In the previous section we used an expression for necessary for normalizing the modes which for completeness we shall now derive. We consider the problem:

where primes denote differentiation with respect to **z**.
We shall write this problem symbolically as,

The Wronskian is defined by:

where are any non-trivial solutions that satisfy the top and bottom boundary conditions respectively. That is,

Let be a solution of the unforced boundary value problem,

Then,

or, equivalently,

This can also be written:

Taking the integral then gives:

We shall need two intermediate results giving the value of the term
in square brackets at **z=0** and **z=D**. To obtain the value at **z=0** we
note that is
constant since,

Thus, we can write:

and solving for one obtains,

This enables us to write,

We can eliminate the derivatives from this equation using the upper boundary condition:

This gives us the value of the term in square brackets in Eq. ()
evaluated at **z=0**.
The value at **z=D** is can be written down directly as:

where we have used the bottom boundary condition,

Using the results of Eqs. () and () in Eq. () we obtain,

where we have added in the term . This is permissible since
, that is, the Wronskian vanishes when **k** is an eigenvalue.

The functions and may all be scaled freely and still satisfy their respective governing equations. Therefore, without loss of generality, we take . Now, dividing both sides of the equation by and taking the limit as we obtain the final result:

Tue Oct 28 13:27:38 PST 1997