For the relatively high frequencies relevant in a turbofan engine duct, the modes of a lined section may be classified in two categories: genuine acoustic 3D duct modes resulting from the finiteness of the duct geometry, and 2D surface waves that exist only near the wall surface in a way essentially independent of the rest of the duct. Per frequency and circumferential order there are at most four surface waves. They occur in two kinds: two acoustic surface waves that exist with and without mean flow, and two hydrodynamic surface waves that exist only with mean flow. The number and location of the surface waves depends on the wall impedance Z and mean flow Mach number. When Z is varied, an acoustic mode may change via small transition zones into a surface waves and vice versa.Compared to the acoustic modes, the surface waves behave-for example as a function of the wall impedance-rather differently as they have their own dynamics. They are therefore more difficult to find. A method is described to trace all modes by continuation in Z from the hard-wall values, by starting in an area of the complex Z-plane without surface waves.

Sound transmission through straight circular ducts with a uniform inviscid mean flow and a constant acoustic lining (impedance wall) is classically described by a modal expansion. A natural extension for ducts with axially slowly varying properties (diameter and mean flow, wall impedance) is a multiple-scales solution. It is shown in the present paper that a consistent approximation of boundary condition and isentropic mean flow allows the multiple-scales problem to have an exact solution. Since the calculational complexities are no greater than for the classical straight duct model, the present solution provides an attractive alternative to a full numerical solution if diameter variation is relevant. A unique feature of the present solution is that it provides a systematic approximation to the hollow-to-annular cylinder transition problem in the turbofan engine inlet duct.

The Ingard–Myers condition, modelling the effect of an impedance wall under a mean flow by assuming a vanishingly thin boundary layer, is known to lead to an ill-posed problem in time domain. By analysing the stability of a linear-then-constant mean flow over a mass-spring-damper liner in a two-dimensional incompressible limit, we show that the flow is absolutely unstable for h smaller than a critical hc and convectively unstable or stable otherwise. This critical hc is by nature independent of wavelength or frequency and is a property of liner and mean flow only. An analytical approximation of hc is given, which is complemented by a contour plot covering all parameter values. For an aeronautically relevant example, hc is shown to be extremely small, which explains why this instability has never been observed in industrial practice. A systematically regularised boundary condition, to replace the Ingard–Myers condition, is proposed that retains the effects of a finite h, such that the stability of the approximate problem correctly follows the stability of the real problem.

Acoustics within mean flows are governed by the linearized Euler equations, which possess a singularity wherever the local mean flow velocity is equal to the phase speed of the disturbance. Such locations are termed critical layers, and are usually ignored when estimating the sound field, with their contribution assumed to be negligible. This paper studies fully both numerically and analytically a simple yet typical sheared ducted flow in order to investigate the influence of the critical layer, and shows that the neglect of critical layers is sometimes, but certainly not always, justified. The model is that of a linear-then-constant velocity profile with uniform density in a cylindrical duct, allowing exact Green’s function solutions in terms of Bessel functions and Frobenius expansions. For sources outside the sheared flow, the contribution of the critical layer is found to decay algebraically along the duct as $O(1/ {x}^{4} )$, where $x$ is the distance downstream of the source. For sources within the sheared flow, the contribution from the critical layer is found to consist of a non-modal disturbance of constant amplitude and a disturbance decaying algebraically as $O(1/ {x}^{3} )$. For thin boundary layers, these disturbances trigger the inherent convective instability of the flow. Extra care is required for high frequencies as the critical layer can be neglected only in combination with a particular downstream pole. The advantages of Frobenius expansions over direct numerical calculation are also demonstrated, especially with regard to spurious modes around the critical layer.

An explicit Wiener-Hopf solution is derived to describe the scattering of sound at a hard-soft wall impedance transition at x = 0, say, in a circular duct with uniform mean flow of Mach number M. A mode, incident from the upstream hard section, scatters at x = 0 into a series of reflected modes and a series of transmitted modes. Of particular interest is the role of a possible instability along the lined wall in combination with the edge singularity. If one of the "upstream" running modes is to be interpreted as a downstream-running instability, an extra degree of freedom in the Wiener-Hopf analysis occurs that can be resolved by application of some form of Kutta condition at x = 0, for example a more stringent edge condition where wall streamline deflection h = O(x 3/2 ) at the downstream side. In general, the effect of this Kutta condition is significant, but it is particularly large for the plane wave at low frequencies and should therefore be easily measurable. For small Helmholtz numbers, the reflection coefficient modulus |R 001 | tends to (1 + M)/(1 − M) without and to 1 with Kutta condition, while the end correction tends to ∞ without and to a finite value with Kutta condition. This is exactly the same behaviour as found for reflection at a pipe exit with flow, irrespective if this is uniform or jet flow. Although the presence of the instability in the model is hardly a question anymore since it has been confirmed numerically, a proper mathematical causality analysis is still not totally watertight. Therefore, the limit of a vortex sheet, separating zero flow from mean flow, approaching the wall has been explored. Indeed, this confirms that the Helmholtz unstable mode of the free vortex sheet transforms into the suspected mode and remains unstable. As the lined-wall vortex-sheet model predicts unstable behaviour for which experimental evidence is at best rare and indirect, the question may be raised if this model is indeed a consistent simplification of reality, doing justice to the double limit of small perturbations and a thin boundary layer. Numerical time-domain methods suffer from this instability and it is very important to decide whether the instability is at least physically genuine. Experiments based on the present problem may provide a handle to resolve this stubborn question.