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

wave theories, it is possible to estimate reasonable

shows curves representing H/h=0.3 and 0.7. For the

values for maximum depth-integrated wave momen-

lower relative wave height of H/h=0.3, there is good

tum flux associated with these waves. Unfortunately,

correspondence between extended linear and Fourier

approximation for values of h/gT2 greater than about

such estimates of (MF)max must be calculated numeri-

cally which could significantly lessen the utility of the

0.03. As the relative depth decreases from 0.03, there

wave momentum flux parameter for design purposes.

is increasing divergence which illustrates the impor-

A simple empirical approximation for the wave

tance of nonlinear wave shape. Linear theory under

momentum flux parameter of finite amplitude waves

predicts extended linear theory by nearly a constant

was developed using a Fourier wave computer

amount.

program. This program was repeatedly run for

For relatively high waves (H/h=0.7), linear and

selected combinations of wave steepness (H/h) and

extended linear estimates clearly underpredict the

relative depth (h/gT2), and the resulting estimates of

correct value of the wave momentum flux parameter.

For example, at a value of h/gT2=0.01, the Fourier

wave kinematics were used to calculate maximum

depth-integrated wave momentum flux according to

approximation estimate of dimensionless (MF)max is

Eq. (7). Results are presented as the set of curves

2.0 times greater then the linear estimate and 1.4 times

shown on Fig. 4. Coding accuracy was checked by

greater than the extended linear estimate. This differ-

assuring that estimates of (MF)max for small ampli-

ence increases as relative depth decreases, emphasiz-

ing the importance of nonlinearities in nearshore

tude, deepwater waves were the same as estimates

waves.

given by the first-order analytical solution. In addi-

Application of the wave momentum flux parameter

tion, it was noted that estimates for very long waves

(small values of h /gT 2) approached the values

to coastal structure design and estimation of coastal

processes will typically involve empirical correlation

obtained from the analytical solitary wave solution

of the parameter with observed responses. It could be

given in the following section. The dashed line on the

argued that the empirical nature of this type of

plot represents the limiting wave steepness given by

application does not depend on absolute values but

rather on relative values of the incorporated wave

The difference between linear, extended linear, and

parameter; and in general, the linear and extended

finite-amplitude theory estimates of the wave momen-

linear curves show similar trends as the finite-

tum flux parameter is illustrated on Fig. 5 which

Fig. 4. Wave momentum flux parameter versus h/gT2 (Fourier wave theory).

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