S.A. Hughes / Coastal Engineering 51 (2004) 10671084

1075

Fig. 3. Wave momentum flux parameter versus h/gT2 (extended linear wave theory).

the wave approaches its limiting relative wave height

given by Williams (1985), is shown by the dashed

(H/h).

line. Percentage contributions from the pressure and

Fourier approximation wave theory (Rienecker

horizontal velocity terms were similar to those noted

and Fenton, 1981; Fenton, 1988; Sobey, 1989)

for linear theory. As expected, estimates of maximum

provides good characterization of steady, finite-

depth-integrated wave momentum flux are greater

amplitude waves of permanent form over the entire

than corresponding estimates from linear theory with

range of water depths from deepwater to nearshore

the largest increase of almost 30% occurring for long

and for wave heights approaching the limiting steep-

waves near the steepness limit. Extended linear theory

ness. This hybrid analytical/computational method-

gives better estimates of (MF)max than linear theory,

ology represents the wave stream function by a

but it still does not account for wave asymmetry about

truncated Fourier series that exactly satisfies the field

the horizontal axis characterized by peaked crests and

equation (Laplace), the kinematic bottom boundary

long, shallow troughs typical of nonlinear waves.

condition, and the lateral periodicity boundary con-

ditions. Nonlinear optimization is used to complete

4.3. Nonlinear (Fourier) wave theory

the solution by determining values for the remaining

unknowns that best satisfy the nonlinear kinematic

Although linear and extended linear wave theories

and dynamic free surface boundary conditions

provide simple analytical estimates of maximum

(Sobey, 1989). Generally, more terms are needed in

depth-integrated wave momentum flux, experience

the truncated Fourier series to represent waves with

tells us that the momentum flux contained in the wave

pronounced asymmetry about the still water line, i.e.,

crest is crucial if we wish to relate this parameter to

steep waves and shallow water waves. Once the

the response of coastal structures in a realistic way.

coefficients of the Fourier series are established for a

The linear theory estimate of maximum wave

particular wave, the kinematics for the entire wave

momentum flux omits that portion of momentum flux

can be easily calculated.

above the still water line, and the extended linear

Because Fourier approximation wave theory pro-

theory neglects the effects of nonsinusoidal wave

vides complete kinematics for finite amplitude waves

forms typical of nonlinear, shallow water waves.

spanning the range covered by Stokes and Cnoidal

These omissions will become more problematic as

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