Exact wave-based Bloch analysis procedure for investigating wave propagation in two-dimensional periodic lattices

This paper presents an exact, wave-based approach for determining Bloch waves in two-dimensional periodic lattices. This is in contrast to existing methods which employ approximate approaches (e.g., finite difference, Ritz, finite element, or plane wave expansion methods) to compute Bloch waves in general two-dimensional lattices. The analysis combines the recently introduced wave-based vibration analysis technique with specialized Bloch boundary conditions developed herein. Timoshenko beams with axial extension are used in modeling the lattice members. The Bloch boundary conditions incorporate a propagation constant capturing Bloch wave propagation in a single direction, but applied to all wave directions propagating in the lattice members. This results in a unique and properly posed Bloch analysis. Results are generated for the simple problem of a periodic bi-material beam, and then for the more complex examples of square, diamond, and hexagonal honeycomb lattices. The bi-material beam clearly introduces the concepts, but also allows the Bloch wave mode to be explored using insight from the technique. The square, diamond, and hexagonal honeycomb lattices illustrate application of the developed technique to two-dimensional periodic lattices, and allow comparison to a finite element approach. Differences are noted in the predicted dispersion curves, and therefore band gaps, which are attributed to the exact procedure more-faithfully modeling the finite nature of lattice connection points. The exact method also differs from approximate methods in that the same number of solution degrees of freedom is needed to resolve low frequency, and arbitrarily high frequency, dispersion branches. These advantageous features may make the method attractive to researchers studying dispersion characteristics, band gap behavior, and energy propagation in two-dimensional periodic lattices.     

Guided wave propagation in single and double layer hollow cylinders embedded in infinite media

Millions of miles of pipes are being used for the transportation, distribution, and local use of petroleum products, gas, water, and chemicals. Most of the pipes are buried in soil, leading to the significance of the study on the subject of guided wave propagation in pipes with soil influence. Previous investigations of ultrasonic guided wave propagation in an elastic hollow cylinder and in an elastic hollow cylinder coated with a viscoelastic material have led to the development of inspection techniques for bare and coated pipes. However, the lack of investigation on guided wave propagation in hollow cylinders embedded in infinite media like soil has hindered the development of pipe inspection methods. Therefore the influence of infinite media on wave propagation is explored in this paper. Dispersion curves and wave structures of both axisymmetric and nonaxisymmetric wave modes are developed. Due to the importance of the convergence of numerical calculations, the requirements of thickness and element number of the finite soil layer between hollow cylinder and infinite element layer are discussed, and an optimal combination is obtained in this paper. Wave structures are used for the mode identification in the non-monotonic region caused by the viscoelastic properties of coating and infinite media.     

Analysis of wave propagation in cylindrical pipes with local inhomogeneities

In this paper, we present a numerical approach to study the guided elastic wave propagation in cylindrical pipes with local inhomogeneities. A hybrid wave finite element (WFE) and finite element (FE) technique is introduced to investigate the dispersion and wave scattering in pipes by taking full advantage of the existing FE codes. Dynamic reduction technique is employed to improve the computational efficiency, which is particularly suitable for the pipes with standard local features. Numerical examples indicate that the proposed technique provides an effective way to calculate the dispersion relationship and the scattered field. Both the axisymmetric and non-axisymmetric wave scattering problems are considered.     

Vibration characteristic analysis of buried pipes using the wave propagation approach

A theoretical analysis for the free vibration of simply supported buried pipes has been investigated using the wave propagation approach. The pipe modeled as a thin cylindrical shell of linear homogeneous isotropic elastic material buried in a linear isotropic homogeneous elastic medium of infinite extent. The vibrations of the pipe are examined by using Flüggle shell equation. The natural frequencies are obtained for the pipes surrounded by vacuo or elastic medium. The results are compared with those available in the literature and agreement is found with them. It is found that the free vibration frequency of the pipe does not appear for some of the axial or circular vibration modes and the real natural frequencies of the pipe are significantly dependent on the rigidity of the surrounding medium.     

