Wave Propagation Coefficients Research Articles

Thermal cycling effects on the seismic wave propagation coefficients of granite

The seismic wave propagation properties of granite after thermal cycles were experimentally investigated. The granite specimens were subjected to a series of thermal cycles with different thermal cycling temperatures and numbers. Ultrasonic and pendulum impact tests were performed on the granite after thermal cycles. Subsequently, a theoretical study was introduced to explore the wave propagation coefficients of granite after thermal cycles. The effects of the thermal cycling number on the wave attenuation rate, wave velocity, attenuation coefficient, and wavenumber were discussed. The results indicate that the wave velocity and dynamic elastic modulus decrease as the number of thermal cycles increases and that the damage factor increases as the number of thermal cycles increases. However, the variations in wave velocity, dynamic elastic modulus, and damage factor are concentrated mainly in the first thermal cycle. The results further indicate that the wave propagation coefficients, attenuation coefficient, and wavenumber increase as the number of thermal cycles increases. However, they are not significantly influenced by the amplitude and wavelength of impact loading.

Experimental study of frequency-temperature coupling effects on wave propagation through granite

The present study investigates the frequency-temperature coupling effects on wave propagation through granite. The pendulum impact test was conducted on the thermally treated long granite bars. The wave propagation coefficients, such as attenuation ratio and wave velocity, were determined for the granite under five groups of high-temperature and six groups of incident waves with different frequencies. A relationship between the dynamic modulus of granite, frequency of stress wave and the high temperature was revealed. The frequency-temperature coupling effects on the dynamic modulus were discussed. The results show that the attenuation ratio, wave velocity and dynamic elastic modulus increase with the increase of frequency. The attenuation ratio increases with the increase of temperature, while the wave velocity and dynamic elastic modulus decrease with the increase of temperature. The proposed relationship between the dynamic modulus, frequency and temperature can efficiently describe the frequency-temperature coupling effects on the granite with acceptable error. The present study has potential in the application of wave propagation through granite in the high-temperature environment.

Improved frequency-domain Spectral Element Method for vibration analysis of nonuniform pipe conveying fluid

The well-known Spectral Element Method (SEM) has a broad application in vibration analysis for various configurations, such as flat plates, beams and straight pipes. Considering the fluid–structure coupling effects in variable cross-section, it has one limitation in dealing with the nonuniform pipes conveying fluid. In the present work, an improved frequency-domain SEM (IFSEM) is proposed for the analysis of the nonuniform fluid-conveying pipes by introducing structural wave propagation. In the IFSEM, the wave propagation coefficient plays a critical role to construct the equivalent dynamic stiffness, improving the analysis accuracy and efficiency. And there is a rule that the propagation coefficients are mainly related to the pipe diameter, flow rate and hydrodynamic pressure. A Simplification is put forward to obtain these coefficients in advance, which can significantly facilitate the vibration analysis. Finally, the IFSEM is validated through several numerical examples and the results show that it can analyze the vibration of the liquid-conveying nonuniform pipes efficiently and accurately.

Determination of Wave Propagation Coefficients of the Granite by High-Speed Digital Image Correlation (HDIC)

Determination of Wave Propagation Coefficients of the Granite by High-Speed Digital Image Correlation (HDIC)

Microscopic analysis of sound attenuation in low-temperature amorphous solids reveals quantitative importance of non-affine effects.

Sound attenuation in low-temperature amorphous solids originates from their disordered structure. However, its detailed mechanism is still being debated. Here, we analyze sound attenuation starting directly from the microscopic equations of motion. We derive an exact expression for the zero-temperature sound damping coefficient. We verify that the sound damping coefficients calculated from our expression agree very well with results from independent simulations of sound attenuation. Small wavevector analysis of our expression shows that sound attenuation is primarily determined by the non-affine displacements' contribution to the sound wave propagation coefficient coming from the frequency shell of the sound wave. Our expression involves only quantities that pertain to solids' static configurations. It can be used to evaluate the low-temperature sound damping coefficients without directly simulating sound attenuation.

