288

L. Erikson et al. / Coastal Engineering 52 (2005) 285302

up-rush, usu, and back-wash, usb, shoreline (swash

timespace history of the leading edge of the swash

front) velocities,

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cosF G

2yhu

2gyhusinb

dxs

xsut

ln

usut

tanF G

cosG

fu

f

dt

2yhb

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

xsbt

lncoshF

5

fb

2gyhbsinb

dxs

usbt

tanhE

2

f

dt

The maximum run-up height, zm, can be found by

manipulation of Eqs. (2) and (5) and trigonometry:

where usu (t=0)=u0, u0=initial shoreline velocity, t=0

2yhusinb

when the swash front is at the initial shoreline position

ln cosG

6

zm

or still water shoreline, SWS, and usb (t=0)=0 with the

futanb

swash front at the maximum landward position.

Eq. (5) describes a parabolic motion skewed by the

Surprisingly, the time-dependent back-wash equation

friction factors. Following Puleo and Holland (2001)

does not appear to have been published previously.

and as suggested by Hughes (1995), two different

The initial shoreline velocity, u0, is assumed to occur

formulations are used to calculate the friction factors,

at the SWS, and is estimated with (Svendsen and

2

Madsen, 1984),

f

7

h

i

pffiffiffiffiffiffi

10yh 2

2:5ln D

u0 C gH

3

90

where H is the wave height, taken at the SWS, and C

and

is an empirical coefficient describing the resistance.

2

Theoretically, C ranges from 1 to a maximum of 2 for

f

8

2

5:32yhqgs1

no bed resistance, whereas a typical value for a dry

2:5ln

s

sand bed is C=1.83 (Cross, 1967; Miller, 1968; Yeh et

al., 1989). A value of 1.83 for C was used for all

where D90=90th percentile on the cumulative grain

simulations in this study.

size curve, s=ratio of sediment to water density,

The sign in Eq. (1) indicates the direction of the

q=fluid density, and s=shear stress term related to bed

velocity (positive for up-rush and negative for back-

roughness (s=1/2qf|us|us).

wash) and the terms E, F, and G are given by,

Eq. (7) is commonly referred to as the dLaw of the

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

WallT and Eq. (8) is a sediment-laden sheet flow

2gf sinb

1

E t

formulation (e.g., Van Rijn, 1982; Wilson, 1988;

2

yh

Hughes, 1995). The former equation is used when a

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

clear-fluid flow without much sediment transport is

gf sinb

F t

predicted, whereas Eq. (8) is used to describe flow

2yh

resistance with significant sediment transport. In both

"

#

pffiffi

cases yh takes on either yhu or yhb depending on the

u0 f

G tan1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

flow direction. Wilson (1989) showed that sheet flow

4

2gyhsinb

conditions (Eq. (8)) predominate when the value of

the Shield's parameter, H, is greater than 0.8 where H

where yh takes on the value of yhu or yhb depending

is defined as,

on direction and with t=0 at the start of each swash

s

H

9

phase. Integrating Eq. (2) with respect to t using the

qgD50s 1

boundary condition that the shoreline displacement is

where D50=median grain size diameter. The shear

zero (xs=0) at t=0 for the up-rush and xs is equal to the

stress term in the numerator is calculated using Eq. (7)

run-up length at t=0 for the back-wash, yields the

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