Natural beam focusing of non-axisymmetric guided waves in large-diameter pipes

The propagation of non-axisymmetric guided waves in larger diameter pipes is studied in this paper by treating the guided waves as corresponding Lamb waves in an unwrapped plate. This approximation leads to a simpler method for calculating the phase velocities of hollow cylinder guided waves, which reveals a beam focusing nature of non-axisymmetric guided waves generated by a partial source loading. The acoustic fields in a pipe generated by a partial-loading source includes axisymmetric longitudinal modes as well as non-axisymmetric flexural modes. The circumferential distribution of the total acoustic field, also referred as an angular profile, diverges circumferentially while guided waves propagate with dependence on such factors as mode, frequency, cylinder size, propagation distance, etc. Exact prediction of the angular profile of the total field can only be realized by numerical calculations. In particular cases, however, when the wall thickness is far less than the cylinder diameter and the wavelength is smaller than or comparable to the pipe wall thickness, the acoustic field can be analyzed based on the characteristics of Lamb waves that travel along a periodic unwrapped plate. Based on this assumption, a simplified model is derived to calculate the phase velocities of non-axisymmetric flexural mode guided waves. The model is then applied to discussions on some particular characteristics of guided-wave angular profiles generated by a source loading. Some features of flexural modes, such as cutoff frequency values are predicted with the simpler model. The relationship between the angular profiles and other factors such as frequency, propagation distance, and cylinder size is obtained and presented in simple equations. The angular profile rate of change with respect to propagation distance is investigated. In particular, our simplified model for non-axisymmetric guided waves predicts that the wave beam will converge to its original circumferential shape after the wave propagates for a certain distance. A concept of "natural focal point" is introduced and a simple equation is derived to compute the 1st natural focal distance of non-axisymmetric guided waves. The applicable range of the simplified equation is provided. Industrial pipes meet the requirement of wall thickness being far less than the pipe diameter. The approximate analytical algorithms presented in this paper provides a convenient method enabling quick acoustic field analysis on large-diameter industrial pipes for NDE applications.     

Modelling of linear wave propagation in spatial fluid filled pipe systems consisting of elastic curved and straight elements

This paper is concerned with the theoretical analysis of time harmonic dynamics of compound elastic pipes with and without internal fluid loading. Compound pipes are assembled as a sequence of segments, each of which has a constant curvature. As a prerequisite for the wave propagation analysis, dispersion equations are solved, Green’s matrices are formulated and Somigliana’s identities are derived for an isolated curved segment. The governing equations of wave motion of a compound pipe are obtained as an ensemble of the boundary integral equations for individual segments and the continuity conditions at their interfaces. The proposed methodology is validated in several benchmark problems and then applied for analysis of the periodicity effects. The results obtained for piping systems with a variable number of identical curved segments are put into the context of the classical Floquet theory. Brief parametric studies suggest that the curved inserts can be employed as a tool for the passive control of wave propagation in fluid-filled pipes, and their stop band characteristics may be tailored to reach desirable attenuation levels in prescribed frequency ranges.     

Free wave propagation in two-dimensional periodic plates

The propagation of flexural waves in a two-dimensional periodic plate which rests on an orthogonal array of equi-spaced simple line supports has been investigated. A type of plane wave motion has been considered. An energy method has been developed to predict the frequency of wave propagation in terms of the propagation constants. A Galerkin type of analysis has been used, incorporating assumed complex modes of wave motion for the identical rectangular elements of the periodic plate. Expressions for the frequency have been obtained firstly by using simple polynomial modes for the plate displacements, and then (alternatively) by using characteristics beam function modes. The use of these different modes has first been demonstrated by applying them to the analysis of wave propagation in periodic beams. A single polynomial mode which satisfies the geometric and wave-boundary conditions of the periodic plate element leads to an elegant expression relating the frequency and the wave propagation constants in the first propagation band. The frequencies so obtained compare well with those found from a multi-mode, characteristic beam function analysis. The latter involves much more algebra, is solved as an eigenvalue problem, and yields the frequencies in as many propagation bands as are desired. The bounding frequencies and corresponding wave motions in the first and higher propagation bands have been identified, and it has been shown that the propagation bands can overlap. Consideration has been given to one-dimensional “strip” structures which are equivalent to the two-dimensional plate when a plane wave in a general direction is propagating. Furthermore, it is shown that the natural frequencies of finite rectangular periodic plates can be obtained very simply from the results of the wave propagation analysis.     

