Sound propagation in straight ducts with nonuniform mean flow is of great practical interest (aircraft engines, airco-units). If the mean flow and duct wall boundary conditions don't change axially (inviscid parallel flow, uniform wall impedance), the prevailing equations (LEE) allow a discrete set of self-similar time-harmonic solutions called "duct modes", which satisfy an eigenvalue equation, named the Pridmore-Brown equation. (N.B. This set does not form a complete basis. It is a subset of all possible solutions.)

In part 1 we present a (Galerkin-type) numerical solution, and an approximate solution. We discuss the possible shielding of the (then ineffective) lining, due to refraction by the shear flow, and aspects of the limit for thin boundary layers.

Then in part 2 we consider the WKB solution of a slowly varying PB-mode in a slowly varying duct with shear flow. In the WKB approach it is assumed that the wave retains its shape. If the duct+medium change slowly enough (meaning little change over one wavelength), the modal order, i.e. the index of the solution of the dispersion relation that defines the mode, remains the same, and the mode does not jump to another eigenstate. The wave function may gradually stretch or compress but the number of nodes and anti-nodes will not change. Under suitable conditions this may be represented mathematically by conserved quantities called adiabatic invariants. If an adiabatic invariant can be found (this may be far from trivial, or not possible at all) the WKB solution is almost as transparent as its straight duct counterpart. For sound in shear flow adiabatic invariants are unexpected (although not impossible) because there is no energy conservation. Indeed no (non-trivial) ones have been found yet in 3D. In 2D, on the other hand, we found an incomplete adiabatic invariant for general flow profiles, and a complete one for a linear flow profile. For illustration some examples are produced by the method of part 1.

Numerical and Asymptotic Solutions of the Pridmore-Brown Equation, AIAA Journal, 58(7), 2020, 3001-3018

Slowly Varying Modes in a 2D Duct with Shear Flow and Lined Walls, JFM 906, 10 January 2021, (Open Access)