7.4. NONLINEAR WAVE GENERATION
385
0c
ht
h
t
H
T
L
Xo
K{rn)
cn
m
cnoidal wave sea surface elevation
distance between cnoidal wave trough
and bottom
water depth
time
cnoidal wave height
cnoidal wave period
cnoidal wavelength
wave board position as a function of time
complete elliptic integral of the first kind
Jacobian elliptic function
elliptic parameter
If we assume that tjc — Hf(0), Eqn. 7.169 gives
/P) = (htrr h>) +cn\dc,m)
(7.171)
n
and substitution of Eqn. 7.171 for f(0) in Eqn. 7.153 gives
(7.172)
/X. \^TTijh Jq
n
which is integrated to get an implicit equation for wave board displacement,
i.e.,
X°^
2K(m)h (/it - h)0c + - (£(0C, m) - (1 - m)0c)
(7.173)
where E(0c,m) is the second incomplete elliptic integral. Goring (1979)
used Newton’s Rule to derive the following iterative equation for 0C
[+ (ht
+ ILE(0&, m)l
0(«+i) _ &(i) _ _L___T______ _______ m__ ______ m________ £ (7.174)
ht + 77cn2(0c’\ m)j
After determining 0C for a specified time, the wave board displacement is
found from Eqn. 7.170 as
/1
0C
Xo(t) = L\^-(7.175)
Typical cnoidal wave board trajectories have a distinct asymmetrical
sawtooth” shape characterized by a rapid forward displacement to the