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

1077

Fig. 5. Comparison of linear, extended linear, and finite-amplitude wave momentum flux parameters.

amplitude curves. Thus, the simpler, analytical linear

curves. The resulting, purely empirical equation

theories could be used without significant loss of

representing the curves shown on Fig. 4 is given as

A1

validity, provided the same linear theory was used for

MF

h

all cases. However, one of the stated criteria for this

A0

20

qgh2 max

gT 2

new parameter is that it must be useful for regular

waves, irregular waves, and nonperiodic waves. By

where

using the best estimate of (MF)max for each category

2:0256

of wave, it may be possible in the future to relate

H

A0 0:6392

21

design guidance established for one type of wave to

h

similar circumstances involving other wave types. For

0:391

example, if rubble-mound armor stability can be

H

A1 0:1804

22

related to (MF)max for regular waves, then it may be

h

possible to extend the stability prediction to structures

exposed to transient ship-generated waves or solitary

Although the empirical coefficients and exponents in

waves simply by estimating the maximum depth-

Eqs. (21) and (22) are expressed to four decimal places,

integrated wave momentum flux for the other type of

corresponding accuracy is not implied. Rounding to

wave. For this reason, it is suggested that estimates of

two decimal places should be reasonably adequate for

(MF)max for regular periodic waves be made using the

practical application of these empirical equations.

Fourier approximation method.

Goodness-of-fit of Eq. (20) compared to the

An empirical equation for estimating the wave

computed values given on Fig. 4 is shown on Fig.

momentum flux parameter for finite amplitude steady

6. For smaller values of nondimensional (MF)max,

waves was established using the calculated curves of

there is reasonable correspondence except for the

constant H/h shown in Fig. 4. A nonlinear best-fit of a

left-most points of each curve (shown below the line

two-parameter power curve was performed for each

of equivalence). This divergence was caused by

calculated H/h curve. Next, the resulting power curve

the power curve tending toward positive infinity as

h/gT2Y0. However, greater deviation begins to occur

coefficients and exponents were plotted as a function

of H/h, and fortunately, both the coefficients and

for dimensionless (MF)maxN0.6. Nevertheless, the

exponents could be reasonably represented by power

only poorly fitted curve is for H/h=0.8 which is at

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