7.3. FIRST-ORDER WAVE GENERATION
351
which gives
PR _ irpgS2 (
tanh kh
\ .
(1 - cosh khY\2
P’~ 2
kT (.sinhUh + 2khJ [S'n kh+
k(h+lj
(7.81)
or in nondimensional form
tanh kh
sinh 2kh 4- 2kh
....
(1 — cosh kh} 12
sinh kh +
k(h + /)
(7.82)
when Eqn. 7.55 is substituted for A and the linear dispersion relationship
was substituted for a2. Notice that the inertia and static power terms make
no net contribution to the mean power over a wave cycle.
For water on both sides of the wave board,
Poo
2%
tanh kh
sinh 2kh 4- 2kh
....
(1 — cosh fc/i)l2
smht;i+ k(h + D ]
(783)
Expressions for piston-type and flap-type wave boards are found by letting
I —♦ oo and I = 0, respectively.
Figure 7.3 is a plot of the dimensionless mean wave power per unit width
as a function of relative depth for a wave board with water on one side.
A similar plot was given by Dean and Dalrymple (1984). In very shallow
water the long wave approximation yields straight line approximations for
the dimensionless mean wave power given by Eqn. 7.82 as
Po
( pghS*\
\ T )
(for a piston)
(7-84)
irkh
ItT
(for a flap)
(7-85)
Dean and Dalrymple (1984) noted that piston-type wave generators were
more efficient in shallow water because the piston motion more closely approximates the nearly uniform vertical distribution of the horizontal fluid
particles. The flap-type wave board is more efficient for deep water waves.