24 The Initial-Value Problem

In this section we shall digress a little from the topic of normal modes, which forms the basis of the development of this chapter. In Chapter 1 we introduced the idea of seeking the development in time of an arbitrary small disturbance of some basic flow to test whether the flow was stable. This led to the linearization of the equations of motion and to the resolution of the arbitrary disturbance into independent wave components, each a normal mode. Even if one accepts the approximation of linearization without reservation, however, one would expect to see in practice not only one normal mode but some superposition of many normal modes which is determined by the nature of the initial disturbance. Here we shall elaborate these ideas in the context of the linear stability of steady parallel flows.

We model this situation first by supposing that at some instant, say t=0, there are given arbitrary smooth distributions of u ? and p ? in space subject to the constraint of incompressibility. We then seek to trace the subsequent development of the solutions u ?( x, t) and p ?( x, t) of the linearized equations for t>0 . This is the classic initial-value problem. If u ? and p ? remain bounded for all time we deem the disturbance stable. Of course the initial-value problem is closely related to the spectrum of normal...