1 % ν2

1&

2

2

π*H*(

π *f *(ν)

exp &

1%

4

4

ν2

(II-1-139)

2 (1 % ν2)

;

;

ν

ν%

1 % ν2

with *ν *as the spectral width parameter. The period *T*z is the mean zero-upcrossing period and its relation to

the mean wave period *T *and mean crest period *T*c defined in terms of moments of spectrum is as follows:

;

(II-1-140)

; *T*c ' 2π

(d) The *most probable maximum period *associated with any given *H** is

2 1 % ν2

max

(II-1-141)

'

2

16ν

1%

1%

2

π*H*(

(e) Chakrabarti and Cooley (1977) investigated the applicability of the joint distribution and determined

that it fits field data provided the spectrum is narrow-banded and has a single peak. A different theoretical

model has been suggested by Cavanie et al. (1978), and it also compares well with the field data.

(1) Introduction.

(a) In the period 1950-1960, Rice's (1944-1945) work on signal processing was extended to ocean

waves (Kinsman 1965; Phillips 1977). In pinciple, the time-history of surface elevation (such as in Figures II-

1-31 and II-1-32) was recognized to be similar to a noise record. By assuming that it is a discrete sample of

a continuous process, the principles of Fourier analysis could be extended to describe the record. The power

of Fourier representation is such that given a series of time snapshots of measurements of a three-dimensional

surface, a full mathematical representation of the surface and its history may be obtained. Unfortunately, this

is a lot of information. As an example, the image in Figure II-1-22 of the entrance to San Francisco Bay is

one snapshot of the surface current field and represents nearly 1 million sample points. To understand the

time variation of the field it would be reasonable to do this every 2 sec or so for an hour. The result is about

1.8 billion sample points that would need to be Fourier transformed. Although, this is computationally

feasible such a measurement cannot be made on a routine basis and it is not clear how the information could

be condensed into a form for practical engineering. However, the utility of the spectral analysis approach is

that it uses a reduced dimensional approach that is powerful and useful. This section will discuss the

Water Wave Mechanics

II-1-77

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