Numerical simulation of characteristics of wave propagation and reflection coefficient in a helix-acoustic black hole

A new type of the acoustic black hole beam—a helix-acoustic black hole—is proposed to overcome the spatial restriction on modular acoustic black hole structures. The modular acoustic black hole structure, consisted of a base and several number of acoustic black hole beams, has potential to apply into real engineering world. There are two main sections in an acoustic black hole beam: (1) a uniform thickness part and (2) an acoustic black hole region of which the thickness decreases according to the power-law profile. Conventional acoustic black hole beams can be ultimately assembled as 8–10 acoustic black hole beams on a modular acoustic black hole structure. In this article, a different shape of an acoustic black hole beam is newly designed to allow the assembly of more acoustic black hole beams on the modular acoustic black hole structure. The shape of the helix-acoustic black hole is such that the thickness of the acoustic black hole region smoothly decreases, just like a conventional acoustic black hole beam, as well as twisting along the longitudinal direction. The normal direction of the bottom surface in the uniform thickness is the same along the longitudinal axis in the conventional acoustic black hole beam. However, the normal direction of the helix-acoustic black hole of the bottom surface in the acoustic black hole region is different along the longitudinal direction. It is necessary to use numerical simulations to explore the performance of the helix-acoustic black hole beam because the shape of acoustic black hole region is different from the conventional one. Two types of numerical simulations were conducted: transient analysis and modal frequency analysis. From the transient analysis, the acoustic black hole effect was investigated by comparing the travel time which is dependent on the variation of the thickness. Using modal frequency analysis, the reflection coefficients between the conventional acoustic black hole beam and helix-acoustic black hole beam are also compared. It is noted that reflection coefficients were additionally compared depending on how “sharply” or “smoothly” they were twisted in the acoustic black hole region of the helix-acoustic black hole. Those results demonstrate that the helix-acoustic black hole has a dynamic characteristic similar to that of conventional acoustic black hole beams, which means that more helix-acoustic black holes can be assembled on the modular acoustic black hole structure by resolving the spatial restriction and leading to expectations of better performance.

A High-Frequency Model of a Rectilinear Beam with a T-Shaped Cross Section

This paper derives an analytical model of a straight beam with a T-shaped cross section for use in the high-frequency range, defined here as approximately 1 to 35 kHz. The web, the right part of the flange, and the left part of the flange of the T-beam are modeled independently with two-dimensional elasticity equations for the in-plane motion and Mindlin flexural plate equation for the out-of-plane motion. The differential equations are solved with unknown wave propagation coefficients multiplied by circular spatial domain functions. These algebraic equations are then solved to yield the wave propagation coefficients and thus produce a solution to the displacement field in all three directions. An example problem is formulated and compared with solutions from fully elastic finite element modeling, a previously derived analytical model, and Timoshenko beam theory. It is shown that the accurate frequency range of this new model is significantly higher than that of the analytical model and the Timoshenko beam model, and, in the frequency range up to 35 kHz, the results compare very favorably to those from finite element analysis.

A High-Frequency Model of a Circular Beam with a T-Shaped Cross Section

This paper derives an analytical model of a circular beam with a T-shaped cross section for use in the high-frequency range, defined here as approximately 1 to 50 kHz. The T-shaped cross section is composed of an outer web and an inner flange. The web in-plane motion is modeled with two-dimensional elasticity equations of motion, and the left portion and right portion of the flange are modeled separately with Timoshenko shell equations. The differential equations are solved with unknown wave propagation coefficients multiplied by Bessel and exponential spatial domain functions. These are inserted into constraint and equilibrium equations at the intersection of the web and flange and into boundary conditions at the edges of the system. Two separate cases are formulated: structural axisymmetric motion and structural non-axisymmetric motion and these results are added together for the total solution. The axisymmetric case produces 14 linear algebraic equations and the non-axisymmetric case produces 24 linear algebraic equations. These are solved to yield the wave propagation coefficients, and this gives a corresponding solution to the displacement field in the radial and tangential directions. The dynamics of the longitudinal direction are discussed but are not solved in this paper. An example problem is formulated and compared to solutions from fully elastic finite element modeling. It is shown that the accurate frequency range of this new model compares very favorably to finite element analysis up to 47 kHz. This new analytical model is about four magnitudes faster in computation time than the corresponding finite element models.