Attenuation characteristics of the fundamental modes that propagate in buried iron water pipes

The attenuation of the fundamental non-torsional modes that propagate down buried iron water pipes has been studied. The mode shapes, mode attenuation due to leakage into the surrounding medium and the scattering of the modes as they interact with pipe joints and fittings have been investigated. In the low frequency region the mode predicted to dominate over significant propagation distances approximates a plane wave in the water within the pipe. The established acoustic technique used to locate leaks in buried iron water pipes assumes that leak noise propagates as a single non-dispersive mode at a velocity related to the low frequency asymptote of this water borne mode. Experiments have been conducted on buried water mains at test sites in the UK to verify the attenuation and velocity dispersion predictions.     

Propagation characteristics of highly elliptical core optical waveguides: a perturbation approach

A simple first-order perturbation approach to study the propagation characteristics of highly elliptical core waveguides is presented. The unperturbed index profile is taken to be a pseudo-rectangular core waveguide for which exact scalar wave modes are known. The method is used to obtain the various propagation characteristics of elliptical core, single mode waveguides. Results for various propagation characteristics obtained by the present analysis agree very well with those reported by other authors obtained by accurate numerical techniques.     

Energy concentration at the center of large aspect ratio rectangular waveguides at high frequencies

Waveguides in non-destructive evaluation (NDE) applications are commonly of a regular geometry (e.g., circular and ring cross section) for which analytical solutions exist. In this paper, wave propagation in infinitely long strips of large rectangular aspect ratio is discussed. Due to the finite width of strips, a large number of modes exist within the structure. This complicates the analysis and usually discourages the use of strip waveguides in NDE sensors. However, it is shown that among the many modes of a strip, there are some with very desirable properties. This is highlighted by the example of two guided wave modes of a large aspect ratio rectangular strip whose dispersion characteristics approach those of the fundamental modes of an infinitely wide plate at high frequencies. The energy of these modes concentrates in the central region of the strip and decays toward the edges so that the strip waveguide can easily be mechanically attached to other components without influencing the wave propagation. Dispersion curves and mode shapes were derived by using a semianalytical finite element technique and are presented over a range of frequencies. It is shown that selective excitation of both modes is possible in practice and the experimental setup is described.     

Phased array focusing with guided waves in a viscoelastic coated hollow cylinder

Guided wave phased array focusing has shown many advantages in long-range pipeline inspection, such as, longer inspection distance, greater wave penetration power and higher detection resolution. Viscoelastic coatings applied to a large percentage of pipes for protection purposes created some challenges in terms of focusing feasibility and inspection ability. Previous studies were all based on bare pipe models. In this work, guided wave phased array focusing in viscoelastic coated pipes is studied for the first time. Work was carried out with both numerical and experimental methods. A three-dimensional finite element model was developed for quantitatively and systematically modeling guided waves in pipes with different viscoelastic materials. A method of transforming measured coating properties to finite element method inputs was created in order to create a physically based model of guided waves in coated pipes. Guided wave focusing possibilities in viscoelastic coated pipes and the effects from coatings were comprehensively studied afterwards. A comparison of focusing and nonfocusing inspections was also studied quantitatively in coated pipe showing that focusing increased the wave energy and consequently the inspection ability tremendously. This study provides an important base line and guidance for guided wave propagation and focusing in a real field pipeline under various coating and environmental conditions.     

