Dispersion relation and growth expression

By substituting, we have obtained a solution for c in terms of known variables.  Note that , which is arbitrary, is multiplying every term linearly and will cancel.

This can be written more concisely as

This is the dispersion relation for the wave speed.  
The solutions are:

Which should be a set of upstream and downstream traveling waves.  
c1 will be the upstream wave and will not appear in a flow system
c2 will be the downstream wave that we would like to study the behavior of