1098

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

and thus, were excluded from the empirical

formulation as indicated by the range of applic-

ability on Eq. (40). This data set is evident on Fig. 8.

One possible explanation is that these shorter waves

on the steep 1:1 slope produced a run-up wedge with a

concave sea surface profile that was not well

approximated by the straight-line water surface

hypothesized in Fig. 2. Thus, the derived run-up Eq.

(37) is not appropriate.

(41) to Ahrens' and Mase's observed 2% run-up

values. Mase's data are the hollow circles clustered

toward the lower left corner of the plot. With the

exception of data for slope cota=1.01 and Hmo/

LpN0.0225 (shown by the X-symbol), the prediction

is reasonable. For comparison, Fig. 10 plots Ahrens'

(1981) measurements of Ru2%/Hmo versus estimates

using the prediction equations for steeper slopes given

in the CEM (Eqs. (16) and (17)). The new irregular

Fig. 10. Comparison of Ahrens' (1981) data to predictions using

wave run-up equations exhibit less scatter for the

Eqs. (16) and (17).

Ahrens' data set then the prediction equations

recommended in the CEM.

quite mild (tana=1/30) to fairly steep (tana=2/3).

A single equation, given by either Eq. (40) or

(41), was found to work reasonably well for

Previously, separate equations based on the Iribarren

breaking waves over the entire range of slopes from

number were needed to cover this range of slopes for

breaking waves. Also note that the slope function,

F(a), in Eq. (40) or (41), i.e., (tana)0.7, has the tangent

of the slope angle raised to essentially the same power

as given by Mase (1989) in Eq. (20). As seen on Fig.

8, the 0.7 exponent of tana lessens the influence of tan

a on run-up as slope decreases. This agrees with the

observation of Douglass (1992) that slope angle

plays a less important role for wave run-up on mild

beaches.

3.4. Solitary wave run-up

Run-up of solitary waves on impermeable plane

slopes has been well studied, producing both

theoretical/empirical formulas for maximum run-

up and numerical models of the entire run-up

sequence (e.g., Synolakis, 1986; Li and Raichlen,

(2003) crafted an analytical formulation for tsu-

mani run-up, and they noted the location and

direction of maximum wave momentum flux for

the cases of initial positive and negative wave

Fig. 9. Comparison of Ahrens' (1981) and Mase's (1989) data to

forms.

predictions using Eqs. (39)(41).

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