110
The terms in Eq. (A.17) that are designated by an overbrace are Uf, times the kinematic
free surface condition and those designated by an underbrace are u_j,0 times the bottom
boundary condition. These are both dropped leaving
, d r dr, dr
(LHS) = u udz+d~x u udzJrd~y u uv dz ^
Substituting the definitions given for u and v in Eqn. (A.8) and (A.9) and recognizing that
the turbulent fluctuating components are of a high frequency relative to the wave frequency
so that the time averages of integrals involving the product of a turbulent component and
a wave induced component are zero gives
d~r d~r d ~r d~r
LHS = / udz+ / dz + u?dz+ tfdz (A.19)
dtj~h0 dt]-K0 dxJ-K dxj-h0
d ~r~T~Z d ~r d ~r d ~r
+ -£- (u')2dz + / 2uu dz + uvdz + uiidz (A.20)
dxJ-ho dxj-h0 dyJ-h0 dyJ-ho
d ~r d ~r d ~r
+ 3-/ vdz+ ui)dz + u'v'dz (A.21)
oyj-ho oyJ-h0 dy J-h0
using the definitions of Eqs. (A.12) and (A.13) this becomes
wv{h"+,i)+rvLrdz-y\w^)LrdLriz
(A.22)
where the two terms designated by underbraces result from being added to complete the
U2 and UV terms.
The right hand side after integrating over the depth assuming a constant density and
assuming an inviscid fluid such that no horizontal viscous stresses exist, i.e. rtx and Tyx are
zero, is
RHS =
ir p:+ir a-
p J-ho dx p Jh0 dz
(A.23)
Using the Leibnitz Rule and averaging over a wave period the right hand side becomes
1 d r 1 dn 1 dhn 1 1
RHS = / pdz+-p,,-!-+-p-ho+-Txx --Tiiir; (A.24)
pdxj-h0 p dx p dx p p v '