Mechanism for slow waves near cutoff frequencies in periodic waveguides

Abstract

Slow waves are of considerable topical interest; we show that they can be created within a simple waveguiding structure, a planar waveguide with periodic corrugations, and we describe the physical mechanism responsible. We show that this is a general feature of waveguides in both scalar systems, we use the Helmholtz equation within a guide with either Dirichlet or Neumann wall conditions as our main pedagogic example, and in fully coupled systems such as in-plane elasticity. The presence of slow modes within elastic waveguides has remained unexplored and so potential applications have not been exploited. We demonstrate that the same physics is responsible for the elastic slow modes with a subtle nuance connected to the presence of negative group-velocity modes within the elastic system. In addition to describing the mechanism we derive the frequencies at which they arise using a high-frequency asymptotic algorithm. These asymptotics predict the existence of slow modes associated with a countable set of discrete eigenfrequencies around the cutoff frequencies of an unperturbed waveguide. The asymptotic scheme leads to a natural physical explanation for the presence of such slow modes. Finite element computations for both dispersion curves and associated eigenmodes provide further evidence of the existence of slow modes and allow for comparison with the asymptotics. Many new applications are possible, particularly in elasticity, to generate analogies of optical delay lines. Furthermore the guiding structure we describe is simple to construct.