On solutions of the burns condition (which determines the speed of propagation of linear long waves on a shear flow with or without a critical layer)

Abstract

The work described here is concerned with the solution of the classical Burns condition, written in the form

where c is the speed of propagation of linear long waves on the surface of a given shear flow, U(z). The bar on the integral denotes that the finite part of the integral is to be taken if critical-layer solutions arise. The main aim of this paper is to collect together some results which may prove useful in the study of critical-layer problems of this type. A few simple examples which give two, three (or even four) solutions are presented, for which direct integration is possible; a model boundary-layer profile is also analysed in the limit as the boundary-layer thickness approaches zero. One example for which exact integration is possible involves a parameter which allows the profile to vary between a simple quadratic function of (1-z) and a simple cubic function. This profile admits either two or three solutions for c, depending on the value of the parameter. (There are always at least two solutions if U(z) is a monotonic function: one in c Umin and one in c Umax.)

These particular results are extended to the class of profiles which satisfy U'(z)>O, U”(z) <O, O≤z< l, and U(0) = 0, U(l) = 0. In our formulation the power-law profiles

are rather special; they are discussed in some detail and it is shown that critical-layer solutions exist if N > 2 but not if l < N ≤ 2. It is further shown in general, for the above class, that if U″(l) < 0 and U″(1) > O then at least one critical-layer solution exists. An example which makes use of this result is included.

Key Words:

• Burns condition
• critical layer
• water waves.