R.J. Sobey, S.A. Hughes r Coastal Engineering 36 (1999) 1736
24
Squaring now and using in the f1 equation establishes a single implicit algebraic
equation in the local wave number k:
2
r g cosh kZ h q z P .
gk
F Z k . s Z uabs y Ua . Z uabs y Ua .
o
o
s 0.
Z 17.
y
pd bs cosh kZ h q z UV .
o
tanh kh
Some special cases of this equation are very revealing. When z P s z UV , Eq. Z17.
becomes:
2
rg
kh
1
X
uabs y Ua
o
uabs y Ua
o
F Z k. s
s 0.
Z 18.
y
Z
.Z
.
pd bs
o
gh
tanh kh
The function khrtanh kh is 1 at kh s 0 Z` wave shallow' water., and approaches kh
asymptotically from above for large kh Z` wave deep' water.. There are solutions only
for:
2
rg
1
uabs y Ua
o
uabs y Ua
o
) 1.
Z 19.
Z
.Z
.
pd bs
o
gh
Solutions do not exist under quite a wide range of conditions. This is a potentially fatal
impediment.
It may be possible to reformulate the locally linear problem to avoid these difficul-
ties. Those time steps where there is no solution might be accommodated by the solution
of Eq. Z14. in the least-squares sense rather than an exact solution. It may also be
advisable to use neighboring observations in the local solutions. From the experience of
the Sobey Z1992. locally nonlinear solution from water surface traces, both of these
measures might help in dealing with observational error bands. All of these, however,
increase the complexity of the locally linear formulation and negate the sole advantage
of a linear formulation, simplicity. A locally nonlinear analysis is viable and is
introduced in Section 4.
4. A local nonlinear analysis
Nonlinear and irregular waves follow a field equation, the Laplace equation:
E 2f
E 2f
s 0,
Z 20.
q
E z2
E xa E xa
together with kinematic and dynamic free surface boundary conditions:
Eh
Eh
X
s 0 at z s h ,
fK swy
y ua
Z 21.
Et
E xa
Ef
1
Z ub ub q w 2 . q gh s 0 at z s h ,
fD s
Z 22.
q
Et
2
and a kinematic bottom boundary condition on a sloping bed:
Eh
w q ua
s 0 at z s yh.
Z 23.
E xa
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