1A. NONLINEAR WAVE GENERATION
371
-3aff2
32 sinh4 Ar/i
(7.129)
and the periodic velocity potential is
Mx,z,t) -
3aH2 cosh 2k(h 4- z)
32
sinh4 kh
sin 2(fcx — at)
(7.130)
Madsen (1971) also listed an additional term related to the depth of the
water prior to wave generation and a steady current term that satisfies
the requirement for no net mass transport. These terms were subsequently
neglected in the approximate wavemaker solution, so they are not included
here.
Solution Near the Wave Board (<$22)
Near the wave board, Madsen (1970) noted that neglecting the standing
wave summation terms in the first-order expressions for is not
justified unless we restrict the solution to wavelengths where h/L < 0(0.1).
Making this assumption, and substituting Eqns. 7.125 and 7.126 into the
wave board boundary condition given by Eqn. 7.123 gives
dfai
-3aH2k cosh2fc(/i 4- z)
dX02
~Â~ = —17--------- • ,4..— cos2at + W ~li~ +
Ox
16
sinh kh
dt
gHkS0 (
sinhk(h 4- z)\
19n
+ 7------ TTT f(z)k cosh k(h 4- z) 4-----, , —---- cos 2a/ (7.131)
oacoshfc/i \
(n 4-I)
J
where Eqn. 7.47 has been substituted for XO1. In arriving at the above
expression, the trigonometric identity sin2 at = 1/2(1 — cos2ai) was used,
resulting in a nonperiodic term that was subsequently dropped as per Madsen (1970).
Restricting the solution to relatively long waves provided Madsen (1971)
justification for eliminating the z-dependence by depth-averaging Eqn. 7.131
which yields (after substituting Eqn. 7.124 for f(z))
— = fl -
h
A dX°2 4- aH2
3
cos2af (7.132)
dx
\
2(h 4- /) J dt
I6/1 \mi
sinh3 kh J
For convenience, the first-order relationship between wave height and wave
board stroke (Eqn. 7.57) was abbreviated to
H
(7.133)