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

1073

where x is circular wave frequency (=2p/T), and T is

A convenient expression for nondimensional max-

wave period. Strictly, the dynamic pressure term

imum depth-integrated wave momentum flux arises

by dividing Eq. (11) by (qgh2) and recognizing wave

should also include a vertical velocity component

(qw2), so the dynamic pressure is at the same level

height H is twice the wave amplitude, a, i.e.,

of approximation as the horizontal velocity term.

2

MF

1 H tanhkh 1 H

However, at the wave crest, w=0 throughout the water

qgh2 max 2 h

kh

8 h

column and contributes nothing to the dynamic

!

pressure; therefore, the term is not included here.

2kh

1

12

Substituting Eqs. (8) and (9) into Eq. (7) and

sinh2kh

integrating from the bottom to the still water level

For convenience, the nondimensional depth-integrated

yields the analytically continuous expression

maximum wave momentum flux, given as

qa2x2 sinh2kh 2kh

qga sinhkh

MFmax

MF

sinh2kh

k coshkh

4k

qgh2 max

10

will be referred to as simply the bwave momentum

The second term in Eq. (10) is simplified by

flux parameter.Q

Eq. (12) expresses the wave momentum flux pa-

gk tanh kh and making use of the identity sinh

rameter as a function of relative wave height (H/h)

khcosh kh=1/2 sinh 2kh. Thus, the first-order

and relative depth (kh). Fig. 2 presents the nondimen-

theory approximation for maximum depth-integrated

sional parameter for a family of curves representing

wave momentum flux is given by

constant values of H/h. The abscissa on the plot is the

!

commonly used relative depth, h/gT2. For a constant

qga2

qga

2kh

tanhkh

MFmax

11

1

water depth, wave period increases toward the left and

k

sinh2kh

2

decreases to the right. The range of relative depths

covers most coastal applications. The dashed line

which, like Sxx, has units of force per unit length of

gives the steepness-limited wave-breaking criterion

wave crest.

Fig. 2. Wave momentum flux parameter versus h/gT2 (linear wave theory).

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