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

1087

the formula should be restricted to values of nopb 2,

up for wind generated waves is Rayleigh distributed,

V

the SPM proposed a Rayleigh cumulative distribution

and they demonstrated good correspondence to Eq.

for estimating run-up given by

(8) for large-scale laboratory run-up experiments with

plunging breakers on slopes between tana=1/31/8.

1=2

RP

ln P

5

2

Rs

account for various rough slopes (rock, concrete

armor, etc.) and his table was included in the SPM.

up elevation associated with P and R s is the

of irregular wave run-up on smooth, impermeable

significant wave run-up. In other words, the run-up

plane slopes with slope angles ranging between

level exceeded by 2% of the run-ups would be

tana=1/1 and tana=1/4. For the mildest 1-on-4 slope

estimated with P=0.02 and denoted as R0.02. The

where most of the waves broke on the slope as

SPM recommended Rs be estimated as the regular

plunging breakers, Ahrens proposed the run-up

wave run-up value determined from the existing

elevation exceeded by 2% of the run-ups be estimated

nomogram procedures.

using the Hunt formula (Eq. (3)), i.e.,

For many years, the Netherlands used a simple

Ru2%

formula for estimating irregular wave run-up given by

1:6nop

9

Hmo

Ru2% 8H1=3 tan a

6

where

tana

where Ru2% is the vertical elevation from SWL

nop pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

10

exceeded by 2% of the run-ups and H1/3 is the

Hmo=Lop

significant wave height (average of the highest 1/3

and Hmo is the energy-based zeroth-moment wave

waves) at the toe of the structure slope. This formula

height. The coefficient 1.6 is near the high limit given

was valid for milder slopes with tanaV1/3.

by Battjes (1974b). Similar formulas were given for

Battjes (1974b) demonstrated the applicability of

significant run-up and mean run-up. For steeper

the Hunt formula (Eq. (3)) for irregular waves

slopes, Ahrens gave an expanded equation in the form

breaking as plungers on the slope for the 2% run-up

!

!2

level with the formulation

RX

Hmo

Hmo

C1 C2

11

C3

tana

Ru2%

2

2

Hmo

gTp

gTp

nom pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Cmnom

7

where

H1=3

H1=3=Lom

where RX is a place-holder for run-up exceedence

levels (2%, significant and mean), and regression

the mean irregular wave period, Tm. Battjes reported

coefficients C1, C2 and C3 are tabulated for different

the coefficient C m varied from 1.49 for fully

slopes and exceedence levels.

developed seas to 1.87 for seas in the initial stages

In Ahrens (1981), significant wave height was

of development. Prototype measurements by Grune

denoted as Hs, but this value was calculated from the

(1982) expanded the range of Cm to between 1.33 and

measured wave spectra according to the definition for

2.86. Van der Meer and Stam (1992) converted Eq. (7)

Hmo (Ahrens, personal communication). This raises an

to a slightly different form

interesting point with respect to design formula that

use the notation Hs to represent irregular waves.

tana

Ru2%

Cpnop where noV pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

8

V

Unless explicitly stated, the user cannot be certain

p

H1=3

H1=3=Lop

whether Hs means the time-series wave parameter H1/3

the peak spectral wave period, Tp. The range given by

band spectra where the wave heights can be assumed

Battjes for the coefficient Cm was converted to

Rayleigh distributed, H1/3cHmo, and it matters not

1.3VCpV1.7 by assuming the ratio Tp/Tm is approx-

which parameter is used for Hs in the design formula.

imately 1.11.2. Van der Meer and Stam also noted

However, as waves approach incipient breaking, H1/3

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