109
The time averages of the vertically integrated wave induced velocities and v are not zero.
These quantities are known as the wave induced mass flux.
Defining
U = u + u
and
V = v + 5
(A.12)
where
u =
1 ft
r / pdz and
o + fj)J-k0
p{h0 + *?)
the time averaged depth integrated continuity equation is
%¡ + §u(ho + ti) + ^v(h0 + f¡)
1 ft
v ~Ti rr / pi> dz
p{h0 + ri)J-ho
(A.13)
(A.14)
The assumption that the density is constant in both time and space has used in obtaining
both Eq. (A.11) and Eq. (A.14). The assumption of a constant bottom has also been used
in obtaining (A.14). If the bottom is not constant in time a term would be included in
Eq. (A.14).
The x and y momentum equations are handled in the same manner as the continuity
equation above. The x and y direction momentum equations are
du du2 duv duw 1 dp 1 rdrxx drvx drxx.
Tt +17 + air + IT = w* + +17 + -aT}
dv t duv t dv2 t dvw 1 dp 1 (drxv drvv drZ]/
dt + dx + dy + dz
= zir + 1A-
+
}
(A.15)
(A.16)
p dy p dx dy dz
The same procedure is followed for both the x and y equations. Only Eq. (A.15) will be
dealt with here. Integrating the left hand side over the depth and applying the Leibnitz
Rule to take the derivative outside the integral gives
t Tip\ d ft dr] dh0
(LHS> = ~aT +
if
dxj-h0 ndx dx
dy J-
, X dr) . dho
uvdz (ut/) (uv).fco +
h0 dy dy
(UW)l ~ (UW)-ho
(A.17)