A high-frequency model of a circular beam with a T-shaped cross section

This paper derives an analytical model of a circular beam with a T-shaped cross section for use in the high-frequency range, defined here as approximately 1 to 50 kHz. The T-shaped cross section is composed of an outer web and an inner flange. The web in-plane motion is modeled with two-dimensional elasticity equations of motion, and the left portion and right portion of the flange are modeled separately with Timoshenko shell equations. The differential equations are solved with unknown wave propagation coefficients multiplied by Bessel and exponential spatial domain functions. These are inserted into constraint and equilibrium equations at the intersection of the web and flange and into boundary conditions at the edges of the system. Two separate cases are formulated: structural axisymmetric motion and structural non-axisymmetric motion and these results are added together for the total solution. The axisymmetric case produces 14 linear algebraic equations and the non-axisymmetric case produces 24 linear algebraic equations. These are solved to yield the wave propagation coefficients, and this gives a corresponding solution to the displacement field in the radial and tangential directions. The dynamics of the longitudinal direction are discussed but are not solved in this paper. An example problem is formulated and compared to solutions from fully elastic finite element modeling. It is shown that the accurate frequency range of this new model compares very favorably to finite element analysis up to 47 kHz. This new analytical model is about four magnitudes faster in computation time than the corresponding finite element models.

A Comprehensive Analytical Dynamic Model of a T-Beam

This paper derives a comprehensive analytical dynamic model of a T-shaped beam that includes in-plane and outof-plane vibrations for mid-frequency range analysis, defined here as approximately 1 kHz to 10 kHz. The web, right part of the flange, and left part of the flange of the T-beam are modelled independently with two-dimensional elasticity equations for the in-plane motion and the classical flexural plate equation for the out-of-plane motion. The differential equations are solved with unknown wave propagation coefficients multiplied by circular spatial domain functions, which are inserted into equilibrium and continuity equations at the intersection of the web and flange and into boundary conditions at the edges of the system resulting in 24 algebraic equations. These equations are solved to yield the wave propagation coefficients and this produces a solution to the displacement field in all three dimensions. An example problem is formulated and compared to solutions from Bickford beam theory and finite element analysis. Higher order branch waves are discussed and a simplified symmetric model is presented.

Elastic Response of Acoustic Coating on Fluid-Loaded Rib-Stiffened Cylindrical Shells

Reinforced cylindrical shells are used in numerous industries; common examples include undersea vehicles, aircraft, and industrial piping. Current models typically incorporate approximation theories to determine shell behavior, which are limited by both thickness and frequency. In addition, many applications feature coatings on the shell interior or exterior that normally have thicknesses which must also be considered. To increase the fidelity of such systems, this work develops an analytic model of an elastic cylindrical shell featuring periodically spaced ring stiffeners with a coating applied to the outer surface. There is an external fluid environment. Beginning with the equations of elasticity for a solid, spatial-domain displacement field solutions are developed incorporating unknown wave propagation coefficients. These fields are used to determine stresses at the boundaries of the shell and coating, which are then coupled with stresses from the stiffeners and fluid. The stress boundary conditions contain double-index infinite summations, which are decoupled, truncated, and recombined into a global matrix equation. The solution to this global equation results in the displacement responses of the system as well as the exterior scattered pressure field. An incident acoustic wave excitation is considered. Thin-shell reference models are used for validation, and the predicted system response to an example simulation is examined. It is shown that the reinforcing ribs and coating add significant complexity to the overall cylindrical shell model; however, the proposed approach enables the study of structural and acoustic responses of the coupled system.

Influence of high loss liquid on the bandwidth of a circular two-layer waveguide at microwaves

ABSTRACTThe electromagnetic waves propagation in a circular two-layer metal waveguide with a central low-loss dielectric rod surrounded by high loss liquid is studied in the microwave range. The dependence of the complex wave propagation coefficient on frequency and liquid layer thickness is obtained. The presence of high loss liquid surrounding the dielectric rod allows expanding the range of operating wavelengths of the standard waveguide from the millimeter to centimeter range, because the wave attenuation coefficient remains relatively small. We find that owing to a high loss liquid, such a waveguide does not have fixed values of cut-off frequencies. An extra low value of the wave attenuation coefficient is found in the centimeter wavelength range at certain liquid layer thicknesses. The proposed structure can be applied as a measuring cell for the determination of the complex permittivity of high loss liquids in small volumes and in a wide range of wavelengths.

