R.J. Sobey, S.A. Hughes r Coastal Engineering 36 (1999) 1736
21
in which ZGrosskopf et al., 1983; Horikawa, 1988. the coefficients are frequency-depen-
dent:
Ep p Z v .
a0 Z v . s
Z 8.
2
2p K p Z v ; zP .
Ep u1Z v .
Eu1 u1Z v . y Eu2 u2Z v .
a1Z v . s
a2 Z v . s
Z 9.
2
p K p Z v ; z P . K u Z v ; z UV .
p K u Z v ; z UV .
Ep u2Z v .
2 Eu1 u2Z v .
b1Z v . s
b2 Z v . s
.
Z 10.
2
p K p Z v ; z P . K u Z v ; z UV .
p K u Z v ; z UV .
Ep pZ v ., Eu1 u1Z v . and Eu2 u2Z v . are the auto-variance spectra estimated from the p, u1
and u2 burst samples. Ep u1Z v ., Ep u2Z v . and Eu1 u2Z v . are the cross-variance spectra from
the same burst samples.
This algorithm has the advantages of convenience, relative simplicity and many years
of routine use. The fundamental reliance on linear theory and on spectral analysis are
both potentially significant constraints. Linear theory compromises the identification and
extraction of nonlinear influences in the data. The practice of spectral analysis insists
that the local kinematics at time t be dependent with almost equal weight on all data
observations at t s 0,D t,2 D t,3D t, . . . ,Z N y 1.D t in the burst sample. For PUV data, D t
is typically 1 s and the ND t duration of the data is typically 20 min, for which N
exceeds 1000 observations. This is a linear and very global Zburst-sample duration.
interpretation. To preserve the physical integrity of the data, a very local interpretation
would be preferable. Locally linear and then locally nonlinear theories are presented in
the following sections.
3. A local linear analysis
A linear but locally focused interpretation is a useful intermediate step, in the spirit of
the Nielsen Z1989. locally linear approximation to irregular wave kinematics from a
water surface trace. The basis is linear theory estimates of dynamic pressure and
horizontal velocity components:
cosh k Z h q z .
pd Z xa , z , t . s r g
hZ xa , t . ,
Z 11.
cosh kh
ka
cosh k Z h q z .
Z v y ka Ua .
hZ xa , t . ,
uaZ xa , z , t . s Ua q
Z 12.
k
sinh kh
in which:
hZ xa , t . s a cosZ ka xa y v t . ,
Z 13.
is the linear water surface, and k and v are related through the linear dispersion
relationship ZEq. Z3...
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