# Localization of wave motions in a fluid-filled cylinder

TL;DR: In this article, the properties of harmonic surface waves in a fluid-filled cylinder made of a compliant material are studied, and the wave motions are described by a complete system of dynamic equations of elasticity and the equation of motion of a perfect compressible fluid.

Abstract: The properties of harmonic surface waves in a fluid-filled cylinder made of a compliant material are studied. The wave motions are described by a complete system of dynamic equations of elasticity and the equation of motion of a perfect compressible fluid. An asymptotic analysis of the dispersion equation for large wave numbers and a qualitative analysis of the dispersion spectrum show that there are two surface waves in this waveguide system. The first normal wave forms a Stoneley wave on the inside surface with increase in the wave number. The second normal wave forms a Rayleigh wave on the outside surface. The phase velocities of all the other waves tend to the velocity of the shear wave in the cylinder material. The dispersion, kinematic, and energy characteristics of surface waves are analyzed. It is established how the wave localization processes differ in hard and compliant materials of the cylinder

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TL;DR: In this paper, the amplitude and frequency characteristics of an orthotropic cylindrical shell with fluid flowing inside were analyzed at fundamental parametric resonance, and a method to calculate the characteristics of the parametric vibrations of the shell when the velocity of the fluid is close to critical was proposed.

Abstract: The paper addresses the dynamic interaction of an orthotropic cylindrical shell with the fluid flowing inside. Its velocity has a constant component and low-amplitude pulsations. A method to calculate the characteristics of the parametric vibrations of the shell when the velocity of the fluid is close to critical is proposed. The amplitude–frequency characteristics of the shell–fluid system at fundamental parametric resonance are determined

12 citations

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TL;DR: In this paper, the authors studied the interaction of a rigid spherical body and a cylindrical cavity filled with an ideal compressible fluid in which a plane acoustic wave of unit amplitude propagates.

Abstract: The paper studies the interaction of a rigid spherical body and a cylindrical cavity filled with an ideal compressible fluid in which a plane acoustic wave of unit amplitude propagates. The solution is based on the possibility of transforming partial solutions of the Helmholtz equation between cylindrical and spherical coordinates. Satisfying the interface conditions between the cavity and the acoustic medium and the boundary conditions on the spherical surface yields an infinite system of algebraic equations with indefinite integrals of cylindrical functions as coefficients. This system of equations is solved by reduction. The behavior of the system is studied depending on the frequency of the plane wave

10 citations

### Cites background from "Localization of wave motions in a f..."

...The nonlinear interaction of different vibration modes is addressed in, e.g., [ 10 ]....

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...Introduction. Many applied problems necessitate studying the dynamic behavior of solids (either rigid or elastic) filled (either partially [ 10 ] or completely [3, 13–15]) with a fluid....

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TL;DR: In this article, the behavior of a mechanical system consisting of an infinite thin cylindrical shell filled with a flowing compressible liquid and containing a pulsating spherical inclusion is analyzed using linear potential flow theory and the theory of thin elastic shells based on the Kirchhoff-love hypotheses.

Abstract: The paper proposes a method to analyze the behavior of a mechanical system consisting of an infinite thin cylindrical shell filled with a flowing compressible liquid and containing a pulsating spherical inclusion. This coupled problem is solved using linear potential flow theory and the theory of thin elastic shells based on the Kirchhoff–Love hypotheses. Use is made of the possibility to represent the general solutions of equations of mathematical physics in different coordinate systems. This makes it possible to satisfy the boundary conditions on both spherical and cylindrical surfaces and to obtain a solution in the form of a Fourier series. Some numerical results are given

7 citations

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TL;DR: In this paper, the motion and localization of bubbles in a liquid subject to two-frequency excitation are studied and the conditions of their stability are analyzed. Plane and spherical waves are considered.

Abstract: The motion and localization of bubbles in a liquid subject to two-frequency excitation are studied. Plane and spherical waves are considered. Stationary solutions are obtained and the conditions of their stability are analyzed

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TL;DR: In this paper, the authors report experimental data on the nonlinear dynamic deformation of the elastic bottom of a cylindrical shell and the formation of bubbles and their clusters under two-frequency excitation.

Abstract: The paper reports experimental data on the nonlinear dynamic deformation of the elastic bottom of a cylindrical shell and the formation of bubbles and their clusters under two-frequency excitation

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TL;DR: In this article, the possibility of the existence of a seismic disturbance along the junction of strata, or within a certain stratum, so that the energy is dissipated by internal viscosity without the occurrence of any appreciable surface displacement is investigated.

