series, Fenton (1979) developed a generalized recursion relationship for the KdV solution of any order.

Fenton's fifth- and ninth-order approximations are frequently used in practice. A summary of formulas of

the cnoidal wave theory are provided below. See Fenton (1979), Fenton and McKee 1990), and Miles (1981)

for a more comprehensive theoretical presentation.

(e) Long, finite-amplitude waves of permanent form propagating in shallow water may be described by

cnoidal wave theory. The existence in shallow water of such long waves of permanent form may have first

been recognized by Boussinesq (1871). However, the theory was originally developed by Korteweg and de

Vries (1895).

(f) Because local particle velocities, local particle accelerations, wave energy, and wave power for

cnoidal waves are difficult to describe such descriptions are not included here, but can be obtained in

graphical form from Wiegel (1960, 1964). Wave characteristics are described in parametric form in terms

of the modules *k *of the *elliptic integrals*. While *k *itself has no physical significance, it is used to express the

relationships between various wave parameters. Tabular presentations of the elliptic integrals and other

important functions can be obtained from the above references. The ordinate of the water surface ys measured

above the bottom is given by

,* k*

(II-1-77)

&

where

(g) The argument of *cn*2 is frequently denoted simply by ( ); thus, Equation II-1-77 above can be written

as

(II-1-78)

(h) 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 0 and 1. When *k *= 0, the wave profile becomes a

sinusoid, as in the linear theory; when *k *= 1, the wave profile becomes that of a solitary wave.

(i) The distance from the bottom to the wave trough *y*t, as used in Equations II-1-77 and II-1-78, is given

by

16*d * 2

(II-1-79)

'

&

'

2

3*L*

where *y*c is the distance from the bottom to the crest, and *E(k) *the complete elliptic integral of the second kind.

Wavelength is given by

II-1-38

Water Wave Mechanics

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