A multimodal method for non-uniform open waveguides

We present an original method for the simulation of ultrasonic waves in a varying cross section elastic waveguide embedded in a homogeneous matrix. The so-called multimodal methods consist in expanding the acoustic field in each transverse section as a series which involves the modes of a perfectly cylindrical waveguide with this particular section. For such a cylindrical waveguide, the modal components of the acoustic field are uncoupled. But a varying cross section yields a coupling of these components. A multimodal approach consists in solving the system of differential equations which couples these components instead of the propagation equations.Such methods have been extensively studied for closed waveguides, that is, when no wave can radiate in the transverse direction. In our case, radiation waves occur, because of the presence of the matrix. Actually, such an open waveguide is characterized by a continuum of radiation modes together with a possible finite set of guided modes. The multimodal method proposed here does not take into account this continuum, but uses instead the concept of leaky modes. These modes appear naturally when the computational domain is artificially limited by Perfectly Matched Layers (PMLs). The main advantage of using leaky modes instead of radiation modes is that the formers compose a discrete set, which simplifies considerably the implementation of a multimodal method. In the case of a simple two-dimensional scalar model, the basic ideas of such a method are presented and illustrated by numerical results.