subject to boundary condition

The problem is symmetric with respect to the two boundaries so the eigen functions fall into two distinct classes: (even mode) one with vertical velocity symmetry with respect to the mid plane and the (odd mode) other with vertical velocity antisymmetry. Even mode has one row of cells along the vertical while odd has two row of cells. Let us assume the solution of (56) of the form

Let then the roots of (58) are given by

Taking the square roots again, the roots are given by

where

re | |||

im |

Here denote the complex conjugate of . From these we have the following relations

**Even solution**

The even solution is given by

From (61) we get

For the nontrivial solution (after some manipulations) we must have

which on simplification gives

which can be written as (by further simplification)

Taking in we can find and .

**Odd solution**

The odd solution is given by

Proceeding as before the we obtain