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
does not appear to have been published previously.
The initial shoreline velocity, u0, is assumed to occur
formulations are used to calculate the friction factors,
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
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
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
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