ArticlePDF Available

Trapped modes of the Helmholtz equation in infinite waveguides with wall indentations and circular obstacles

Article

Trapped modes of the Helmholtz equation in infinite waveguides with wall indentations and circular obstacles

Abstract

Trapped modes of the Helmholtz equation are investigated in infinite, 2D acoustic waveguides with Neumann or Dirichlet walls. A robust boundary element scheme is used to study modes both inside and outside the continuous spectrum of propagating modes. An effective method for distinguishing between genuine trapped modes and spurious solutions induced by the domain truncation is presented. The method is also suitable for the detection and study of ‘nearly trapped modes’. These are of great practical importance as they display many features of trapped modes but do not require perfect geometry. An infinite, 2D channel is considered with one or two discs on its centreline. The walls may have rectangular, triangular or smooth cavities. The combination of a circular obstacle and a rectangular cavity, in both Neumann and Dirichlet guides is studied, illustrating the possible use of a movable disc to detect wall irregularities. The numerical method is validated against known results and many new modes are identified, both inside and outside the continuous spectrum. Results obtained suggest that at least one symmetry line is an important condition for the formation of trapped mode-type resonances. The addition of a symmetry-preserving geometric parameter to a problem which has a discrete embedded trapped mode solution for a specific geometry, tends to lead to a continuous set of trapped modes.
A preview of the PDF is not available
... Duan et al. [15] employed a semi-analytical approach in a study of acoustic trapped modes around a variety of bodies and cavities, again in a two-dimensional duct. Very recently, Sargent and Mestel [16] devised a boundary element-based approach, with which they analyzed a variety of two-dimensional waveguide configurations. ...
... Such solutions are known as embedded trapped modes since the wavenumber is located (embedded) within the continuous wavenumber spectrum. They were found initially for non-symmetric geometries, such as by a finite strip placed off the centerline in a two-dimensional waveguide [26] but later, examples of existence of such trapped modes were found also for symmetric geometries [16,27]. Since such solutions exist only for very specific values of the geometric parameters they are more difficult to find than 'classic,' non-embedded trapped modes. ...
Article
Full-text available
The paper is concerned with a partly analytical, partly numerical study of acoustic trapped modes in a cylindrical cavity (expansion chamber), placed in between two semi-infinite pipes acting as a waveguide. Trapped mode solutions are expressed in terms of Fourier–Bessel series, with the expansion coefficients determined from a determinant condition. The roots of the determinant, expressed in terms of the real wavenumber k, correspond to trapped modes. For a shallow cavity and for low values of the circumferential mode number it is found that there is just one trapped mode in the allowable wave number domain, and this mode is symmetric about a radial axis in the center of the cavity. As the circumferential mode number is increased, more and more trapped modes, placed between two cutoff frequencies, come into play, and they alternate between symmetric and antisymmetric modes. An analytical explanation of the mechanism behind the mode increasing and mode alternation is given via asymptotic expressions of the determinant condition. Numerical computations are done for verification of the analytical results and for consideration of less shallow cavities. Also for these cases, similar phenomena of an increasing number of trapped modes, and alternation between symmetric and antisymmetric modes, are found.
... A more relevant perspective is that the finite slit domain degenerates to the semi-infinite one in the limit ǫ → 0 if the position along the slit is held fixed. The degenerate geometry is a special case of the more general obstacle-in-channel class of configurations that have been extensively searched for BIC in acoustics, water waves and photonics [3][4][5][26][27][28][29][30][31][32][33][34][35][36][37][38]. In particular, the semiinfinite geometry shown in Fig. 2 is known to support BIC due to symmetry. ...
Preprint
Full-text available
Localised wave oscillations in an open system that do not decay or grow in time, despite their frequency lying within a continuous spectrum of radiation modes carrying energy to or from infinity, are known as bound states in the continuum (BIC). Small perturbations from the typically delicate conditions for BIC almost always result in the waves weakly coupling with the radiation modes, leading to leaky states called quasi-BIC that have a large quality factor. We study the asymptotic nature of this weak coupling in the case of acoustic waves interacting with a rigid substrate featuring a partially partitioned slit -- a setup that supports quasi-BIC that exponentially approach BIC as the slit is made increasingly narrow. In that limit, we use the method of matched asymptotic expansions in conjunction with reciprocal relations to study those quasi-BIC and their resonant excitation. In particular, we derive a leading approximation for the exponentially small imaginary part of each wavenumber eigenvalue (inversely proportional to quality factor), which is beyond all orders of the expansion for the wavenumber eigenvalue itself. Furthermore, we derive a leading approximation for the exponentially large amplitudes of the states in the case where they are resonantly excited by a plane wave at oblique incidence. These resonances occur in exponentially narrow wavenumber intervals and are physically manifested in cylindrical-dipolar waves emanating from the slit aperture and exponentially large field enhancements inside the slit. The asymptotic approximations are validated against numerical calculations.
