This shows that the conditions for marginal stability are independent of the Prandtl number . Let us introduce . Then from (38) and (39) we have

Now we prove that for marginally stable disturbances

where the minimum is taken over all real functions for which the integrals exist, and or vanish on boundary which might be rigid or free. For a given value of the minimum is attained when is true eigen functions at marginal state.

First we prove that the is stationary when is an eigen function. Let be variation in for a small variation in of . Here is compatiable with the BCs i.e. and and either or depending on the nature of the suraface at . From (42) we have

where

and

Using integration by parts (44), (45) give

Thus

and conversely if for an arbitrary compatiable with the BCs of the problem then (47) must hold and must be a eigen function of the problem.

Next we show that the lowest eigen value of is a true mimimum. We shall assume that there is a countable infinity of marginal states for each real wave number and we denote them by corresponding to the ( ) with . We first show that is orthogonal to . To show this we multiply

by symmetry in and .

Multiplying

Interchanging and in (49) and subtracting the new equation from (49) gives (using (49)

Let be any pair of functions which satisfies the BCs of the problem (not necessarily eigen functions). We represent as

where

Also assuming uniform convergence of the series we can differentiate term by term to
give
Now using orthogonality relation of and we get

Thus

Equality holds iff all the except are zero. Therefore are the eigen functions that gives the minimum of (42).