisfying two nonlinear free-surface boundary conditions and the dispersion relationship. Finding the

coefficients requires that wave height, wave period, water depth, and either the Eulerian current or the depth-

averaged mass transport velocity be specified.

(f) In Fenton's theory, the governing field equation describing wave motion is the two-dimensional (x,z

in the Cartesian frame) Laplace's equation, which in essence is an expression of the conservation of mass:

M2Ψ

M2Ψ

(II-1-99)

2

LΨ '

'0

%

2

2

M*x*

M*z*

where Ψ is the stream function. *Ψ *is a periodic function that describes wave motion in space and time, which

also relates to the flow rate.

(g) Wave motion is a boundary-value problem, and its solution requires specifying realistic boundary

conditions. These boundary conditions are usually imposed at the free surface and sea bottom. Since the

seabed is often impermeable, flow rate through the sea bottom must be zero. Therefore, the bottom boundary

condition may be stated in terms of Ψ as

Ψ(*x*,&*d*) ' 0 at z ' &d

(II-1-100)

(h) Two boundary conditions, *kinematic *and *dynamic*, are needed at the free surface. The kinematic

condition states that water particles on the free surface remain there, and consequently, flow rate through the

surface boundary must be zero. The net flow *Q *between the sea surface and seabed may be specified as

Water Wave Mechanics

II-1-51

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