Tunnelling effects for acoustic waves in slowly varying axisymmetric flow ducts

Abstract

a b s t r a c tThe multiple-scales Wentzel-Kramers-Brillouin (WKB) approximation is used to model the propagation of acoustic waves in an axisymmetric duct with a constriction in the presence of mean flow. An analysis of the reflection/transmission process of modes tunnelling through the constriction is conducted, and the key mathematical feature is the presence of two turning points, located at either real axial locations or in the complex plane. The resulting asymptotic solution consists of WKB solutions in regions away from the constriction and an inner solution valid in the near vicinity of the constriction. A solution which is uniformly valid throughout the duct is also derived. A range of test cases are considered, and the importance of accounting for the inner region, even in cases in which the turning points lie away from the real axis, is demonstrated.

The acoustic Green's function for swirling flow in a lined duct

“…If we let Z j → ∞ then the final terms in Eq. (21) and Eq. (22) become zero and we recover the hard wall boundary conditions from Posson and Peake [25].…”

“…Our method in the previous subsection to find the eigenmodes is only applicable when q n (r, κ) had at most a single zero close to the duct. There is no simple analogue to the uniformly-valid Langer solution when q n (r, κ) has two or more zeros; a uniformly-valid solution can be found using parabolic cylinder functions [21], but it is difficult to implement.…”

Wave propagation in slowly varying waveguides using a finite element approach

“…Uniformly valid approximations, i.e., solutions valid close and far from the critical section can be derived. A single turning point can happen in different positions for different frequencies [13,14]), or it is possible to have the presence of more than one turning point, creating the effect of wave tunnelling, if two turning points are close together [8]. Assuming a time harmonic solution, , e , it is possible to define a local wavenumber .…”

“…Moreover, randomly varying material and geometric properties along the axis of propagation play a significant role in the so-called mid-frequency region. Wave solutions for non-homogeneous waveguides can be found by applying the classical WKB approximation [5][6][7][8]. Named after Wentzel, Kramers and Brillouin, it was initially developed for solving the Schrödinger equation in quantum mechanics.…”

“…In this region, the WKB approximation is no longer valid and a uniformly valid solution is required (e.g. [8,13,14]). In engineering applications, manufacturing variability introduces variability in homogeneous or periodic structures affecting the dynamic performance of the designed structure [15].…”