1088
S.A. Hughes / Coastal Engineering 51 (2004) 10851104
becomes larger than Hmo and the design formulas will
shallow water structures. Thus, there could be
give different answers. Thus, it is important to
uncertainty regarding wave run-up formula accuracy
determine which parameter is meant by the notation
under these conditions.
Hs.
In the recently available Coastal Engineering
Manual (CEM) (Burcharth and Hughes, 2002), two
data for smooth, impermeable slopes ranging between
sets of design guidance are presented for irregular
tana=1/11/4, and they proposed design formulas for
wave run-up on smooth, impermeable slopes. For
steeper slopes in the range tana=1/11/4, the data of
significant and 2% run-up elevations. These formulas
were then compared to other published data sets, and
Ahrens (1981) was represented by
modifications to the formulas were made where
Ru2%
1:6nop
16
necessary. The final recommended equation for
nopV2:5
for
Hmo
breaking waves (plunging or spilling on the slope)
was given as
and
2:26nop
Ru2%
Ru2%
12
nopV2:5
for
4:5 0:2nop
17
2:5bnopb9
for
1 0:324nop
Hmo
Hmo
Once again, the irregular wave deepwater Iribarren
with nop based on Hmo and Tp. The appearance of nop
number is based on Tp and Hmo at, or near, the toe of
in both the numerator and denominator has no
the slope. For milder structure slopes in the range
physical meaning, it is simply an empirical fit to the
tana=1/31/8, the CEM recommends the guidance of
data. For nonbreaking waves (surging/collapsing),
De Waal and Van der Meer (1992), given by
Ahrens et al. recommended
Ru2%
1:5noV for
18
0:5bnoV V2:0
Ru2%
p
p
H1=3
1:6F0:24
13
nopz4:0
for
Rs
and
with the significant run-up estimated by
Ru2%
3:0noV for
19
2:0bnoV b4
h
i
p
p
H1=3
Rs
3:5
exp 2:48Xp 0:446cosa 0:19P
Hmo
with nop calculated using Tp and H1/3. De Waal and
V
for nopz4:0
14
Van der Meer stated water depth at the toe of the
structure was at least three times H1/3 for all data used
where
to establish Eqs. (18) and (19) so they assumed waves
were Rayleigh distributed. Therefore, it should be
h cota 2
h cota
Hmo=Lp
reasonable to apply Eqs. (18) and (19) using Hmo
Xp
and P
!3
Lp
Lp
2ph
tanh
also developed guidance for run-up on composite
Lp
slopes (berm), and they gave reduction factors for
15
slope roughness, shallow water and incident wave
angle.
and h is the water depth at the toe of the structure, and
All of the wave run-up studies discussed above
pertain to run-up on coastal structures with slopes as
peak period Tp. In the transitional range 2.5bnopb4.0,
mild as tana=1/8. Not as many laboratory experiments
a weighted average technique was proposed. A key
have been conducted for more gentle slopes similar to
point made by Ahrens et al. (1993) was the lack of
those found on natural beaches.
wave run-up data representing severe wave conditions
(0.33VHmo/hV0.60) relative to water depth at the
run-up experiments on mild impermeable plane
slopes of tana=1/5, 1/10, 1/20 and 1/30. He
structure toe which may be the design condition for
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