7.4. NONLINEAR WAVE GENERATION
369
Governing Equations
The governing second-order equations corresponding to the wave board
depicted in Figure 7.1 are listed in Table 7.1. These equations are solved
by assuming the second-order velocity potential can be represented as
<^2 — ^21 + 022
(7.116)
Substitution of Eqn. 7.116 into the second-order governing equations allows
us to separate the original equations into two sets of differential equations.
The first set of equations is given by
<92021
<92021
n
/•
.,x
/7 1 1 71
dxz + dz^ =0
(mflrad)
(7117>
=0
(at z = —h)
(7.118)
dz
d2(/>2i
dt2
,
dfa _
( d2 d3(J>i
+ ■ ’ dz “ 71 V dz *
+ dzdi? J
- 2
+
(at z = 0)
(7.119)
\ dx dxdt
dz dzdt )
The second-order potential 021 must satisfy the inhomogeneous free surface
boundary condition given by Eqn. 7.119, but the solution disregards the
boundary condition at the wave board.
The second set of differential equations corresponding to 022 is given as
+
(in fluid)
(7.120)
dx2
dz2
(at z = —h)
(7.121)
dz
= 0
(at z = 0)
(7.122)
dt2
dz
d22 _
dx
d2\ , ,/ x dXO2
—ar+zw dt
_Xn [ff.lÊZÉl____ L_ÊÉil
(atx = 0) (7.123)
with