S.A. Hughes / Coastal Engineering 51 (2004) 10851104

1091

Fig. 1. Wave momentum flux parameter versus h/gT2 (Fourier wave theory).

under the crest when waves become nonlinear. For

the above expression has a maximum error of 0.0014

both linear and extended-linear theory, the relative

over the range of Williams' table. Williams' (1985)

contribution of the horizontal velocity term (qu2) to

tabulation of limit waves is more accurate than the

the total wave momentum flux varies between about

traditional limit wave steepness given by

5% for low amplitude, long period waves to nearly

30% for waves approaching limiting steepness.

Hlimit

0:412 tanhkh

27

To better represent the maximum depth-integrated

L

wave momentum flux in nonlinear waves, Hughes

(2004) used Fourier approximation wave theory for

which overestimates limiting steepness for long

regular steady waves over a horizontal bottom. He

waves and underestimates limiting steepness for short

determined wave kinematics and calculated values of

waves.

the dimensionless wave momentum flux parameter for

An empirical equation for estimating the wave

selected combinations of relative wave height (H/h)

momentum flux parameter for finite amplitude

and relative water depth (h/gT2). Results were plotted

steady waves was established using the calculated

as a family of curves representing constant values of

curves of constant H/h shown in Fig. 1. A nonlinear

H/h as shown on Fig. 1.

best-fit of a two parameter power curve was per-

The dashed line on Fig. 1 gives the steepness-

formed for each calculated H/h curve, and the re-

limited wave breaking criterion tabulated by Williams

sulting coefficients and exponents for each fitted

power curve were also approximated as power

approximation

curves. The resulting, purely empirical, equation

representing the curves of constant H/h shown on

x2Hlimit

a1r a2r2 a3r3

Fig. 1 is given as

co tanh

26

1 b1r b2r2

g

A1

MF

h

2

where r=x h/g, a1=0.7879, a2=2.0064, a3=0.0962,

A0

28

qgh2

gT 2

b1=3.2924, b2=0.2645 and co=1.0575. Sobey noted

max

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