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

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steepest 1:1 slope. Existing regular wave run-up

increasing wave nonlinearity, and thus, it is

formulas based on Iribarren number did better than

anticipated that the new run-up formulas for

the new formula for mild slopes but poorer on

irregular waves might give better estimates for very

steeper slopes.

nonlinear waves arriving at the toe of the slope.

Irregular wave data from Ahrens (1981) and

Likewise, there is hope that the wave momentum

flux parameter will prove equally useful for

slope functions for breaking and nonbreaking wave

estimating wave run-up on rough and permeable

run-up corresponding to the 2% run-up elevation.

structure slopes. However, both of these hypotheses

The wave momentum flux parameter was defined in

remain unproven at this time.

terms of the frequency-domain wave parameters

Hmo and Tp. For waves that break as plunging or

Notation

spilling breakers on the slope, a single equation

a1, a2, a3 empirical coefficients

was found covering the slopes in the range 1/30

A0

empirical coefficient

VtanaV2/3. In this formula, the influence of

A1

empirical exponent

structure or beach slope on wave run-up decreases

b1, b2 empirical coefficients

with slope in agreement with observations made by

co

empirical coefficient

C

empirical coefficient

surging waves on steep slopes was limited to the

Cm empirical run-up coefficient

range 1/4VtanaV1/1. Comparison of predictions to

Cp

empirical run-up coefficient

measurements for both breaking and nonbreaking

C1, C2, C3 empirical coefficients

F(a) empirical function of structure or beach slope

irregular wave run-up were good with the exception

of short-period waves breaking on the 1:1 slope. It

e

base of natural logarithm

was hypothesized that the sea surface profile of the

g

gravitational acceleration

run-up wedge was no longer a straight line in this

h

water depth from bottom to the still water

instance, so the crude run-up formula was no

level

longer valid. Estimation of irregular run-up on

H

uniform steady wave height

structure slopes using the formulas given in the

Hlimit steepness limit wave height

Coastal Engineering Manual produced generally

Hmo zeroth-moment wave height related to the area

poorer comparisons to the measurements of Ahrens

beneath the spectrum

(1981).

Ho

deepwater uniform wave height

Maximum run-up of breaking and nonbreaking

Hs

significant wave height for irregular wave

solitary waves on smooth, impermeable plane slopes

train

was adequately predicted using the wave momentum

H1/3 average of the highest 1/3 waves in an irregular

flux parameter for solitary waves. This illustrates the

wave train

wave number [=2p/L]

utility of the wave momentum flux parameter for

k

nonperiodic waves.

KM unknown constant of proportionality

The premise that wave run-up can be estimated

KP

reduction factor to account for slope porosity

as a function of the wave momentum flux parameter

(KP=1 for impermeable slopes)

appears valid based on the data used to develop the

L

local wavelength

empirical formulas in this paper. As noted by

Lo

deepwater wavelength

run-up laboratory data for severe shallow water

irregular wave period Tm

conditions of near depth-limited breaking (0.33V

Hmo/hV0.60) relative to water depth at the structure

spectral period Tp

toe. So we are not really certain how the

Lp

wavelength associated with peak spectral

laboratory-based wave run-up formulas perform for

period Tp

what may be the design run-up condition. The wave

M

coefficient for solitary wave theory (function of

momentum flux parameter includes the effect of

H/h)

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