1094

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

4.0, 6.0 and 10.0 for a total of 100 observations. For

assumed in the derivation of Eq. (37). The root-mean-

purposes of this paper, wave height at the structure toe

squared error between measured and predicted R/h

was estimated by linear shoaling of the measured

was 0.19, excluding Granthem's 1:1 slope data.

wave height recorded at the offshore measurement

For comparison, the same data have been plotted

location. Generally, waves were significantly larger

on Fig. 4 using Hunt's Eqs. (1) and (3). (Note that the

than in Granthem's tests and only those waves with H/

ordinate axis is run-up nondimensionalized by wave

hb1.0 at the toe after shoaling were used for the

height rather than water depth.) The Iribarren number

present analysis.

characterizes run-up very well for mild slopes that

produce low no values implying plunging or spilling

The wave momentum flux parameter was estimated

for all 152 observations using the formula given by Eq.

wave breaking on the slope. As slopes get steeper and

no exceeds 2.0, scatter becomes greater. Granthem's

(28), even though application to waves with H/hz0.8

is not strictly appropriate. A best-fit of Eq. (37) to the

1:1 slope data are problematic in this plot as well.

run-up data set yielded a reasonably simple equation,

3.3. Irregular wave run-up

1=2

R

MF

3:84 tana

38

qgh2

h

Two published irregular wave laboratory data sets

were used to test the simple run-up relationship given

where (3.84 tana) represents the empirical function

by Eq. (37) for irregular wave run-up. Both data sets

CF(a) in Eq. (37). The results are shown in Fig. 3 with

represent normally incident wave run-up on smooth,

impermeable plane slopes. Fig. 5 shows irregular

the straight line representing Eq. (38). Granthem's

data are solid markers and Saville's data are hollow

breaking and nonbreaking wave run-up data for 275

markers. Most of the data follow the trend given by

values of 2% run-up elevation measured by Ahrens

the straight line with the exception of Granthem's

(1981) versus Iribarren number for steeper slopes

ranging between 1/4VtanaV1/1. The solid lines on

results for a 1:1 slope, which have much lower run-up

values than estimated. Waves on this steep slope were

(Eqs. (16) and (17)) given in the Coastal Engineering

probably surging breakers or nonbreaking waves,

Manual. Data corresponding to milder slopes are

which do not conform to the straight-line sea surface

Fig. 3. Regular wave relative run-up versus Eq. (38).

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