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

estimating nonbreaking wave run-up. For irregular

(2004) derived formulas for estimating the maximum

waves breaking on the slope, a single formula proved

depth-integrated wave momentum flux for periodic

sufficient for all slopes in the range 2/3VtanaV1/30. A

(regular) waves and solitary waves.

slightly different formula is given for nonbreaking

wave run-up. In addition, two new equations for

2.1. Estimates for periodic waves

breaking and nonbreaking solitary wave run-up are

presented.

An estimate of wave momentum flux in periodic

waves was given by Hughes for first-order wave

theory in nondimensional form as

2. Wave momentum flux parameter

MF

1 H tanhkh

qgh2 max 2 h

kh

2

!

representing the maximum depth-integrated wave

1 H

2kh

24

1

momentum flux occurring in a wave of permanent

8 h

sinh2kh

form, i.e., the maximum over the wave of the integral

The dimensionless parameter to the left of the equal

Z gx

sign represents the nondimensional maximum depth-

pd pu2 dz

MFx; t

23

integrated wave momentum flux, and it is referred to

h

as the bwave momentum flux parameterQ.

Eq. (24) expresses nondimensional maximum

where MF(x,t)--depth-integrated wave momentum

wave momentum flux as a function of relative wave

flux at x and t; pd--instantaneous wave dynamic

height (H /h ) and relative depth (kh ). However,

pressure at a specified position; u--instantaneous

integration over the water depth stopped at the still

horizontal water velocity at the same specified

water level, and Eq. (24) does not include that part of

position; q--water density; h--water depth; x--

the wave above the still water level where a

horizontal direction perpendicular to wave crests;

significant portion of the wave momentum flux is

z--vertical direction, positive upward with z=0 at

found.

still water level; g(x )--sea surface elevation at

An improved estimate of (MF)max was obtained

location x; t--time.

using extended-linear theory in which expressions for

Maximum depth-integrated wave momentum flux

linear wave kinematics are assumed to be valid in the

has units of force per unit wave crest, and Hughes

crest region so the integration could be continued up

speculated this wave parameter may prove useful in

to the free surface at the crest. This resulted in a

empirical correlations relating waves to nearshore

slightly different expression given by

coastal processes occurring on beaches and coastal

structures. He also noted that integration of Eq. (23)

MF

over a uniform periodic wave results in radiation

qgh2

max

Stewart (1964). However, values of MF vary over a

H sinhkh H =2Š

1

wave from large positive values at the wave crest to

kh coshkh

2

h

large negative values in the trough, whereas the value

2

!

of Sxx is relatively small in comparison to the

sinh2kh H =2 2kh H =2Š

1 H

maximum. When considering force loading on coastal

8 h

sinh2kh

structures, perhaps better correlations can be made

25

using a parameter representative of the maximum

force in the wave instead of one corresponding to the

integration over the entire wavelength.

However, the wave form is still sinusoidal rather

Estimates of (MF)max can be made for any wave for

than having peaked crests and shallow troughs typical

of nonlinear shoaled waves and, consequently, the

which sea surface elevation and wave kinematics are

extended-linear theory under-predicts momentum flux

known either through theory or measurement. Hughes

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