Potential flow for waves

A linearized model for the waves can be used for calculating de total force induced by the potential flow, as the integral of the pressure field.

Equation of Continuity:

As the fluid considered is a liquid, it can be assumed ρ=constant , and thus:

Balance of momentum:

Assuming ρ=constant , and neglecting viscous effects (irrotational movement), the previous equations result as:

For solving these equations the boundary conditions to consider are:

Velocity normal to the surface equal to zero, and tangent velocity non-zero:

Calling:

The momentum equations result:

A solution deriving from a potential Φ is taken. It must satisfy Laplace equation

Momentum equation now reduces to Bernoulli equation, taken a constant C as integration constant.

For integration, imposing the boundary conditions, the problem simplifies:

A solution for the Laplace problem of the type Φ=f(z).g(x) can be found:

With:

  • K=L/2π ; wave number
  • ω ; angular frequence
  • ω 2 = K.g.tanh(K.d)

In our particular case, the expression that seems to suit better for the pressures created by the wave over the ship´s hull is:

"z " coordinates will be negative in this case (i.e. the reference system origin is the wave surface, with the Z axis pointing downwards).