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Derivation of the Normalization Formula

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,


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:

Thus Eq. (gif) becomes:


This gives us the value of the term in square brackets in Eq. (gif) 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. (gif) and (gif) in Eq. (gif) 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:

Michael B. Porter
Tue Oct 28 13:27:38 PST 1997