1092

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

where

with the coefficients M and N approximated by the

2:0256

empirically determined functions

H

A0 0:6392

29

h

!'0:44

&

H

0:391

32

M 0:98 tanh 2:24

H

h

A1 0:1804

30

h

Even though the empirical coefficients and exponents

!

H

in Eqs. (29) and (30) are expressed to four decimal

N 0:69 tanh 2:38

33

h

places, corresponding accuracy is not implied.

Rounding to two decimal places should be reasonably

adequate for practical application of these empirical

Note that maximum depth-integrated wave momen-

equations.

tum flux for solitary waves is a function of only

The empirical equation represented by Eq. (28),

relative wave height, H/h. The first bracketed term in

along with Eqs. (29) and (30), provides an easy

Eq. (31) arises from the dynamic pressure, and the

method for estimating maximum wave momentum

second term represents the contribution of horizontal

flux for finite amplitude, steady, regular waves. This

velocity to the maximum wave momentum flux.

formulation gives more accurate estimates of the true

The solitary wave estimates of the wave momen-

maximum depth-integrated wave momentum flux

tum flux parameter represent the upper limit of the

nonlinear (Fourier) wave case when h/(gT2) ap-

than linear and extended linear theory because it

better represents the momentum flux in the wave

proaches zero (see Fig. 1). At a value of H/h=0.1,

crest, which is expected to be critical for most

the velocity term contributes only about 7% of the

applications to coastal structures.

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

For irregular wave trains, Hughes recommended

percentage increases to around 38% of the total.

that the wave momentum flux parameter be repre-

sented by substituting frequency-domain irregular

wave parameters Hmo (zeroth-moment wave height)

3. Wave run-up as a function of wave momentum

and T p (peak spectral period) directly into the

flux parameter

empirical Eqs. (28)(30). While this might not be

the best set of irregular wave parameters to use, these

In the following sections the maximum, depth-

frequency-domain parameters are commonly reported

integrated wave momentum flux parameter is corre-

for laboratory and field measurements, and numerical

lated to existing available data of normally incident,

irregular wave hindcast and forecast models output

breaking and nonbreaking wave run-up on smooth,

frequency-domain parameters.

impermeable plane slopes. Included are data for

regular waves, irregular waves and solitary waves.

2.2. Estimates for solitary waves

3.1. Wave run-up deviation

nondimensional wave momentum flux parameter

using first-order solitary wave theory given as

correlation between the time series of wave run-up on

!

2

a beach and the time series of depth-integrated mass

MF

1

H

H

flux within the swash zone. They also noted that the

2

qgh2 max 2

h

h

local depth-integrated momentum flux was balanced

&

!

mainly by the weight of water in the swash zone,

N2

H

M

H

which was approximated as a triangular wedge. Their

1

1

tan

h

2

h

2M

observation suggests that maximum wave run-up on

!'

an impermeable slope might be directly proportional

1

M

H

to the maximum depth-integrated wave momentum

tan3

1

31

3

2

h

flux contained in the wave before it reaches the toe of

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