Question

For the free surface elevation η = A cos(kx − ωt) and associated velocity potential φ = B sin(kx − ωt) cosh k(z + H), use the linearized free surface boundary conditions

∂φ/∂z=∂η/∂t at z=0

∂φ/∂t=−gη at z=0

to find:

(a) The relationship between A and B.

(b) The dispersion relationship.

Answer 1

Given
   
And
  
And so,
   
  
And

And the first condition gives, at z = 0, we have
  
  
    .....................(1)
And the 2nd condition implies, at z = 0, we have

   
      ......................(2)

Equations (1) and (2) are the relationships between A and B.
b)
From equation (1)
  
And from equation (2)
  
And so,
     
  
   
  
This is the dispersion relationship.