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.     

Circumferential normal modes in an empty elastic cylinder

The so-called circumferential normal modes propagating in an empty elastic cylinder are considered. A dispersion equation for the wave numbers of these waves, an equation for the critical frequencies, and expressions for the eigenfunctions of such a waveguide are derived. Solutions to these equations are obtained by numerical methods for different values of the parameter d representing the relative thickness of the cylinder. An analysis of the solutions is performed, and the main properties of the dispersion curves are described, including those for the low-frequency waves of the new type, which correspond to the branches in the form of open loops. Individual normal modes are identified on the basis of the calculations and subsequent analysis of eigenfunctions.     

Vibration and parametric excitation in asymmetric circular plates under moving loads

The natural frequencies and modes of transverse vibration of circular plates containing small imperfections are determined through a perturbation method. Incision of equally spaced, equal-size radial slots at the rim of the plate creates asymmetry in some, but not all, of the vibration modes, and it causes the repeated natural frequencies of these modes in the symmetric plate to split into two distinct values. These vibration modes are called the split modes, and those associated with the repeated natural frequencies are called the repeated modes. A relationship identifying the split and repeated modes for any configuration of slots is presented. The vibration of a plate containing any number of thin slots cut into it at the rim and with any number of rotating linear springs is analyzed. Parametric instability can be excited in the split modes of the plate by the springs rotating below critical speed, but it cannot be excited in the repeated modes. The response of the plate in forms such as traveling or standing waves at parametric resonance is discussed. The theoretical predictions of split and repeated vibration modes and of the excitation of parametric instability are confirmed by experiments.     

A new multichannel time reversal focusing method for circumferential Lamb waves and its applications for defect detection in thick-walled pipe with large-diameter

This paper proposes a new multichannel time reversal focusing (MTRF) method for circumferential Lamb waves which is based on modified time reversal algorithm and applies this method for detecting different kinds of defects in thick-walled pipe with large-diameter. The principle of time reversal of circumferential Lamb waves in pipe is presented along with the influence from multiple guided wave modes and propagation paths. Experimental study is carried out in a thick-walled and large-diameter pipe with three artificial defects, namely two axial notches on its inner and outer surface respectively, and a corrosion-like defect on its outer surface. By using the proposed MTRF method, the multichannel signals focus at the defects, leading to the amplitude improvement of the defect scattered signal. Besides, another energy focus arises in the direct signal due to the partial compensation of dispersion and multimode of circumferential Lamb waves, alongside the multichannel focusing, during MTRF process. By taking the direct focus as a time base, accurate defect localization is implemented. Secondly, a new phenomenon is exhibited in this paper that defect scattered wave packet appears just before the right boundary of truncation window after time reversal, and to which two feasible explanations are given. Moreover, this phenomenon can be used as the theoretical basis in the determination of defect scattered waves in time reversal response signal. At last, in order to detect defects without prior knowing their exact position, a large-range truncation window is used in the proposed method. As a result, the experimental operation of MTRF method is simplified and defect detection and localization are well accomplished.     

Resonant freak microwaves

The Helmholtz equation describing transverse magnetic modes in a closed flat microwave resonator with 60 randomly distributed discs is numerically solved. At lower frequencies, the calculated wave intensity spatially distributed obeys the universal Porter-Thomas form if localized modes are excluded. A superposition of resonant modes is shown to lead to rare events of extreme intensities (freak waves) at localized “hot spots”. The temporally distributed intensity of such a superposition at the center of a hot spot also follows the Porter-Thomas form. Branched modes are found at higher frequencies. The results bear resemblance to recent experiments reported in an open cavity.     

Thickness-shear modes and magnetoelastic waves in a longitudinally magnetized ferromagnetic plate

Magnetoelastic (ME) waves and thickness-shear modes in the ferromagnetic plate are studied. Coupled vibrations of magnetization and shear elastic deformations excited simultaneously by a variable magnetic field propagate in two mutually perpendicular directions: parallel and normal to a surface. For parameters characteristic of isotropic ferromagnet with the sample magnetization and Zeeman field parallel to the surface, resonant frequencies of shear modes are computed and their dispersion law is examined. It is shown that the dependence of dimensional resonances frequencies on wave number k_z of ME wave propagating along saturating field direction occurs. The possibility of excitation of ME waves with different k_z explains multimode character of thickness ME resonances.     

On the criterion for effectiveness of wave linear transformation in a smoothly inhomogeneous medium and of nonadiabatic transitions during atomic collisions

A universal criterion for effectiveness of linear transformation of waves with locally close characteristic exponents in smoothly inhomogeneous media is obtained. The same criterion is applicable for estimating the effectiveness of nonadiabatic transitions in slow atomic collisions. The formalism developed for an analysis of the linear interaction of waves is based of the WKB asymptotic form of the solution of a scalar nth order ordinary differential equation. The obtained criterion can be applied in any practical problem for drawing a conclusion about the effectiveness of the linear interaction of modes if only the characteristic equation of waves in a homogeneous medium and the coefficients of the initial differential equation are known. In this case, the solution of the problem is reduced to elementary arithmetic calculations.     

