John W. Miles

On the stability of heterogeneous shear flows

Small perturbations of a parallel shear flow U(y) in an inviscid, incompressible fluid of variable density ρ 0 (y) are considered. It is deduced that dynamic instability of statically stable flows (

Surface-wave damping in closed basins

\rho ^{\prime}_0 (y)\; \textless \; 0

Note on a heterogeneous shear flow

) cannot be other than exponential, in consequence of which it suffices to consider spatially periodic, travelling waves. The general solution of the resulting differential equation is considered in some detail, with special emphasis on the Reynolds stress that transfers energy from the mean flow to the travelling wave. It is proved (as originally conjectured by G. I. Taylor) that sufficient conditions for stability are

Obliquely interacting solitary waves

U^{\prime}(y) \not= 0

On the generation of surface waves by shear flows. Part 5

and

Resonantly interacting solitary waves

J(y)\; \textgreater \frac {1} {4}

On the Reflection of Sound at an Interface of Relative Motion

throughout the flow, where

Surface-wave scattering matrix for a shelf

J(y) = -g \rho^{\prime}_0(y)|\rho (y)U^{\prime 2}(y)

Resonantly forced surface waves in a circular cylinder

is the local Richardson number. It also is pointed out that the kinetic energy of a normal mode in an ideal fluid may be infinite if

Nonlinear Faraday resonance

0 \; \textless \; J(y_c) \; \textless \; \frac {1}{4}