An analytical model of a curved beam with a T shaped cross section

This paper derives a comprehensive analytical dynamic model of a closed circular beam that has a T shaped cross section. The new model includes in-plane and out-of-plane vibrations derived using continuous media expressions which produces results that have a valid frequency range above those available from traditional lumped parameter models. The web is modeled using two-dimensional elasticity equations for in-plane motion and the classical flexural plate equation for out-of-plane motion. The flange is modeled using two sets of Donnell shell equations: one for the left side of the flange and one for the right side of the flange. The governing differential equations are solved with unknown wave propagation coefficients multiplied by spatial domain and time domain functions which are inserted into equilibrium and continuity equations at the intersection of the web and flange and into boundary conditions at the edges of the system resulting in 24 algebraic equations. These equations are solved to yield the wave propagation coefficients and this produces a solution to the displacement field in all three dimensions. An example problem is formulated and compared to results from finite element analysis.

An Insight into Creeping Electromagnetic Waves around the Human Body

The electromagnetic wave propagation around human body torso is modelled by considering elementary electric and magnetic dipoles over an infinite muscle-equivalent cylinder. The poles in the spectral domain Green’s function with smallest imaginary part are found to correspond to creeping wave propagation coefficients which predict the general trend in propagation around human body. In addition, it was found that axial magnetic field component is crucial for communication via creeping waves since it generates modes with smaller field decay compared to axial electric field. The developed model may thus serve as a practical guideline in design of on-body wearable antennas. The theoretical considerations are verified with simulations and measurements on the prototype of PIFA antenna placed on the human body.

A higher order shear deformation model of a periodically sectioned plate

This talk develops a higher order shear deformation model of a periodically sectioned plate. A parabolic deformation expression is used with periodic analysis methods to calculate the displacement field as a function of plate spatial location. The problem is formulated by writing the transverse displacement field and the in-plane rotations as a series solution of unknown wave propagation coefficients multiplied by an exponential indexed wavenumber term in the direction of varying structural properties multiplied by an exponential constant term in the direction of constant structural properties. These expansions, along with various structural properties written using Fourier summations, are inserted into the governing differential equations that were derived using Hamilton’s principle. The equations are now algebraic expressions that can be orthogonalized and written in a global matrix format whose solution is the wave propagation coefficients, thus yielding the transverse and in-plane displacements of the...

Three dimensional wave propagation in time-varying materials: A mathematical model based on the weak solutions of continuity in the moving property interface

Abstract In this work, the elastic wave propagation through the moving property interface (MPI) in time-varying materials has been investigated. First, a general description of the MPI in time-varying materials has been given. Then, the elastic wave propagation through MPI has been systematically explored. In general, the wave propagation behaviors through MPI can be divided into two types depending on moving velocity of MPI and incident angle. For the first type, it has similar wave propagation behavior of that in static property interface, that is the reflection and refraction waves can be determined by the displacement and stress continuity in MPI. While, for the second type, it may have two transmitted waves, or only one transmitted wave or one reflected wave. Therefore, the transmitted waves or reflected wave cannot be solved by the commonly used continuous conditions, i.e. the displacement and stress continuities in MPI, due to the over constrained or under constrained problem. In order to deal with the wave propagation through MPI, a novel mathematical model is proposed based on the weak form of continuity in MPI. The results have shown that the wave propagation coefficients depend on not only the wave impedance radio and incident angle, but also the moving velocity of MPI. Besides, the transmitted and reflected wave frequencies and wavelengths are also significantly influenced by the moving velocity of MPI. When MPI catches up or exceeds the reflected or transmitted waves, the wavelength of the reflected or transmitted waves degenerates to zero, so that a shock wave is found in MPI and the displacements at the two sides of MPI are discontinuous. The wave propagation behaviors across MPI are further simulated via COMSOL Multiphysics software which shows very good agreement with the theoretical predictions. Moreover, the energy balance of SH waves in MPI is discussed. This work indicates that the novel mathematical model proposed in this paper is applicable to high dimensional wave phenomenon with moving boundaries, and is capable for studying complex wave equations, especially for wave equations with moving boundaries.

