1100

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

triangular wedge derivation presented earlier is not

and predicted R/h was 0.12 for all data, and the RMS

appropriate for steeper slopes and a more accurate

error dropped to 0.034 when the three out-lying points

description of the wedge volume is needed. Never-

were discarded. Also note that structure slope has a

theless, Eq. (43) yielded reasonable estimates for

relatively minor influence for nonbreaking solitary

slopes between 1/30 and 1/10.

waves.

A theoretical run-up equation for nonbreaking

3.4.2. Nonbreaking Solitary Wave Run-up

solitary waves was presented by Li and Raichlen

Nonbreaking wave run-up data from Synolakis

(2001) as

(1986, 1987) for a plane impermeable slope with

5=4

cota=2.08, and data from Hall and Watts (1953) for

R

H

1=2

2:831cota

slopes with cot a=1.0, 2.14 and 3.73 were used to

h

h

Li

examine the utility of the wave momentum flux

9=4

H

parameter for estimating nonbreaking solitary wave

3=2

0:293cota

45

run-up. In this case, relative wave run-up R/h was

h

shown to be directly proportional to the maximum

Predictions using Eq. (45) are compared to the

depth-integrated wave momentum flux parameter with

a very good fit to the 122 data points provided by the

nonbreaking run-up data in Fig. 13. Good corre-

spondence is seen except for the milder slope

with cota=3.73. It is interesting to note that

R

MF

substitution of Eq. (31) for MF/(qgh2) in Eq. (44)

1:82cota1=5

44

qgh2

h

results in an expression containing terms with H/h

raised to powers that are approximately the same as in

Goodness-of-fit is shown in Fig. 12 for the nonbreaking

Eq. (45).

wave run-up data. With the exception of a few outlying

points, the empirical Eq. (44) does remarkably well.

Overall root-mean-squared error between measured

4. Summary and conclusions

The goal of this study was to develop new

formulas for wave run-up on smooth, impermeable

plane slopes based on a new parameter representing

the maximum depth-integrated wave momentum flux

occurring in a wave. These formulas should be as

good as existing formulas for estimating run-up due to

waves that break on the slope and better at estimating

nonbreaking wave run-up.

A crude run-up formula was derived based on the

simple argument that the weight of water contained in

the run-up wedge above still water level at maximum

run-up is proportional to the maximum depth-inte-

grated wave momentum flux in the wave at or near the

toe of the slope. The derived general formula included

an unknown function of slope that needed to be

determined empirically.

Existing published wave run-up data for regular,

irregular and solitary waves were used to establish

the empirical slope functions for the new wave run-

up formulas. Reasonable predictive capability for

Fig. 13. Measured versus predicted nonbreaking solitary wave run-

regular waves was demonstrated for all but the

up--Li formula.

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