管道弯头对低频纵向导波传播特性影响分析

管道弯头显著改变了导波传播特性,影响了对检测信号的解读,研究弯头对导波传播特性的影响是实现复杂管道系统导波检测的基础。采用半解析有限元法计算弯管导波频散曲线,分析了弯管导波频散曲线所呈现的不同特征,并基于弯管导波频散曲线,以低频L(0,1)模态导波为研究对象,实验研究了低频L(0,1)模态导波通过管道弯头时的模态变换特征。研究结果发现,当L(0,1)模态导波通过管道弯头时,不仅会发生L(0,1)到F(1,1)的模态变换,还会模态变换出反向L(0,1)模态导波,即弯头反射现象,且随着激励频率的降低和弯头弯曲半径的减小,弯头反射现象愈发明显。研究结果将深化对弯管导波传播特性的认识,推动导波检测技术在复杂管道系统检测中的应用。      

Electromagnetic Wave Propagation in a Magneto-Plasma Filled Coaxial Structure: I, Theoretical

This study is concerned with the problem of electromagnetic wave propagation in a magneto-plasma filled coaxial structure. The problem is formulated using the classical boundary value problem approach. Based on this formulation an extensive numerical investigation is performed. The numerical investigation shows the existence of propagating slow modes, backward modes, a quasi-TEM mode, and waveguide-type modes in a magneto-plasma filled coaxial structure. Dispersion curves for these different modes are presented. The dependence of the different modes on various plasma parameters is considered.     

Whistling of a pipe system with multiple side branches: Comparison with corrugated pipes

Corrugated pipes are widely used because they combine local rigidity with global flexibility. Whistling induced by flow through such pipes can lead to serious environmental and structural problems. The whistling of a multiple side branch system is compared to the whistling behavior of corrugated pipes. The study has been restricted to cavities with sharp edges which are convenient for theoretical modeling. The side branch depth is chosen to be equal to the side branch diameter, which corresponds to cavity geometries in typical corrugated pipes. The low frequency resonance modes of the multiple side branch system have been predicted by means of acoustic models, of which the validity has been tested experimentally. Several experiments have been carried out for characterizing the whistling behavior of the system. While the behavior of a multiple side branch system is interesting on its own it can be compared to that of corrugated pipes. These experiments show that the multiple side branch system is in many aspects a reasonable model for corrugated pipes. Advantage of the multiple side branch system is that it is an experimental setup allowing easy modification of cavity depth. We used this feature to identify the pressure nodes of the acoustic standing wave along the main pipe as the regions where sound is produced. This contradicts recent publications on corrugated pipes. Another interesting aspects is that the system appears to whistle at the second hydrodynamic mode of the cavities rather than at the first hydrodynamic mode. A prediction model for the whistling behavior is proposed, consisting of an energy balance, based on the vortex sound theory. The model predicts the observed Strouhal number but overestimates the acoustic fluctuation amplitude by a factor four.     

NETWORK CHARACTERIZATION OF STEP DISCONTINUITY WITH GENERALIZED CONSERVATION OF COMPLEX POWER TECHNIQUE AND ITS APPLICATION TO ANALYZING PERIODIC DIELECTRIC WAVEGUIDES

The propagation of electromagnetic waves along open periodic, dielectric waveguides is formulated in the case that surface wave is guided and propagates normally to the corrugation. Our approximate analysis with the propagation characteristics is to consider a corresponding bounded waveguide problem in which perfect electric or magnetic walls are introduced, and the periodic corrugation is regarded as consisting of step discontinuities connected by a length of uniform slab waveguide. By properly taking into account of both surface modes and only a few non-surface-modes, a novel network approach is proposed for characterizing step discontinuity based on the generalized conservation of complex power technique (GCCPT). Employing solution selection rule (SSR), we can readily derive propagation characteristics in the Bragg interaction region. A number of numerical results are shown to demonstrate the usefulness of our approach.     

Free vibration of thin circular rings on periodic radial supports

Natural frequencies and normal modes are obtained for in-plane, inextensional vibrations of a thin circular ring with equi-spaced, identical radial supports. A wave approach is used. Natural frequencies are determined from the propagation constants of the ring by considering it as an endless periodic structure. Normal modes are obtained by superposition of a pair of opposite-going free wave groups. Numerical results have been presented for both rigid and circumferentially guided supports. It has been shown that at certain frequencies two different natural modes can exist. This has been verified experimentally.     

