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S.A. Hughes / Coastal Engineering 51 (2004) 10671084
4.2. Extended linear wave theory
(1998) as the rational approximation
x2Hlimit
a1r a2r2 a3r3
kinematics are valid above the still water level, it is
cotanh
13
1 b1r b2r2
g
possible to derive a somewhat more accurate estimate
of maximum depth-integrated wave momentum flux
where r=x2h/g, a1=0.7879, a2=2.0064, a3=0.0962,
at the wave crest. This technique has been referred to
b1=3.2924, b2=0.2645, and co=1.0575. Sobey noted
as extended linear theory or one-and-a-half-order
the above expression has a maximum error of 0.0014
wave theory.
Substituting Eqs. (8) and (9) for p d and u ,
tabulation of limit waves is more accurate than the
respectively, in Eq. (7), integrating from z=h to
traditional limit steepness given by
z=a (wave crest), applying the dispersion relation
x2=gk tanh kh, and making use of the identity sinh kh
Hlimit
0:142 tanhkh
14
cosh kh=1/2 sinh 2kh as before yields
L
qga sinhkh aŠ qga2
Eq. (14) overestimates limiting steepness for long
MFmax
coshkh
k
2
waves and underestimates limiting steepness for short
!
waves.
sinh2kh aŠ 2kh a
17
The relative contribution of the velocity term (qu2)
sinh2kh
to the total depth-integrated wave momentum flux
varies between about 5% for low-amplitude long-
Dividing Eq. (17) by (qgh2) and substituting a=H/2
period waves to nearly 30% for waves approaching
gives the nondimensional form of the maximum
limiting steepness. Linear theory estimates of max-
depth-integrated wave momentum flux parameter for
imum depth-integrated wave momentum flux are
extended linear theory, i.e.,
lower than actual because the momentum flux above
the still water level is neglected.
2
H sinhkh H =2Š 1 H
MF
1
As the wave period increases, and the wave length
qgh2
kh coshkh
2
h
8 h
becomes very long (shallow water waves), the wave
max
!
number approaches zero, and Eq. (12) approaches a
sinh2kh H =2Š 2kh H =2
limiting value for the wave momentum flux parameter
sinh2kh
given by
18
2
MF
1
H
1
H
The asymptotic long-wave limit of Eq. (18) as
qgh2
2
h
4
h
max
kY0 is given by
for very short waves
15
2!
MF
1
H
1 H
This limit is evident on the ordinate axis of Fig. 2.
qgh2
2
h
4 h
Similarly, Eq. (12) approaches an asymptotic form for
max
!
1 H
very short period waves given by
1
2 h
2
1
MF
1
H
h
1 H
for very long waves
19
qgh2 max 8p2 h
gT 2
8 h
for very short waves
16
which is seen to be an extension of the linear theory
long-wave limit.
However, this deepwater limit is of little interest when
Fig. 3 plots the extended linear theory solution for
considering nearshore coastal processes or coastal
the wave momentum flux parameter as a function of
relative depth h/gT2. The limiting wave steepness, as
structures.