4.2. SHORT-WAVE HYDRODYNAMIC MODELS
111
wave period the mean energy flux through the infinitesimal width is zero,
thus the total mean rate of energy loss is balanced by the work done on the
fluid by the pressure forces on the narrow cross-section, i.e.,
The average rate of energy loss due to the viscous boundary layer under
a small amplitude linear wave was given by Boussinesq in 1878 as
dE dW
ïï+t = Ci
<4-5I>
where
de
p /ttp
dt = ~2 V ~T ^U° + V°^
(4‘52)
v - fluid kinematic viscosity
T - wave period
P ~ fluid density
and u0 and v0 are the sinusoidal velocity maxima given as
with
agk cosh k(z + h)
u°=
lu
<4’53)
a
cosh kh
agk sinh k(z 4- h)
. t
v° =
k u
(4-54)
a
cosh kh,
n
q ?s~
«a
a
1
1
1
1
1
1
wave amplitude
gravity
wave numb er
circular wave frequency
water depth
vertical coordinate positive upward with z — 0
at still water level and z = — h at the bottom
For a channel of width, B, the mean viscous dissipation over the bottom
area, B dx, is
evaluated at z = — h. Recognizing that vo is zero at the bottom, and
substituting for u0 from Eqn. 4.53, the expression for the mean bottom
energy dissipation becomes
d -^^d ±Bdx = -P -J^{ul + iT)Bdx
(4.55)
at
at
z y J