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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