3 Hydrodynamics
279
convection term in the vertical equation:
∂w
∂t
= −
1
ρ
∂ p
∂z
− g
In case with horizontal bottom, the water depth H is constant and the
normal velocity at the bottom is zero. That is w b = 0. However, on the
water surface, the vertical velocity of the water mass particle is w s ≈ ∂η
∂t.
Assume that the vertical velocity varies linearly with water depth, then
w(x, z, t) =
z
H + η
∂η
∂t
where η is the height of the wave. Differentiate the above equation with time
t, then ignore the high-order terms and substitute them into the previous
formula to obtain
z
H + η
∂ 2 η
∂t 2 = −
∂
∂z
p
ρ
+ gz
Integral above equation from z to (H + η ) is obtained
p
ρ
= g(H + η − z) +
(H + η) 2 − z 2
2(H + η)
∂ 2 η
∂t 2
This formula is the correction of the distribution law of hydrostatic
pressure at water particles by considering vertical acceleration derived by
Businniques. The equation is substituted into the equation of motion in the
horizontal direction and the small terms are omitted to obtain
du
dt
= −g
∂η
∂ x
−
(H + η) 2 − z 2
2(H + η)
∂ 3 η
∂t 2 ∂ x
Considering the characteristics of shallow water waves, we Integrate the
above equation along the water depth and change u(x, z, t ) into the average
velocity of water depth v(x, t ), that is
v(x,t) =
1
H + η
H +η
0
u(x, z, t)dz