TRAPPING OF A WAVE IN A CURVED CYLINDRICAL ACOUSTIC WAVEGUIDE WITH CONSTANT CROSS-SECTION

Cylindrical acoustic waveguides with constant cross-section. are considered, specifically, a straight waveguide Omega = R x omega subset of R-d and a locally curved waveguide Oe that depends on a parameter epsilon is an element of (0, 1]. For d > 2, in two different settings (epsilon = 1 and epsilon << 1), the task is to find an eigenvalue lambda(epsilon) that is embedded in the continuous spectrum [0,+infinity) of the waveguide Omega(epsilon) and, hence, is inherently unstable. In other words, a solution of the Neumann problem for the Helmholtz operator Delta + lambda(epsilon) arises that vanishes at infinity and implies an eigenfunction in the Sobolev space H-1(Omega(epsilon)). In the first case, it is assumed that the cross-section omega has a double symmetry and an eigenvalue arises for any nontrivial curvature of the axis of the waveguide Omega(epsilon). In the second case, under an assumption on the shape of an asymmetric cross-section., the eigenvalue lambda(epsilon) is formed by scrupulous fitting of the curvature O(epsilon) for small epsilon > 0.