1072

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

The instantaneous flux of horizontal momentum

that occurs during passage of a wave, i.e., the

(mf) across a unit area of a vertical plane oriented

maximum of

parallel to the wave crests is given by

Z gx

pd qu2dz

MFx; t

7

mf x; z; t pd qu2

4

h

that occurs at the wave crest when g(x)=a. Note that

where pd--instantaneous wave dynamic pressure at a

on the surface of a perfectly reflecting, impermeable

specified position; u--instantaneous horizontal water

vertical wall, the horizontal velocity u is zero, and

velocity at the same specified position; q--water

Eq. (7) becomes simply the integral over the water

density.

depth of the dynamic pressure exerted by the wave

Longuet-Higgins and Stewart (1964) defined the

on the wall, or the total instantaneous wave force

component of bradiation stressQ perpendicular to the

on the wall (excluding the hydrostatic pressure

wave crest as the wave momentum flux integrated

component).

over the water depth and averaged over the wave,

Maximum depth-integrated wave momentum flux,

i.e.,

as defined by Eq. (7) with g(z)=a, can be determined

Z

Z

for any surface wave form provided the velocity and

gx

L

1

pd qu2dzdx

5

Sxx

pressure field under the crest can be specified. In

L

h

0

theory, this means that a wave parameter based on

momentum flux has the potential of applying to both

They substituted linear (Airy) theory expressions for

periodic and transient wave types, which may be a

pressure and horizontal velocity and completed the

useful property. More importantly, the physical

integration by applying first-order wave kinematics

relevance of wave momentum flux to force loading

above the still water level, which is not strictly first-

on structures seems logical, thus fulfilling an impor-

order theory. This bextended linear theoryQ resulted

tant criterion for the proposed wave parameter.

in the expression

4.1. Linear (Airy) wave theory

1

1

2kh

qga2

6

Sxx

2

2 sinh2kh

In linear wave theory, dynamic pressure and

horizontal wave velocity are in phase with the sea

surface elevation, and the maximum wave momentum

where L--local wave length; h--water depth from

flux occurs at the wave crest. The first-order

bottom to the still water level; g--instantaneous sea

approximation of depth-integrated wave momentum

surface elevation relative to still water level; z--

flux is found by substituting the dynamic pressure and

vertical coordinate directed positive upward with

horizontal wave velocity at the wave crest from Airy

origin at the SWL; x--horizontal coordinate positive

wave theory into Eq. (7) and integrating from the

in the direction of wave propagation; g--gravita-

bottom only up to the still water level because

tional acceleration; a--wave amplitude; k--wave

kinematics are not specified above still water level

number [=2p/L]. Note that Sxx has units of force

in first-order theory. From linear wave theory with no

per unit length of wave crest.

unidirectional current,

There is significant variation of depth-integrated

wave momentum flux over a wave length from large

coshkh z

positive values at the crest to large negative values

8

pdzcrest qga

in the trough. So instead of adopting a wave-

coshkh

averaged value (i.e., Sxx) which is quite small

and

compared to the range of variation, it is logical

when considering some coastal processes, such as

coshkh z

the wave force loading on structures, to focus on the

9

uzcrest ax

sinhkh

maximum, depth-integrated wave momentum flux

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