EXPERIMENTAL STUDY FOR DISPERSION AND ATTENUATION CORRECTION IN VISCOELASTIC SPLIT HOPKINSON PRESSURE BAR ARRANGEMENT

Split Hopkinson pressure Bar (SHPB) arrangement is the most practical and accurate method for studying dynamic mechanical behavior of materials under high strain rate. However in case of studying foams, and soft tissues (low impedance materials) a modified version of SHPB with bars made from a viscoelastic material like Poly methyl methacrylate PMMA acrylate is commonly used .so a good prediction of the stress waves and particle velocity acting at the interface between bars and specimen taking into account the dispersion and attenuation effect is essential. In this study the wave propagation coefficient as a function of the phase velocity and the attenuation coefficient in relation to the frequency are calculated experimentally by a set of free end bar tests using a cylindrical projectile to the circumstances of stepped velocity and different firing pressure. Correction processes for longitudinal stains related to incident and reflected waves are preceded in frequency domain using Fast Fourier Transformer (FFT) technique.

A Higher-Order Shear Deformation Model of a Periodically Sectioned Plate

This paper develops a higher-order shear deformation model of a periodically sectioned plate. A parabolic deformation expression is used with periodic analysis methods to calculate the displacement field as a function of plate spatial location. The problem is formulated by writing the transverse displacement field and the in-plane rotations as a series solution of unknown wave propagation coefficients multiplied by an exponential indexed wavenumber term in the direction of varying structural properties multiplied by an exponential constant term in the direction of constant structural properties. These expansions, along with various structural properties written using Fourier summations, are inserted into the governing differential equations that were derived using Hamilton's principle. The equations are now algebraic expressions that can be orthogonalized and written in a global matrix format whose solution is the wave propagation coefficients, thus yielding the transverse and in-plane displacements of the system. This new model is validated with finite-element theory and Kirchhoff plate theory for a thin plate simulation and verified with comparison to experimental results for a 0.0191 m thick sectional plate.

The influence of initial stress on wave propagation and dynamic elastic coefficients

The governing equations of wave propagation in one dimension of elastic continuum materials are investigated by taking the influence of the initial stress into account. After a short review of the theory of elastic wave propagation in a rock mass with an initial stress, results indicate that the initial stress differentially influences P-wave and S-wave propagation. For example, when the initial stress is homogeneous, for the P-wave, the initial stress only affects the magnitude of the elastic coefficients, but for the S-wave, the initial stress not only influences the elastic coefficients but also changes the governing equation of wave propagation. In addition, the P-wave and S-wave velocities were measured for granite samples at a low initial stress state; the results indicate that the seismic velocities increase with the initial stress. The analysis of the previous data of seismic velocities and elastic coefficients in rocks under ultra-high hydrostatic initial stress are also investigated.

Single rod impact tests: Analytical, experimental, and modelling investigation

This work investigates the single rod impact test technique, from which wave propagation characteristics, i.e. the propagation coefficient and wave speed, can be obtained and used to identify the material modulus as a function of frequency. To accomplish this, an elastic Titanium rod and a viscoelastic Polymethyl Methacrylate (PMMA) rod have been tested. With a known modulus, the Titanium rod modulus was successfully recovered from the wave propagation coefficient in a frequency range of 5 kHz ∼ 30 kHz, which verifies the validity of this technique. The PMMA modulus identified by this technique agrees reasonably with the modulus derived from DMA tests in the range 2 kHz ∼ 20 kHz. The good agreement between the theoretical and simulation-based modulus further confirms the effectiveness of the single rod impact test in this use. It is also demonstrated that the single rod impact technique is capable of detecting the minor mechanical changes induced by temperature variations as small as 2 °C.