Trapped modes in angular joints of 2D waveguides

Abstract

The Helmholtz equation is considered in an angular joint of two semi-infinite planar waveguides with ideal Dirichlet boundary conditions at their walls and with as the junction angle. We examine trapped modes generated by the discrete spectrum of the Dirichlet Laplacian and, in particular, prove the existence of a critical angle such that, for , the total multiplicity of the discrete spectrum equals 1. We also provide an asymptotic lower bound for the multiplicity as is small and give numerical results.

Keywords:

• trapped modes
• discrete spectrum
• broken waveguides

AMS Subject Classifications:

• 35P15
• 35P25
• 47A75

Acknowledgments

The work is supported by RF Government Grant 11.G34.31.0066, “Scientific school” grant 2631.2012.2, RFBR Grant 12-02-00114.