Download Governing Equattions

Chapter II. Review of Linear Wave Theory
2.1 Governing Equations for Surface Gravity Water Waves
A rectangular coordinate system is defined such that the plane of x-o-y is coincident with
the still water surface and the z-axis points upward. Assuming that water is
incompressible and flow is irrotational, a potential function is introduced to describe the
velocity induced by surface gravity waves.
V  
(2.1) .
Due to incompressibility of water, the potential satisfies the Laplace equation.
2  0
(2.2)
If the depth of water, h, is uniform and the bottom is impermeable, then the bottom
boundary condition describes the vertical velocity being equal to zero there.

0
z
at z  h.
(2.3)
If the pressure is constant at the free surface and we neglect surface tension force and
Coriolis force, the dynamic and kinematic boundary conditions at the free surface of
water are given by,.

1
2
 g    C (t )
t
2
 

  h  h  0
z t
at z   ,
(2.4)
at z   ,
(2.5)
where  is the surface elevation and depends on x,y and t, C (t ) the Bernoulli constant
which will be properly chosen to ensure that z = 0 located at the still water level and g the
gravitational acceleration. The notations  and h stand for the gradient and horizontal
gradient operators, respectively.
If there is no presence of lateral physical boundary nearby, then a periodic boundary
condition for the surface waves is invoked. Otherwise, lateral boundary conditions are
properly determined based on the physical descriptions of the lateral boundaries.