Acoustic Resonance in Ducts

Trapped modes in a waveguide with a long obstacle

“…We were to a large extent motivated by , Evans (1992) and Evans & Linton (1992). Though the numerical results of these authors do not extend to very long obstacles, they helped us predict the asymptotics.…”

“…For definiteness we assume throughout the paper (apart from the last section) that the obstacle is a rectangle of length 2a > 0 and width 0 6 2b < 2 (b = 0 corresponds to an infinitely thin rigid plate). Obstacles in the shape of rectangular blocks or plates are standard test cases studied previously by other authors, see, for example, , Evans (1992) and Evans & Linton (1992).…”

Complex resonances and trapped modes in ducted domains

“…Using a numerical relaxation technique Parker (1967) was able to compute the resonant frequencies of these trapped modes and found good agreement with his experimental results. Subsequently the finite-length rigid plate on the centreline aligned with the duct walls became the model problem for trapped modes about slender obstacles and various methods were applied to compute these so-called Parker modes: Franklin (1972) used a variational formulation, Nayfeh & Huddleston (1979), Evans & Linton (1994) and Duan (2004) applied the mode matching method, and Koch (1983), and Woodley & Peake (1999) employed the Wiener-Hopf technique. Very similar problems occur in quantum waveguides, where the trapped modes are known as bound states, cf.…”

Bound states in coupled guides. II. Three dimensions

“…Similar results were also given in Groves (1998) and Davies and Parnovski (1998). The example of a cylindrical sleeve inside a circular cylindrical waveguide with Neumann conditions on all boundaries considered in Evans and Linton (1994) was recalculated using the residue calculus technique and extended to cover different angular variations.…”

“…Using the method of multipole expansions Ursell was able to prove the existence of resonant states with certain angular variation, provided the sphere was sufficiently small. The method presented in this paper is similar to that used by Evans and Linton (1994), who developed an approximate solution for the existence of trapped modes in an infinitely long, rigid, circular cylindrical tube containing a concentric, rigid, open-ended circular cylinder of finite length. Linton and McIver (1998) proved that acoustic resonances can exist when any rigid, thin obstacle is placed in a rigid cylindrical waveguide of constant cross-section in such a way that its normal is everywhere perpendicular to the generators of the cylinder.…”