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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.