EM 1110-2-1100 (Part II)
30 Apr 02
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
Mx
Mz
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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