EM 1110-2-1100 (Part II)
30 Apr 02
1 % ν2
1&
2
2
H(
πH(
T(
π f (ν)
exp &
1%
p(H,T ) '
T(
4
4
ν2
(II-1-139)
2 (1 % ν2)
H
T
H( '
;
f (ν) '
;
T( '
ν
H
T
ν%
z
1 % ν2
with ν as the spectral width parameter. The period Tz is the mean zero-upcrossing period and its relation to
the mean wave period T and mean crest period Tc defined in terms of moments of spectrum is as follows:
m0
Tz ' 2π
;
m2
(II-1-140)
m0
m2
; Tc ' 2π
T ' 2π
m1
m4
(d) The most probable maximum period associated with any given H* is
2 1 % ν2
max
T(
(II-1-141)
'
2
16ν
1%
1%
2
πH(
(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.
c.
Spectral analysis.
(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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