Trapped acoustic modes

A constructive proof is given for the existence of trapped acousticmodes in the vicinity of a strip of length 2a parallel to, andmidway between, the bounding lines of a two-dimensional waveguide.The modes, which are shown to exist for sufficiently large a,satisfy Neumann conditions on the strip and the bounding linesand a Dirichlet condition on the midline outside the strip,and may be either symmetric or antisymmetric about a line throughthe centre of, and perpendicular to, the strip. When a is large,the equation determining the wavenumber of the modes reducesto that proposed by Evans and Linton (1991) using nonrigorousarguments     

Trapped modes for periodic structures in waveguides

The Laplace operator is considered for waveguides perturbed by a periodic structure consisting of N congruent obstacles spanning the waveguide. Neumann boundary conditions are imposed on the periodic structure, and either Neumann or Dirichlet conditions on the guide walls. It is proven that there are at least N (resp. N‐1) trapped modes in the Neumann case (resp. Dirichlet case) under fairly general hypotheses, including the special case where the obstacles consist of line segments placed parallel to the waveguide walls. Copyright © 2004 John Wiley & Sons, Ltd.     

Scheme for interpretation of approximately computed eigenvalues embedded in a continuous spectrum

It is assumed that a trapped mode (i.e., a function decaying at infinity that leaves small discrepancies of order ? ? 1 in the Helmholtz equation and the Neumann boundary condition) at some frequency κ⁰ is found approximately in an acoustic waveguide Ω⁰. Under certain constraints, it is shows that there exists a regularly perturbed waveguide Ω^? with the eigenfrequency κ^? = κ⁰ + O(?). The corresponding eigenvalue λ^? of the operator belongs to the continuous spectrum and, being naturally unstable, requires “fine tuning” of the parameters of the small perturbation of the waveguide wall. The analysis is based on the concepts of the augmented scattering matrix and the enforced stability of eigenvalues in the continuous spectrum.     

Criteria for eigenvalues embedded into the absolutely continuous spectrum of perturbed Stark type operators

In this paper, we consider the perturbed Stark operator

$Hu=H_{0}u+qu=?u^{″}?xu+qu,$

where q is the power-decaying perturbation. The criteria for q such that $H=H_0+q$ has at most one eigenvalue (finitely many, infinitely many eigenvalues) are obtained. All the results are quantitative and are generalized to the perturbed Stark type operator.     

Trapped modes in a cylindrical elastic waveguide with a damping gasket

An infinite cylindrical body containing a three-dimensional heavy rigid inclusion with a sharp edge is considered. Under certain constraints on the symmetry of the body, it is shown that any prescribed number of eigenvalues of the elasticity operator can be placed on an arbitrary real interval (0, l) by choosing suitable physical properties of the inclusion. In the continuous spectrum, these points correspond to trapped modes, i.e., to exponentially decaying solutions to the homogeneous problem. The results can be used to design filters and dampers of elastic waves in a cylinder.     

Trapped modes for an elastic strip with perturbation of the material properties

The elasticity operator, for zero Poisson coefficient, withstress-free boundary conditions on a two-dimensional strip withlocal perturbation of Young's modulus, is considered. We provethe existence of embedded eigenvalues and describe their asymptoticbehaviour.     

Trapped modes along a periodic array of freely floating obstacles

We consider the coupled problem describing the motion of a linear array of three‐dimensional obstacles floating freely in a homogeneous fluid layer of finite depth. The interaction of time‐harmonic waves with the floating objects is analyzed under the usual assumptions of linear water‐wave theory. Quasi‐periodic boundary conditions and a simplified reduction scheme turn the system into a linear spectral problem for a self‐adjoint operator in Hilbert space. Based upon the operator formulation, we derive a sufficient condition for the nonemptiness of its discrete spectrum. Various examples of obstacles that generate trapped modes are given. Copyright © 2014 John Wiley & Sons, Ltd.     

Trapped modes of oscillation of a liquid for surface-piercing bodies in oblique waves

The linear problem of the harmonic oscillations of an ideal incompressible heavy liquid with a free surface in the presence of two and more infinitely long partially submerged cylindrical bodies of arbitrary cross-section is considered. It is proved that there are configurations of the bodies which provide examples of the non-uniqueness of the boundary-value problem in the case of an arbitrary frequency of the oscillations and an arbitrary non-zero angle between the generatrix of the cylinders and the direction of propagation of the surface waves. In the case of these configurations, the homogeneous boundary-value problem has non-trivial solutions with a finite energy integral, which describe trapped modes of oscillation of the liquid.     

Eigenvalues of the Dirac operator in the continuous spectrum

Functionsp(x) andq(x) for which the Dirac operator $$Dy = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ { - 1} \\ \end{array} } & {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \\ \end{array} } \right)\frac{{dy}}{{dx}} + \left( {\begin{array}{*{20}c} {p(x) q(x)} \\ {q(x) - p(x)} \\ \end{array} } \right)y = \lambda y, y = \left( {\begin{array}{*{20}c} {y_1 } \\ {y_2 } \\ \end{array} } \right), y_1 (0) = 0,$$ has a countable number of eigenvalues in the continuous spectrum are constructed.     

Trapped modes in a channel containing three layers of fluids and a submerged cylinder

The problem of existence of trapped waves in fluids due to a cylinder is investigated for the hydrodynamic set-up which involves a horizontal channel of infinite length and depth and of finite width containing three layers of incompressible fluids of different constant densities. The set-up also contains a cylinder which is impermeable, fully immersed in the bottom (lower-most) fluid layer of infinite depth, and extends across the channel with its generators perpendicular to the side walls of the channel. When the ratios of the densities of the adjacent fluids differ from unity by sufficiently small quantities, the underlying mathematical problem reduces to a generalized nonlinear eigenvalue problem involving a cubic polynomial-cum-operator equation. The perturbation analysis of this eigenvalue problem suggests existence of three distinct modes with different frequencies: one of the order of one persisting at the free surface, and the other two of the order of the density ratio (except for modulo one) persisting at the two internal interfaces. The correlation between these results for the three-layer case and very recent numerical results of other authors in the two-layer case has also been addressed. Received: March 3, 2005