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

1079

Dividing both sides by qgh2 yields the nondimen-

and the total pressure, which is given at first order

as hydrostatic, is expressed as a function of zs at

sional expression for the wave momentum flux

parameter

the crest as

!

2

MF

1 H

H

2

PTzs qggs zs qgH h zsŠ

24

qgh2 max 2

h

h

&

!

N2 H

M H

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 tan

1

where C--solitary wave celerity gH hŠ;

2M h

2

h

!'

gs--sea surface elevation measured from the sea

1 3 M H

floor; z s--vertical coordinate-directed positive

tan

1

28

3

2

h

upward with origin at the sea floor; h--water

depth from the bottom to still water level; M, N--

The first bracketed term in Eq. (28) arises from the

coefficients that are functions of H/h.

dynamic pressure, and interestingly, this term is

Because the reference coordinate system has its

exactly twice the value of the long wave approxima-

origin on the sea floor, the depth-integrated momen-

tion of the wave momentum flux parameter derived

tum flux definition equation changes slightly to

from linear wave theory and shown in Eq. (15). The

second term represents the contribution of horizontal

Z

gx

velocity to the wave momentum flux parameter.

pd qu2dzs

MFx

25

The coefficients M and N are typically presented in

0

graphical form (e.g., Wiegel, 1964; Shore Protection

Manual, 1984). To accommodate calculations, a

nonlinear curve fit was applied to the plotted curves

and the maximum depth-integrated wave momentum

to produce the following simple equations that give

flux for a solitary wave is found as

reasonable values for M and N

&

!'0:44

Z

Z

Hh

h

H

M 0:98 tanh 2:24

29

MFmax

qgH h zsŠdzs

qgzsdzs

h

0

0

Z

Hh

C2N 2

!

26

q

!2 dzs

H

N 0:69 tanh 2:38

30

M zs

0

h

1 cos

h

The empirically fit equations (solid lines) are plotted

along with the data points taken from the Shore

The first integral in Eq. (26) is the total pressure, and

the second integral is the hydrostatic pressure between

underprediction and overprediction errors for Eq.

the bottom and the still water level. Subtracting the

(29) are 0.018 and 0.023, respectively. Overall, root-

second integral from the first results in depth-

mean-squared error is 0.010. Eq. (30) has maximum

integrated wave dynamic pressure. Performing the

under- and over-prediction errors of 0.010 and 0.006,

integration and substituting for wave celerity, C,

respectively, with overall root-mean-squared error of

results in the expression

0.0056.

The variation of the wave momentum flux param-

eter for solitary waves as a function of H/h is shown

qg 2

MFmax

H 2H h

in Fig. 8. These values represent the upper limit of the

2

!

&

nonlinear (Fourier) wave case when h /(gT 2)

qg H hN 2h

M H

1

tan

approaches zero (see Fig. 4). At a value of H/h=0.1,

2

M

2

h

!'

the velocity term contributes only about 7% of the

1 3 M H

calculated momentum flux, whereas as at H/h=0.8,

tan

1

27

3

2

h

the percentage increases to around 38% of the total. It

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