23 Flows with Piecewise-Linear Velocity Profiles

The eigenvalue problem defined by equations (21.17) and (21.18) is difficult to solve explicitly when U(z) is a smoothly varying function. When U(z) is piecewise linear, however, the solutions of Rayleigh s equation are simple exponential or hyperbolic functions which must satisfy certain matching conditions at a discontinuity of U(z) or U'(z). The use of piecewise-linear profiles thus provides a simple method of modelling some features of smoothly varying profiles.

Suppose then that U or U ? are discontinuous at z=z ₀ (say) and let ? f= f( z ₀+0) ?f( z ₀ ?0) denote the jump in f(z) at z ₀. To derive the first matching condition rewrite Rayleigh s equation in the form

On integrating this equation across the discontinuity from z ₀ ? ? to z ₀+ ? and then letting ? ?0 we obtain

This condition also follows immediately from equation (21.16) by requiring that the pressure be continuous across the material interface, i.e. ? =0 at z=z ₀+ ?(x, t), where ?(x, t)= ? ₀ exp {i ? (x ? ct)}. To first order in the small perturbation ? ₀ this gives equation (23.2).

To derive the second matching condition divide the pressure equation (21.16) by (U ?c) ² to get

On integrating across the discontinuity, we have

The right-hand side...