1086
S.A. Hughes / Coastal Engineering 51 (2004) 10851104
progressed since 1950. This is by no means a
dimensionally nonhomogeneous equation for maxi-
comprehensive overview of the run-up literature,
mum run-up R given as
and papers describing the more recent advances aimed
tana
R
at modeling wave run-up numerically have been
2:3 rffiffiffiffiffiffi
2
H
H
excluded.
T2
1.1. Regular wave run-up
where T is wave period and a is structure slope angle.
Recognizing that the coefficient 2.3 has units of ft1/2/s,
Among the earlier investigations of wave run-up
Eq. (2) can be expressed as a dimensionally homoge-
are the contributions of Granthem (1953), Saville
neous equation with the introduction of the gravity
constant in imperial units, i.e.,
researchers measured wave run-up caused by regular
wave trains impinging on various types of smooth and
tana
R
R
rough sloping structures, composite slope structures
1:0 pffiffiffiffiffiffiffiffiffiffiffi
1:0no
3
or
H
H
H =Lo
and other variations (stepped, recurved, etc.). Run-up
results were plotted as functions of various wave
parameters and structure slope, but no design formulas
( g/2p)T2, and
were given. Wave run-up estimation guidance was
given in the earliest version of the Corps of Engineers'
tana
Shore Protection Manual (Beach Erosion Board,
no pffiffiffiffiffiffiffiffiffiffiffi
4
H =Lo
1961) as a series of design nomograms, and this
technique for regular wave run-up was propagated for
is defined as the deepwater Iribarren number
over 20 years in essentially the same form (Shore
(Iribarren and Nogales, 1949), also known as the
Protection Manual, 1984), although extensions were
bsurf similarity parameterQ (Battjes, 1974a). Often
made based on a reanalysis of regular wave run-up by
the parameter no is calculated using a finite-depth
local wave height in the vicinity of the slope toe
Early practical formulas for regular wave run-up
rather than a true deepwater Ho. For example, in
on smooth and rough plane slopes and composite
laboratory experiments, it is common to specify H
slopes were presented by Hunt (1959). Curiously,
as the wave height measured over the flat-bottom
Hunt was a Major in the Corps of Engineers
portion of the wave facility before significant wave
stationed in Detroit, MI, but his run-up formulas
transformation occurs due to shoaling. In some
were never included in any version of the Corps'
cases, HcHo, but this is not always assured. For
Shore Protection Manual. Hunt recognized that
the discussion in this paper, we will assume that no
different formulas would be needed to differentiate
is based on the local wave height at or near the toe
run-up caused by nonbreaking waves that surge up
of the slope rather than Ho.
steeper slopes from run-up caused by waves that
break on milder slopes as plunging or spilling
1.2. Irregular wave run-up
breakers.
For surging waves on plane, impermeable slopes,
The capability to predict maximum wave run-up on
a variety of structure slopes and surface types
R
advanced structure design, but the above regular wave
1
c3
methods were not altogether realistic given the
H
irregular character of natural sea states. The impor-
where R is the maximum vertical run-up from SWL
tance of irregular wave run-up on structures was
and H is wave height (assumed to be the deepwater
acknowledged in the 1977 and 1984 editions of the
wave height, i.e., HcHo). Hunt's analysis for the case
SPM (Shore Protection Manual, 1977, 1984). Based
where waves break on the slope resulted in a
on earlier publications suggesting irregular wave run-
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