Bound states in elastic waveguides

We consider numerically the L-, T-, and X-shaped elastic waveguides with the Dirichlet boundary conditions for in-plane deformations (displacements) which obey the vectorial Navier-Cauchy equation. In the X-shaped waveguide we show the existence of a doubly degenerate bound state with frequency below the first symmetrical cutoff frequency, which belongs to the two-dimensional irreducible representation E of symmetry group C-4v. Moreover the next bound state is below the next antisymmetric cutoff frequency. This bound state belongs to the irreducible representation A(2). The T-shaped waveguide has only one bound state while the L-shaped one has no bound states.