Complex resonances and trapped modes in ducted domains

Abstract

Owing to radiation losses, resonances in open systems, i.e. solution domains which extend to infinity in at least one direction, are generally complex valued. However, near symmetric centred objects in ducted domains, or in periodic arrays, so-called trapped modes can exist below the cut-off frequency of the first non-trivial duct mode. These trapped modes have no radiation loss and correspond to real-valued resonances. Above the first cut-off frequency isolated trapped modes exist only for specific parameter combinations. These isolated trapped modes are termed embedded, because their corresponding eigenvalues are embedded in the continuous spectrum of an appropriate differential operator. Trapped modes are of considerable importance in applications because at these parameters the system can be excited easily by external forcing. In the present paper directly computed embedded trapped modes are compared with numerically obtained resonances for several model configurations. Acoustic resonances are also computed in two-dimensional models of a butterfly and a ball-type valve as examples of more complicated geometries.