Internal Waves 



519 



In nature we deal with cases in which the surface of the upper layer must 

 be considered as a free surface. In this case we can give the velocity potentials 

 of the two superposed layers the form 



(XVI. 8) 



<p = Ccosh(z + h)cosxxe' at , | 



cp' = (Acoshxz^ Bsinhxz)cosxxe' al , J 



in which // and /;' are the thickness of the lower and upper layers respectively. 

 Where we deal with internal waves occurring at the boundary surface be- 

 tween two homogeneous layers of different density, the velocity components 



Fig. 213. Streamlines and orbits in a progressive internal wave travelling from left to right 



at the boundary of two fluids. 



must satisfy the kinematic and dynamic boundary conditions both at the 

 free surface and at the internal boundary surface, and also the equation of 

 continuity for both layers. This leads to a quadratic equation for the velocity 

 of propagation c of waves with a wave length X — 2n/x. 



c 4 (QCOthxhcothxh' + Q')-c 2 Q(coth>ch' +cothxh)^ +(q—q')^ = Q. (XVI. 9) 



It is shown by this equation that for each wave length two different types 

 of wave may exist. If the difference in density at the discontinuity surface 

 is small, the two following equations are the approximate roots: 



