21

in which ZGrosskopf et al., 1983; Horikawa, 1988. the coefficients are frequency-depen-

dent:

Z 8.

2

2p *K * p Z v ; *z*P .

Z 9.

2

p *K * p Z v ; *z * P . K u Z v ; *z * UV .

p *K * u Z v ; *z * UV .

2 *E*u1 u2Z v .

.

Z 10.

2

p *K * p Z v ; *z * P . K u Z v ; *z * UV .

p *K * u Z v ; *z * UV .

and *u*2 burst samples. *E*p u1Z v ., *E*p u2Z v . and *E*u1 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 *N*D *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.

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 *.

hZ xa , *t *. ,

Z 11.

cosh *kh*

cosh *k *Z h q *z *.

Z v y *k*a Ua .

hZ xa , *t *. ,

Z 12.

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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