EM 1110-2-1100 (Part II)
31 July 2003
c. Mean water level is governed by the cross-shore balance of momentum
1 dSxx
dη
(II-4-20)
'&
ρgd dx
dx
where Sxx is the cross-shore component of the cross-shore directed radiation stress, for longshore homoge-
neous waves and bathymetry (see Equations II-4-34 through II-4-36 for general equations). Radiation stress
both raises and lowers (setdown) the mean water level across shore in the nearshore region (Figure II-4-7).
d. Seaward of the breaker zone, Longuet-Higgins and Stewart (1963) obtained setdown for regular
waves from the integration of Equation II-4-20 as
2π
H2
1
L
η'&
(II-4-21)
8
4π
sinh
d
L
assuming linear wave theory, normally incident waves, and & = 0 in deep water. The maximum lowering of
the water level, setdown, occurs near the break point &b.
η
η
e. In the surf zone, & increases between the break point and the shoreline (Figure II-4-7). The gradient,
η
assuming linear theory (Sxx = 3/16 ρ g H2), is given by
Figure II-4-7.
Definition sketch for wave setup
d (H 2)
dη
3
1
(II-4-22)
'&
dx
16 h % η dx
of & depends on wave decay through the surf zone. Applying the saturated breaker assumption of linear wave
where the shallow-water value of Sxx = 3/16 ρ g d H2 has been substituted into Equation II-4-20. The value
η
height decay on a plane beach, Equation II-4-22 reduces to
dη
1
tanβ
(II-4-23)
'
8
dx
1%
2
3 γb
Combining Equations II-4-21 and II-4-23, setup at the still-water shoreline &s is given by
f.
η
Surf Zone Hydrodynamics
II-4-13
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