On the effect of viscosity and thermal conductivity on sound propagation in ducts: A re-visit to the classical theory with extensions for higher order modes and presence of mean flow

The dispersion equation for the axisymmetric modes of viscothermal acoustic wave propagation in uniform hard-walled circular ducts containing a quiescent perfect gas is classical. This has been extended to cover the non-axisymmetric modes and real fluids in contemporary studies. The fundamental axisymmetric mode has been the subject of a large number of studies proposing approximate solutions and the characteristics of the propagation constants for narrow and wide ducts with or without mean flow is well understood. In contrast, there are only few publications on the higher order modes and the current knowledge about their propagation characteristics is rather poor. On the other hand, there is a void of papers in the literature on the effect of the mean flow on the quiescent modes of propagation. The present paper aims to contribute to the filling of these gaps to some extent. The classical theory is re-considered with a view to cover all modes of acoustic propagation in circular ducts carrying a real fluid moving axially with a uniform subsonic velocity. The analysis reveals a new branch of propagation constants for the axisymmetric modes, which appears to have escaped attention hitherto. The solution of the governing wave equation is expressed in a modal transfer matrix form in frequency domain and numerical results are presented to show the effects over wide ranges of frequency, viscosity and mean flow parameters on the propagation constants. The theoretical formulation allows for the duct walls to have finite impedance, but no numerical results are presented for lined ducts or ducts carrying a sheared mean flow.     

黏弹介质包裹的充液管道中低频轴对称波传播特性

研究埋地充液管道中低频轴对称波传播特性。将土壤考虑为黏弹介质,结合Kennard薄壳方程和Kelvin-Voigt线性黏弹性模型,引入土壤载荷矩阵,推导出土-管滑移情形下流体主导波和管壁压缩波的相速度表达式。通过数值模拟计算得到流体主导波和管壁压缩波的频散和衰减曲线并进行可靠性验证,分析两种波引起的管壁径向位移之比,讨论厚径比和品质因子对流体主导波传播的影响。结果表明,黏弹介质对流体主导波和管壁压缩波的相速度影响较小,但对衰减影响较大;流体主导波对管壁径向位移有较大的影响,是泄露噪声传播的主要载体;厚径比越大,流体主导波的相速度越大,衰减越小;而品质因子越大,流体主导波的频散和衰减都越小。研究结果可为埋地充液管道的泄漏检测提供一定的理论参考。      

The contribution of a lateral wave in simulating low-frequency sound fields in an irregular waveguide with a liquid bottom

A further development of a previously proposed approach to calculating the sound field in an arbitrarily irregular ocean is presented. The approach is based on solving the first-order causal mode equations, which are equivalent to the boundary-value problem for acoustic wave equations in terms of the cross-section method. For the mode functions depending on the horizontal coordinate, additional terms are introduced in the cross-section equations to allow for the multilayer structure of the medium. A numerical solution to the causal equations is sought using the fundamental matrix equation. For the modes of the discrete spectrum and two fixed low frequencies, calculations are performed for an irregular two-layer waveguide model with fluid sediments, which is close to the actual conditions of low-frequency sound propagation in the coastal zone of the oceanic shelf. The calculated propagation loss curves are used as an example for comparison with results that can be obtained for the given waveguide model with the use of adiabatic and one-way propagation approximations.     

Simulation of low-frequency sound propagation in an irregular shallow-water waveguide with a fluid bottom

A further development of a previously proposed approach to calculating the sound field in an arbitrarily irregular ocean is presented. The approach is based on solving the first-order causal mode equations, which are equivalent to the boundary-value problem for acoustic wave equations in terms of the cross-section method. For the mode functions depending on the horizontal coordinate, additional terms are introduced in the cross-section equations to allow for the multilayer structure of the medium. A numerical solution to the causal equations is sought using the fundamental matrix equation. For the modes of the discrete spectrum and two fixed low frequencies, calculations are performed for an irregular two-layer waveguide model with fluid sediments, which is close to the actual conditions of low-frequency sound propagation in the coastal zone of the oceanic shelf. The calculated propagation loss curves are used as an example for comparison with results that can be obtained for the given waveguide model with the use of adiabatic and one-way propagation approximations.