Preprint
We construct a surface with a cylindrical end which has a finite number of Laplace eigenvalues embedded in its continuous spectrum. The surface is obtained by attaching a cylindrical end to a hyperbolic torus with a hole. To our knowledge, this is the first example of a manifold with a cylindrical end whose number of eigenvalues is known to be finite and nonzero. The construction can be varied to give examples with arbitrary genus and with an arbitrarily large finite number of eigenvalues. The constructed surfaces also have resonance-free regions near the continuous spectrum and long-time asymptotic expansions of solutions to the wave equation.
Article
Full-text available
We consider the numerical solution of scalar wave equations in domains which are the union of a bounded domain and a finite number of infinite cylindrical waveguides. The aim of this paper is to provide a new convergence analysis of both the Perfectly Matched Layer (PML) method and the Hardy space infinite element method in a unified framework. We treat both diffraction and resonance problems. The theoretical error bounds are compared with errors in numerical experiments.
Article
Full-text available
We present an experimental study on the trapped modes occurring around a vertical surface-piercing circular cylinder of radius a placed symmetrically between the parallel walls of a long but finite water waveguide of width 2d. A wavemaker placed near the entrance of the waveguide is used to force an asymmetric perturbation into the guide, and the free-surface deformation field is measured using a global single-shot optical profilometric technique. In this configuration, several values of the aspect ratio a/d were explored for a range of driving frequencies below the waveguide's cutoff. Decomposition of the obtained fields in harmonics of the driving frequency allowed for the isolation of the linear contribution, which was subsequently separated according to the symmetries of the problem. For each of the aspect ratios considered, the spatial structure of the trapped mode was obtained and compared to the theoretical predictions given by a multipole expansion method. The waveguide–obstacle system was further characterized in terms of reflection and transmission coefficients, which led to the construction of resonance curves showing the presence of one or two trapped modes (depending on the value of a/d), a result that is consistent with the theoretical predictions available in the literature. The frequency dependency of the trapped modes with the geometrical parameter a/d was determined from these curves and successfully compared to the theoretical predictions available within the frame of linear wave theory.
Article
Experimental study on water-wave trapped modes – CORRIGENDUM - Volume 862 - P. J. Cobelli, V. Pagneux, A. Maurel, P. Petitjeans
Article
Light bends the wrong way in materials where both ε and μ are negative as was pointed out in 1968, but the absence of natural materials with this property led to neglect of the subject until 1999 when it was shown how to make artificial materials, metamaterials, with negative μ. The rapid advance of the subject since that date, both in theory and experiment, is reflected in the exponential growth of publications now at the 200 per year level and still growing. This interest is explained by the sudden availability of a qualitatively different class of electromagnetic materials combined with the quite startling properties which these materials appear to have; all of which provokes debate as each new facet of their behaviour is revealed. Experiment has been vital to resolution of controversy and has chiefly been in the microwave region of the spectrum though there is potential in the optical region currently being explored by several groups.
Article
The existence of acoustic, Rayleigh–Bloch modes in the vicinity of a one-dimensional (1D) periodic array of rigid, axisymmetric structures is established with the use of a variational principle. Axisymmetric modes at frequencies below the cut-off frequency are shown to exist for all piecewise smooth structures and non-axisymmetric modes are found for a class of structures whose radial dimension is sufficiently large compared to the structure spacing. The theory is illustrated with numerical calculations of the wave numbers of Rayleigh–Bloch modes for an array of circular plates. An integral equation for the acoustic wave field in the neighbourhood of such an array is obtained and solved with the use of a Galerkin technique, which builds in the singularity in the derivative of the field at the rim of the plate.
Article
The vibration of infinite or semi-infinite membrane strip with a local enlargement is studied. An efficient domain decomposition and matching method is used to accurately solve the governing Helmholtz equation. Due to the enlargement, the fundamental frequency is decreased considerably and the corresponding vibration mode is effectively localized near the junction. The results are particularly significant to quantum waveguides.
Article
Perturbations of an eigenvalue in the continuous spectrum of the Neumann problem for the Laplacian in a strip waveguide with an obstacle symmetric about the midline are studied. Such an eigenvalue is known to be unstable, and an arbitrarily small perturbation can cause it to leave the spectrum to become a complex resonance point. Conditions on the perturbation of the obstacle boundary are found under which the eigenvalue persists in the continuous spectrum. The result is obtained via the asymptotic analysis of an auxiliary object, namely, an augmented scattering matrix.