108
Integrating over the depth from z h0 to z = r¡ and using the Leibnitz Rule to take
derivatives outside the integrad, the continuity equation becomes
d f* dr] d n
atLpi-p'irt+TzL'il
ho
f \ dr) r \ dh0 d n
-W,s-W-kj7+aiL/'"1
/ s
+ (?*)-fa0)-**. = 0
(A.5)
The terms with an overbrace are p times the free surface kinematic boundary condition
and the terms with an underbrace are p times the bottom boundary condition. Eliminating
both of these from Eq. (A.5) gives
d [i dfi dr* ,
37 / Pdz+z~ pudz + pvdz- 0
dt J-h0 dxj-k0 dy J-h
(A.6)
Equation (A.6) is now time averaged. Time averaging means averaging over a wave
period. The time average of a term is denoted by an overbar. Thus
^ i rT
-U
Fdt
(A.7)
where F represents any term and T is the wave period. Letting u and v be comprised of a
mean flow, a wave induced flow, and an arbitrary or random turbulence component
u = + + u'
V = V + V + V1
(A.8)
(A.9)
The turbulent fluctuations are assumed to be of a high frequency relative to the wave
frequency so that their time averages over the wave period are zero. By substituting the
above expressions for u and v into Eq. (A.6) and taking time averages the continuity equation
is
{p{K + ?)} + ^ {Piho + ?)} + P*dz+ (A.10)
(All)