Abstract: In considering how the energy of a seismic disturbance is dissipated one is led to enquire into the possibility of the existence of waves, analogous to Rayleigh waves and Love waves, that are propagated in the interior of the earth along the junction of strata, or chiefly within a certain stratum, so that the energy is dissipated by internal viscosity without the occurrence of any appreciable surface displacement. Two surfaces of discontinuity of density and elastic properties are commonly believed to exist below that part of the earth’s crust which is accessible to geologists, namely, the junction of the granitic layer with the basic rocks, and the surface of separation of the Wiechert metallic core from the rocky shell. It becomes of interest to examine whether a wave of the Rayleigh type can be propagated along such an interface; an enquiry may also be made into the circumstances in which a wave of the Love type may exist if a stratum of uniform thickness is bounded on both sides by very deep layers of different materials.

577 citations

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TL;DR: In this paper, the authors describe the modes that have a steady state sin(ωt−βz) dependence on (t,z), with emphasis on the modes and ranges of parameters that are of interest for delay lines.

Abstract: Clad rods have been investigated for use as long delay lines because they offer isolation of the signal from the surface and low dispersion. In addition, single‐mode propagation is achieved with a larger (and hence more conveniently transduced) cross section than is possible with a homogeneous rod at the same frequency. This paper describes the modes that have a steady‐state sin(ωt−βz) dependence on (t,z), with emphasis on the modes and ranges of parameters that are of interest for delay lines. Only rods of circular cross section, and isotropic, linear elastic materials are considered. Attention is drawn to correspondences with homogeneous rods and with the hypothetical case of infinite thickness cladding, which is most useful as a model for understanding the behavior of corresponding modes in an actual clad rod. Written for the nonspecialist, the paper includes a tutorial review of the concepts and results needed to understand wave propagation in rods and clad rods. In addition, the following new results are reported: (1) for homogeneous rods, a representation of the lowest flexural mode dispersion curve that is for practical purposes independent of Poisson’s ratio; (2) for homogeneous rods, displacement distributions of the first three high‐frequency shear modes, i.e., the ’’flexural’’ modes that are asymptotic to the shear velocity (it was found that the distribution previously attributed to the lowest such mode actually belongs to the next); (3) for clad rods, the first demonstration of interface modes of the Stoneley type for nonaxially symmetric waves; (4) classification of clad rods according to the nature and ordering of their asymptotic velocities showing that there are eight types of clad rod, rather than four as stated in the previous literature; (5) for infinitely clad rods having the same shear modulus, proof that the dispersion of torsional waves and their penetration into cladding are universal functions of f/fc∞. The ratio of the shear velocities of the two materials affects the cutoff frequency fc∞, but not the universal penetration and dispersion functions. Subjects on which significant tutorial or descriptive material is given include typical waveguide dispersion, characteristic velocities of an isotropic elastic material, the effect of coupling of dilatational and shear waves at a boundary, waves in homogeneous rods, the connection of isolation to total internal reflection (with Love waves and SH waves in a clad plate as an example), Stoneley waves at a plane interface, and previous results on the clad rod.

284 citations

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TL;DR: In this paper, a high speed computer was used to investigate the problem of wave propagation in an isotropic elastic cylinder, and an approximate solution was found for the L(0,1) mode impinging on a traction-free interface.

Abstract: A high‐speed computer was used to investigate the problem of wave propagation in an isotropic elastic cylinder. Dispersion curves corresponding to real, imaginary, and complex propagation constants for the symmetric and the first four antisymmetric modes of propagation are given. The radial distributions of axial and radial displacements and of shear and normal stresses are given for the symmetric mode. By using a finite number of modes of propagation, an approximate solution is found for the problem of the L(0,1) mode impinging on a traction‐free interface. The reflection coefficient is determined in this way and the accompanying generation of higher order modes at the interface is shown to cause a high‐amplitude end resonance. Experimental results obtained by using the resonance method in conjunction with a long rod are presented to substantiate the calculated reflection coefficient and the frequency of end resonance. Phase velocities, based on measurements of the wavelength of standing waves and resonance frequencies, were obtained for the symmetric and first two antisymmetric modes. These measurements extend into the frequency range of more than one propagating mode. The rms deviation between theoretical and experimental results is in general less than 0.2% with the exception of the dispersion curve for the L(0,2) mode which deviates by 0.7%.

264 citations