significance, it is used to express the relationships between the varions
wave parameters. Tabular présentations of the elliptic intégrais and
other important fonctions can be obtained from the above references. The
ordinate of the water surface ys measured above the bottom is given by
y5 = yf + H en2 2K(k)
(2-59a)
where y^ is the distance from the bottom to the wave trough, en is the
elliptic cosine function, K(k) is the complété elliptic intégral of the
first kind, and k is the modulus of the elliptic intégrais. The argument
of en2 is frequently denoted simply by ( ), thus, Equation 2-59a above
can be written as
ys = yt + H en2 ( ) .
(2—59b)
The elliptic cosine is a periodic function where cn2[2K(k) ((x/L) - (t/T))]
has a maximum amplitude equal to unity. The modulus k is defined over
the range between 0 and 1. When k = 0, the wave profile becomes a
sinusoid as in the linear theory, and when k = 1, the wave profile
becomes that of a solitary wave.
The distance from the bottom to the wave trough, y^, as used in
Equations 2-59a and b, is given by
Yt yc
H
16d2
H
J = T " I =
K(k) [K(k) - Eî + 1 - T ’
(2-60)
a
d
d
JL
d
where y^ is the distance from the bottom to the crest and E(k) is the
complété elliptic intégral of the second kind. Wavelength is given by
(2-61)
and wave period by
d_
kK(k)
yf i + 2L A E(k)\
y,k2 \2 K(ky
Cnoidal waves are periodic and of permanent form thus L = CT.
Pressure under a cnoidal wave at any élévation y, above
dépends on the local fluid velocity, and is therefore complex.
it may be approximated in a hydrostatic form as
P = Pg (y5 - y),
(2-62)
the bottom
However,
(2-63)
2-48