7.4. NONLINEAR WAVE GENERATION
383
Newton’s Rule is applied as
= X™ -
(7.161)
where Xo^ denotes the zth iteration. Performing the necessary operations
yields the iterative equation for Xo
X(‘+1) = x® -
0
q
[Xo’) -
tanh ic(Ct - Xo‘^)j
^1 + ^sech2[«(Ct - Xo‘))j
(7.162)
The maximum positive and negative wave board excursions for a solitary
wave are found by evaluating Eqn. 7.159 at t = ±oo, giving
(7.163)
The total stroke is the distance between the maximum and minimum displacements, i.e.,
IlGHh
\ 3
(7.164)
which gives a wave height-to-stroke ratio for a solitary wave of
H
S~3
3H
16/i
(7.165)
At time t = 0, solution of the iterative Eqn. 7.162 will yield an initial
wave board location given approximately as
x(o)=-4=<7166)
Kfl
V * 3
depending upon when the iteration process is terminated. Goring and
Raichlen (1980) suggested that the total stroke duration time be estimated
as the time it takes the wave board to reach 0.999 times its final position
(given as +H/K.h> ). This requires
/
pp
tanh k ( Ctf — 0.999 —
\ J K,h
where tj is the stroke duration, or
= 0.999
(7.167)
1
tj ~ kC 3.80 +
(7.168)
h J