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

1093

the slope. In this section, a simple derivation is

equation based on the dimensionless maximum wave

performed based on this assumption.

momentum flux parameter, i.e.,

!1=2

1=2

geometry at the point of maximum wave run-up. The

2KPtana

R

MF

!

36

force of the wave has bpushedQ the water up the

qgh2

tana

h

1

KM

impermeable slope. At the instant of maximum run-

tanh

up, the fluid within the hatched area on Fig. 2 has

almost no motion. (Li and Raichlen, 2002 noted this

or more simply

was nearly the case for solitary wave run-up.)

!1=2

Following the lead of Archetti and Brocchini (2002),

R

MF

CF a

37

a simple physical argument is that the weight of the

qgh2

h

fluid contained in the hatched wedge area ABC

where C is an unknown constant and F(a) is a

(W(ABC)) is proportional to the maximum depth-

function of slope angle to be determined empirically.

integrated wave momentum flux of the wave before

For convenience, the bmaxQ subscript has been

it reached the toe of the structure slope, i.e.,

dropped from the wave momentum flux parameter.

KPMFmax KMWABC

34

In the new run-up equation relative run-up (R/h) is

directly proportional to the square root of the wave

where KM is an unknown constant of proportionality

momentum flux parameter. Representing the run-up

and KP is a reduction factor to account for slope

sea surface slope as a straight line is an approximation,

porosity (KP=1 for impermeable slopes).

but, for waves on gentle slopes were wave breaking has

The weight of water per unit width contained in

occurred, this might be a reasonable assumption as

triangle ABC shown on Fig. 2 is given by

!

shown by Li (2000). On steeper slopes where waves

qg R2 tana

behave more like surging breakers, the sea surface

1

35

WABC

2 tana tanh

elevation will have more of a concave shape, also

illustrated by Li. Another simplification in this deriva-

where R is maximum vertical run-up elevation from

tion is the absence of slope friction, and this was shown

SWL, a is structure slope angle, and h is an unknown

by Archetti and Brocchini (2002) to be important for

angle between still water level and run-up water

swash zone run-up processes on mild slopes where the

surface (which is assumed to be a straight line).

wave travels over a much longer distance.

Substituting Eq. (35) into Eq. (34), rearranging and

dividing both sides by h2 yields a new run-up

3.2. Regular wave run-up

The proposed run-up relationship given by Eq. (37)

was empirically fit to existing regular wave run-up

laboratory test results published many years ago by

impermeable-slope tests, waves propagated over a flat

bottom before reaching a linear slope that was varied

between 158 and vertical. Wave heights were some-

what mild with maximum relative wave height of

H/hc0.35. A total of 52 run-up values for slopes

ranging over cota=1.00, 1.43, 1.73, 2.14, 2.75 and

3.73 are used in this reanalysis.

Most of the waves in the experiments reported by

before reaching the impermeable structure slope. Run-

Fig. 2. Maximum wave run-up on a smooth impermeable plane

up values were reported for slopes with cota=2.0, 3.0,

slope.

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