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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