Mode coalescence and the Green’s function in a two-dimensional waveguide with arbitrary admittance boundary conditions

Analytic mode-matching for accurate handling of exceptional points in a lined acoustic waveguide

“…The location of exceptional points in the complex plane has been discussed in the literature, and a thorough description can be found in the recent paper [23] by the present authors. The reader is reminded that EP2s form a continuum [23]. That is, to a given arbitrary complexvalued transverse wavenumber ᾱ, there corresponds a unique pair of wall parameters obtained from…”

“…no longer form a complete set, and this is indicative of the existence of further (non-separable) waveform. Following [20,23], the new wave function can be defined by differentiation with respect to s, that is: and it is easy to see that…”

“…where, following the notation of [23], the overbars indicate that these are EP2 values of the parameters (i.e. the overbar does not indicate the complex conjugate).…”

“…These functions, analogous to Jordan generalized vectors in a finite-dimensional vector space, compensate for the loss of one or more dimensions in the space spanned by the natural eigenmodes. It is worth commenting that, at an EP, the additional wavefunctions introduce a linear (EP2) or quadratic (EP3) spatial growth [20][21][22][23], which moderates the exponential attenuation-although clearly this dominates at large distance from the source. This means that the attenuation depends not only on the source type but also on how it couples to the additional wavefunctions, as discussed in [19] for acoustic waveguides or in [24] for the photonic context.…”

“…To this end, two of the present authors have recently proposed a numerical algorithm which enables the trajectories of the eigenvalues in the vicinity of an exceptional point to be explored in a systematic way for discrete systems, such as those arising after the finite element discretization [25]. This work was followed by a comprehensive analysis of the behaviour of the transverse wavenumbers of a two-dimensional waveguide lined along both sides, as they coalesce in the vicinity of an optimal point [23]. In the latter article, numerical investigations suggest that the first EP3 (resulting from the coalescence of the first three modes) corresponds to maximum modal attenuation.…”