S.A. Hughes / Coastal Engineering 51 (2004) 10851104
1089
expressed the 2%-run-up elevation as the following
equation for breaking wave run-up, we see run-up is
function of nop
directly proportional to wave period, slope and the
square root of wave height, i.e.,
0:71
Ru2%
1:86 nop
20
nopV3:0
for
Hmo
R~H 1=2 T tan a
22
Mase also presented empirical estimation formulas
of the same form for Rmax, R1/10, R1/3 and R with
Thus, variation in incident wave height is less
important and water depth at the toe of the structure
different coefficients and slightly different exponents.
slope is not included. A possible explanation for the
It was noted in the Coastal Engineering Manual
success of the Hunt formulation for breaking waves
(Smith, 2002) that Eq. (20) overpredicts the best-fit
may lie in the assumption that broken waves become
line to Holman's (1986) field measurements of beach
selfsimilar during shoaling. Consider two waves
run-up by a factor of two, but the equation did provide
having significantly different wave heights but the
an approximate upper envelope to the measurements.
same value of wave steepness, H/Lo. Depth-limited
Effects of beach permeability, nonuniform beach
slope, wave breaking over sand bars and other factors
breaking will occur at different water depths on the
related to Holman's data were not quantified, and
slope, and the magnitude of the dimensional flow
these could explain the differences between laboratory
kinematic parameters at breaking will be different
between the two waves. However, the good correla-
and field data.
tion between run-up and deepwater Iribarren number
measurements of beach run-up and argued that beach
suggests that depth of initial wave breaking and
slope was not an important parameter for predicting
breaking wave kinematics are not critical for breaking
wave run-up on natural beaches. Plots of relative
wave run-up because ultimately the two different
maximum run-up versus Iribarren number showed no
waves having the same value of H/Lo become similar
better correlation than plots of maximum relative run-
in the surf zone as observed by Battjes (1974a).
up versus wave steepness. Furthermore, maximum rel-
For nonbreaking wave run-up, we should expect
wave kinematics to be more important, particularly for
ative run-up plotted as a function of beach slope exhi-
shallow water nonlinear waves approaching limiting
bited little correlation. Given the problem of defining
beach slope and the slope variability, Douglass sug-
steepness. Wave steepness contained in the deepwater
gested that beach slope be eliminated from the run-up
Iribarren number (H/Lo) does not adequately charac-
equation when applied to beaches, and he proposed the
terize wave nonlinearity in shallow water, so we might
expect poorer results when using no to estimate
following equation based on Holman's data
nonbreaking wave run-up. It is anticipated that water
Rmax
0:12
depth at the structure toe will become an important
sffiffiffiffiffiffiffiffi
21
parameter for nonbreaking wave run-up such as
Hmo
Hmo
shown by Ahrens et al. (1993) in Eqs. (14) and (15).
Lop
1.4. Present study
1.3. Importance of Iribarren number
This paper re-examines existing wave run-up data
for regular, irregular and solitary waves on smooth,
Practically, all present day wave run-up guidance is
impermeable plane slopes. A crude model is used to
given in terms of the deepwater Iribarren number
derive a new wave run-up equation in terms of a
using local wave height, and there can be no question
dimensionless wave parameter (Hughes, 2004) repre-
about its significance when waves break as plunging
senting the maximum, depth-integrated momentum
or spilling waves on the slope. However, run-up data
flux in a wave as it reaches the toe of the structure
for nonbreaking breaking waves that surge up steeper
slope. The goal of the study was to provide an
slopes does not correlate as well to the Iribarren
estimation technique that was as good as existing
number, and instead run-up appears in this case to be
formulas for breaking wave run-up and better at
directly related to wave height. Rearranging Hunt's
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