text: In many instances, the oscillatory flow in the fluid interior of surface waves can be described accurately using potential flow theory, apart from boundary layers near the free surface and bottom (where vorticity is important, due to viscous effects, see Stokes boundary layer). Then, the flow velocity u can be described as the gradient of a velocity potential :
  Consequently, assuming incompressible flow, the velocity field u is divergence-free and the velocity potential  satisfies Laplace's equation
  in the fluid interior.
  The fluid region is described using three-dimensional Cartesian coordinates (x,y,z), with x and y the horizontal coordinates, and z the vertical coordinate – with the positive z-direction opposing the direction of the gravitational acceleration. Time is denoted with t. The free surface is located at , and the bottom of the fluid region is at .
  The free-surface boundary conditions for surface gravity waves – using a potential flow description – consist of a kinematic and a dynamic boundary condition.
  The kinematic boundary condition ensures that the normal component of the fluid's flow velocity,  in matrix notation, at the free surface equals the normal velocity component of the free-surface motion :
  The dynamic boundary condition states that, without surface tension effects, the atmospheric pressure just above the free surface equals the fluid pressure just below the surface. For an unsteady potential flow this means that the Bernoulli equation is to be applied at the free surface. In case of a constant atmospheric pressure, the dynamic boundary condition becomes:
  where the constant atmospheric pressure has been taken equal to zero, without loss of generality.
  Both boundary conditions contain the potential  as well as the surface elevation η. A (dynamic) boundary condition in terms of only the potential  can be constructed by taking the material derivative of the dynamic boundary condition, and using the kinematic boundary condition:
  At the bottom of the fluid layer, impermeability requires the normal component of the flow velocity to vanish:
  where h(x,y) is the depth of the bed below the datum  and n is the coordinate component in the direction normal to the bed.
  For permanent waves above a horizontal bed, the mean depth h is a constant and the boundary condition at the bed becomes: