Wave trapping in a two-dimensional sound-soft or sound-hard acoustic waveguide of slowly-varying width

In this paper we derive novel approximations to trapped waves in a two-dimensional acoustic waveguide whose walls vary slowly along the guide, and at which either Dirichlet (sound-soft) or Neumann (sound-hard) conditions are imposed. The guide contains a single smoothly bulging region of arbitrary amplitude, but is otherwise straight, and the modes are trapped within this localised increase in width.Using a similar approach to that in Rienstra (2003) [13], a WKBJ-type expansion yields an approximate expression for the modes which can be present, which display either propagating or evanescent behaviour; matched asymptotic expansions are then used to derive connection formulae which bridge the gap across the cut-off between propagating and evanescent solutions in a tapering waveguide. A uniform expansion is then determined, and it is shown that appropriate zeros of this expansion correspond to trapped mode wavenumbers; the trapped modes themselves are then approximated by the uniform expansion. Numerical results determined via a standard iterative method are then compared to results of the full linear problem calculated using a spectral method, and the two are shown to be in excellent agreement, even when ?, the parameter characterising the slow variations of the guide’s walls, is relatively large.     

Finite element computation of trapped and leaky elastic waves in open stratified waveguides

Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. In numerous applications, the guiding structure is surrounded by a solid matrix that can be considered as unbounded in the transverse directions. The physics of waves in such an open waveguide significantly differs from a closed waveguide, i.e. for a bounded cross-section. Except for trapped modes, part of the energy is radiated in the surrounding medium, yielding attenuated modes along the axis called leaky modes. These leaky modes have often been considered in non destructive testing applications, which require waves of low attenuation in order to maximize the inspection distance. The main difficulty with numerical modeling of open waveguides lies in the unbounded nature of the geometry in the transverse direction. This difficulty is particularly severe due to the unusual behavior of leaky modes: while attenuating along the axis, such modes exponentially grow along the transverse direction. A simple numerical procedure consists in using absorbing layers of artificially growing viscoelasticity, but large layers may be required. The goal of this paper is to explore another approach for the computation of trapped and leaky modes in open waveguides. The approach combines the so-called semi-analytical finite element method and a perfectly matched layer technique. Such an approach has already been successfully applied in scalar acoustics and electromagnetism. It is extended here to open elastic waveguides, which raises specific difficulties. In this paper, two-dimensional stratified waveguides are considered. As it reveals a rich structure, the numerical eigenvalue spectrum is analyzed in a first step. This allows to clarify the spectral objects calculated with the method, including radiation modes, and their dependency on the perfectly matched layer parameters. In a second step, numerical dispersion curves of trapped and leaky modes are compared to analytical results.     

Perturbation methods for the analysis of the dynamic behavior of damaged plates

This paper presents an analytical method for the analysis of the dynamic behavior of damaged plates. The proposed approach allows the derivation of mode shapes and corresponding curvature modes for plates with various kinds of defects. Damage is modeled as a localized reduction in the plate thickness. Both point and line defects are considered to model notches or line cracks and delaminations in the plate. Small thickness reductions are considered so that the dynamic behavior of the damage plate can be analyzed through perturbations with respect to the undamaged modes. Results are presented to demonstrate the sensitivity of the curvature modes with respect to the considered low damage levels. Also, the curvature modes are used for the estimation of the strain energy of the plate and for the formulation of a damage index which can be used to provide damage location and extent information.     

Water wave trapping in a long array of bottomless circular cylinders

This study tries to identify wave trapping situations by engaging and properly combining two well established phenomena: (i) the trapped modes induced by arrays of cylinders and (ii) the pumping trapped modes which are known to occur in moonpools. To this end, the fundamental hydrodynamic boundary value problem for arrays of bottomless cylinders was solved using standard domain decomposition. The method employed expansions of the solutions for the velocity potentials in polar harmonics combined with the eigenfunction expansions technique. The solution sought for the velocity potentials is achieved using the “direct” method of approach which accordingly requires the employment of a sophisticated matrix manipulation process.The elaboration of the concerned concept was motivated by three basic tasks: (i) to identify whether arrays of truncated and bottomless cylinders indeed preserve the occurrence of Neumann, Dirichlet and near trapped modes, extensively investigated for bottom-seated cylinders; (ii) to examine whether the expected pumping modes in moonpools modify the characteristics of the hydrodynamic resonance regimes (trapped modes) in the open liquid space between the cylinders and vice versa and (iii) to explore the possibility to suggest relevant configurations as parts of integrated mechanisms for practical applications, focusing a fortiori to clusters of hydrodynamically interacting Oscillating Water Columns (OWCs).The method developed is generic and can be employed for arbitrary configurations of multi-body arrays accommodating bottomless cylinders with uneven geometrical characteristics. Trapped modes are identified numerically as peaks in loading and this fact has been explicitly demonstrated in rows of cylinders. Therefore, the numerical results shown and discussed in the present are based on a specific in-line array that has been investigated in the past for bottom-seated cylinders. The investigated subject, i.e. whether the combined wave trapping induced by the examined configuration could be conceived as an efficient water wave power extraction mechanism is approached and discussed through dedicated computations of the free-surface displacements in the moonpools.     

THICKNESS-SHEAR VIBRATION OF A RECTANGULAR QUARTZ PLATE WITH PARTIAL ELECTRODES

We study free vibration of a thickness-shear mode crystal resonator of AT-cut quartz. The resonator is a rectangular plate partially and symmetrically electroded at the center with rectangular electrodes. A single-mode, three-dimensional equation governing the thickness-shear displacement is used. A Fourier series solution is obtained. Numerical results calculated from the series show that there exist trapped thickness-shear modes whose vibration is mainly under the electrodes and decays rapidly outside the electrodes. The effects of the electrode size and thickness on the trapped modes are examined.     

Wave propagation in circular cylindrical shells with periodic axial curvature

The propagation of longitudinal and flexural waves in axisymmetric circular cylindrical shells with periodic circular axial curvature is studied using a finite element method previously developed by the authors. Of primary interest is the coupling of these wave modes due to the periodic axial curvature which results in the generation of two types of stop bands not present in straight circular cylinders. The first type is related to the periodic spacing and occurs independently for longitudinal and flexural wave modes without coupling. However, the second type is caused by longitudinal and flexural wave mode coupling due to the axial curvature. A parametric study is conducted where the effects of cylinder radius, degree of axial curvature, and periodic spacing on wave propagation characteristics are investigated. It is shown that even a small degree of periodic axial curvature results in significant stop bands associated with wave mode coupling. These stop bands are broad and conceivably could be tuned to a specific frequency range by judicious choice of the shell parameters. Forced harmonic analyses performed on finite periodic structures show that strong attenuation of longitudinal and flexural motion occurs in the frequency ranges associated with the stop bands of the infinite periodic structure.     

A connection between linear normal modes and lines of curvature on the potential surface

For the N-degree-of-freedom of linear conservative vibratory systems, the corresponding potential functions can be viewed as N-hypersurfaces in (N + 1)-dimensional space. In this paper, a connection between the geometrical properties (principal curvatures, curvature lines) of potential surfaces and the vibratory characteristics (natural frequencies, linear modes) of the system is built. It is proved that the linear normal modes are exactly the projections of the lines of curvature on the potential surface onto the configuration space with metric m_ij (the mass-matrix); and that the squared natural frequencies are exactly the principal curvatures, at the origin of the configuration space, of the potential surface.     

Analysis of thickness-shear and thickness-twist modes of AT-cut quartz acoustic wave resonator and filter

The thickness-shear(TS) and thickness-twist(TT) vibrations of partially electroded AT-cut quartz plates for acoustic wave resonator and filter applications are theoretically studied. The plates have structural variations in one of the two in-plane directions of the plates only. The scalar differential equations derived by Tiersten and Smythe for electroded and unelectroded AT-cut quartz plates are used, resulting in free vibration resonant frequencies and mode shapes for both fundamental and overtone families of modes. The trapped modes with vibrations, mainly confined in the electroded areas, are found to exist in both the resonator and the filter structures. The numerical results for the trapped modes are presented for different aspect ratios of electrodes and material properties, providing a reference to the design and optimization of quartz acoustic wave resonators and filters.     

Effects of middle plane curvature on vibrations of a thickness-shear mode crystal resonator

We study the effects of a small curvature of the middle plane of a thickness-shear mode crystal plate resonator on its vibration frequencies, modes and acceleration sensitivity. Two-dimensional equations for coupled thickness-shear, flexural and extensional vibrations of a shallow shell are used. The equations are simplified to a single equation for thickness-shear, and two equations for coupled thickness-shear and extension. Equations with different levels of coupling are used to study vibrations of rotated Y-cut quartz and langasite resonators. The influence of the middle plane curvature and coupling to extension is examined. The effect of middle plane curvature on normal acceleration sensitivity is also studied. It is shown that the middle plane curvature causes a frequency shift as large as 10⁻⁸ g⁻¹ under a normal acceleration. These results have practical implications for the design of concave–convex and plano-convex resonators.     

Curvature-induced oscillations of the loss of leaky modes in optical fibers: a ray analysis

A new curvature loss formula for leaky modes is derived from a direct application of the volume current method. This formula is confirmed in the particular case of a leaky LP₀₁ mode in a DIC fiber with a moderate curvature, by means of combination of the standard analytical perturbation analysis of DIC fibers with a ray analysis of the field in the outer cladding. Curvature induced oscillations of the leakage loss are predicted, due to an interference effect. The validity of the new loss formula for larger curvatures is discussed.     

Stability Analysis of a Vertical Curved Channel Flow with a Radial Temperature Gradient

The linear stability analysis of a Newtonian incompressible fluid in a vertical curved channel formed by two coaxial cylindrical surfaces with a radial temperature gradient and an azimuthal pressure gradient shows that critical modes are oscillatory and non-axisymmetric. We have derived a generalized Rayleigh discriminant which includes both the curvature and buoyancy effects. Centrifugal buoyancy induces weak asymmetry of the dependence of the control parameter critical values on the sign of the temperature gradient. The critical parameters depend on the temperature gradient, the radius ratio and the nature of the fluid. For a wide curvature channel flow, there are two critical modes: oscillatory Dean modes for small temperature gradients and oscillatory centrifugal-thermal modes for relatively large temperature gradients. Received 14 November 2001 and accepted 29 March 2002 Published online: 2 October 2002 Communicated by H.J.S. Fernando     

Hybrid Green's function for SH motion in a low velocity layer

Wave motion in an elastic layer has been analyzed conventionally either in terms of normal (discrete and continuous) mode spectra or in terms of ordinary or generalized ray fields. Either of these representations is inconvenient for describing the motion over the entire range of observations from early to late arrivals. A recently developed hybrid formulation, wherein modal fields, ray fields and a remainder are combined according to specified criteria within a single rigorous framework, overcomes these difficulties, Combinations may be chosen that ensure smooth passage from the ray phase at early times to the modal phase at later times. The method is illustrated here on the two-dimensional time-dependent SH Green's function for a homogeneous sediment layer above a higher velocity homogeneous semi-infinite bedrock, with the source located in the sediment. The hybrid fields may include various combinations of trapped modes, leaky modes, trapped rays, leaky rays, and lateral rays (head waves). This study complements an earlier investigation wherein the source was located in the bedrock.     

Length dependence of critical measures in single-walled carbon nanotubes

This paper presents an investigation on the buckling behaviour of single-walled carbon nanotubes under various loading conditions (compression, bending and torsion) and unveils several aspects concerning the dependence of critical measures (axial strain, bending curvature and twisting angle) on the nanotube length. The buckling results are obtained by means of an atomistic-scale generalized beam theory (GBT) that incorporates local deformation of the nanotube cross-section by means of independent and orthogonal deformation modes. Moreover, some estimates are also obtained by means of non-linear shell finite element analyses using Abaqus code. After classifying the buckling modes of thin-walled tubes (global, local and distortional), the paper addresses the importance of the two-wave distortional mode (flattening or ovalization mode) in their structural behaviour. Then, the well known expression to determine the critical strain of compressed nanotubes, which is based on Donnell theory for shallow shells, is shown to be inadequate for moderately long tubes due to warping displacements appearing in the distortional buckling modes. After that, an in-depth study on the buckling behaviour of nanotubes under compression, bending and torsion is presented. The variation of the critical kinematic measures (axial strain, bending curvature and twisting angle) with the tube length is thoroughly investigated. Concerning this dependence, some uncertainties that exist in the specific literature are meticulously explained, a few useful expressions to determine critical measures of nanotubes are proposed and the results are compared with available data collected from several published works (most of them, obtained from molecular dynamics simulations).     

