S.A. Hughes / Coastal Engineering 51 (2004) 10851104
1101
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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