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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