Waves, normal modes and frequencies in periodic and near-periodic rods. Part I

Longitudinal wave motions and localized normal modes in a rod system with periodically-alternating material properties are investigated in this paper. The energy injected into the rod system is shown either to be transported through the whole rod system in pass-bands or to be trapped near the excitation source in stop-bands. For this one-dimensional continuous model, the full power of linear system theory is utilized and a new transfer matrix method is proposed to get closed-form normal mode solutions. Localized normal modes in stop-bands in perfectly-periodic rods with asymmetric bays are identified. It is shown that for this strongly-coupled elastic system, a single small disorder may produce one or two additional modes in each stop-band, these modes are localized around the disordered bay. By understanding basic behavior of such a system, it is hoped ultimately that some insights can be achieved where closed-form results are not possible.     

RESONANCES IN WAVE DIFFRACTION/RADIATION FOR ARRAYS OF ELASTICALLY CONNECTED CYLINDERS

As shown by Maniar & Newman in 1997, for a long array of bottom-mounted cylinders in the open sea, resonant modes occur as “near-trapping” and large diffraction forces are excited on the cylinders. The mechanism of such a resonant phenomenon was subsequently explained by the present authors in connection with the Dirichlet trapped modes for an array of cylinders aligned perpendicular to the walls in a wave channel. This paper examines similar resonant phenomena for radiation problems. Considered is an array of elastically connected cylinders in a wave channel. The cylinders are surface-piercing and extend to the sea-bottom. They constitute an array in a line, and each cylinder is allowed to oscillate only in the direction parallel to the line. Nonradiating wave modes, which cause only added mass force and no hydrodynamic damping are demonstrated to exist for an array of cylinders across the wave channel. Each mode corresponds to a “dry-mode” for the periodic array of elastically connected cylinders. This result leads to the existence of pure-resonant modes for a periodic array of elastically connected cylinders across the channel. Trapped modes for the corresponding diffraction problem are obtained as the limiting case when the stiffness of the springs has an infinite value.     

Experimental study of disk impact onto shallow water

A problem of the impact of circular disks with a flat, convex, or concave onto a finite-depth fluid layer is experimentally studied. The influence of the small curvature of the lower surface of the body on the added mass, characteristic time of the impact on the free surface, parameters of the air cavity trapped during the impact, and type of the body impact on the bottom of the hydrodynamic test tank is examined     

Wave Localization in Hydroelastic Systems

The phenomenon of wave localization in hydroelastic systems leads to the strength concentration of radiation fields. The linear method considers the process of localization to be the formation of nonpropagation waves (trapped modes phenomenon). The presence of such waves in the total wave packet points to the existence of mixed natural spectrum of differential operators describing the behaviour of hydroelastic systems. The problem of liquid and oscillating structure interaction caused by the trapped modes phenomenon has been solved (membranes, dies). The interaction of the liquid and elastic structures with inclusions can lead to localized mode formation. In the case of solitary wave motion in nonlinear elastic media, contacting with the liquid, these solitons can be interpreted as “moving inclusions”. The analytical solution for solitary waves has been found. If the soliton speed v0 is more than the velocity of sound c0 in the liquid, the solitary waves strongly slow down. If c0 is close to v0, then a resonance can be observed and solitons move without any resistance. If the soliton speed is less than c0, the solitary wave slow-down is negligible, compared to the case v0 > c0. This revised version was published online in July 2006 with corrections to the Cover Date.     

Vibration of a membrane strip with a segment of higher density: analysis of trapped modes

The Helmholtz equation governs the dynamics of membranes and waveguides. Using the mode matching method, trapped modes are found for an infinite strip with a segment of inhomogeneity. The exact frequencies and mode shapes are determined as a function of density ratio and length of the segment. Nonuniqueness and nonexistence are demonstrated.     

Localized elastic fields in periodic waveguides with defects

The variational method for determining localized waves (trapped modes) is modified for periodic elastic waveguides with partially clamped surfaces. Two sufficient conditions for the existence of localized fields in waveguides with defects (cavities with positive volume and cracks) are established. In the presence of elastic and geometrical symmetries, localized fields were also found in periodic elastic waveguides with surfaces free of external loads.     

Vibrations of ribbed shallow rectangular shells on an elastic foundation

A procedure to determine the natural frequencies and modes of ribbed shallow shells with rectangular planform on an elastic foundation is developed. The effect of the radius of curvature and the Winkler and Pasternak coefficients of subgrade reaction on the natural frequencies and modes of a shallow spherical shell with a square planform is analyzed. It is revealed that not only the natural frequencies but also the modes of the shell depend on the coefficients of subgrade reaction