Nonlinear development of instabilities in supersonic vortex sheets: I. The basic kink modes

Classical linearized stability analysis predicts (neutral) stability of supersonic vortex sheets for compressible flow with normalized Mach numbers, M > √2, while recent detailed numerical simulations by Woodward indicate the nonlinear development of instabilities for M > √2 through the development and interaction of propagating kink modes in the slip-stream. These kink modes are discontinuities in the slip-stream bracked by shock waves and rarefaction waves which grow self-similarly in time. In this paper, the apparent paradox is resolved by developing appropriate small amplitude high frequency nonlinear time-dependent asymptotic perturbed solutions which yield the response to a very small amplitude nonlinear planar sound wave incident on the vortex sheet. The analysis leads to three specific angles of incidence depending on M > √2 where nonlinear resonance occurs. For these three special resonant angles of incidence the perturbation expansions automatically yield simplified equations. These equations involve an appropriate Hamilton-Jacobi equation for the perturbed vortex sheet location; the derivative of the solution of this Hamilton-Jacobi equation provides boundary data for two nonlinear Burgers transport equations for the sound wave emanating from the two sides of the vortex sheet. These equations are readily solved exactly and lead to the quantitative time-dependent nonlinear development of three different types of kink modes with a structure similar to that observed by Woodward.     

Dispersion relation for surface-localized magnetic polaritons and magnetostatic waves in antiferromagnetic superlattice

In addition to the general dispersion equation for surface-localized magnetic polaritons and magnetostatic waves, which propagate in the system antiferromagnetic superlattice-antiferromagnet, we derive a simplified result for the long-wavelength limit λ?L (L is the period of superlattice) when the superlattice is found to behave like an anisotropic bulk medium (effective-medium approach). The dispersion curves and frequency region of the existence of the surface magnetic polaritons and magnetostatic waves are presented numerically for several values of the external magnetic field and for different antiferromagnetic materials.     

管道结构中的类兰姆波层析成像

理论和实验上研究了层析成像在管道结构导波检测中的适用性。在管的两个圆周带上分别布置发射和接收换能器,形成多点发射-多点接收的几何结构。将管中绕管道沿最短路径的螺旋方向到达接收换能器的模式作为感兴趣的模式,它们看起来像在弯曲的板中传播,称之为类兰姆波模式。给出了类兰姆波各射线路径走时的理论计算方法,并采用联合迭代重建技术(SIRT)对走时数据进行慢度图像重建。数值模拟和实验研究了管中横穿孔缺陷层析成像的输出并对健康管和缺陷管中类兰姆波的走时数据进行比较,发现穿过缺陷区域的类兰姆波因为绕过缺陷传播而呈现出更长的走时。文中也给出了两缺陷管道类兰姆波层析成像的数值模拟和实验结果,相比单缺陷的情况,射线较少穿过区域缺陷的重建图像质量下降。研究结果为采用类兰姆波模式进行管道缺陷无损评价提供了理论依据。      

Ultrasonic guided wave parameters for detection of axial cracks in feeder pipes of PHWR nuclear power plants

The dispersion curves for the feeder pipes in PHWR nuclear power plants were determined. The wave modes used for the detection of notches in the feeder pipe were confirmed as F(m,2) and/or L(0,1) by an analysis of short time Fourier transformation (STFT). The axial notches in the straight pipe were not detectable, but an axial notch in a bent pipe was detected with the mode at the frequency of 500 kHz. Initial F(m,2) and/or L(0,1) modes contains a circumferential displacement and might be converted to certain complicated modes in the bent region, which is sensitive to the axial notch. The circumferential guided wave technique was also applied for quantitative evaluation of the axial notches. The waves generated by a rocking motion of the transducer along the circumferential direction were estimated as the circumferential guided waves after a review of the acquired data and the dispersion curves.     

Waveguide properties of a plane ring-shaped plate. I. flexural waves

A new type of the so-called “angular” waves for flexural normal modes propagating in a ringshaped plate is studied. A dispersion equation for wave numbers, an equation for critical frequencies, and expressions for the eigenfunctions of such a waveguide are derived. Solutions to these equations are obtained by numerical methods for various values of parameter d, which represents the relative width of the ring. The solutions are analyzed, and the main properties of dispersion curves are described. Individual normal modes are identified on the basis of the calculation and further analysis of eigenfunctions.     

An anisotropic model for frequency analysis of arterial walls with the wave propagation approach

An anisotropic model for calculating natural frequency of arterial walls is proposed in this paper. The first-order shear deformation theory (FSDT) is used for the arterial walls, and the wave propagation approach is applied that can easily handle the boundary conditions. Results obtained using this model have been evaluated against those available in the literature and the agreement has been found to be good. Experiments were carried out on a natural rubber latex tube. The relative differences of the first four natural frequencies between the experiment and the theory are less than 7%. The variation of the natural frequency of this tube with the longitudinal and circumferential modes m and n is studied which suggests the first four natural frequencies are with n = 1 and m = 1-4. Simulations show that classical Donnell’s, Love’s and beam theories are not suitable for this thick tube while FSDT results closely agree with the experiment. The anisotropy of circumferential elastic modulus on natural frequencies of the tube is analyzed.     

Frequency and amplitude alterations of sound modes in a space-dependent random mass density field

Abstract

We investigate the effect of a space-dependent random mass density field on small amplitude acoustic modes that are settled in a semi-infinite medium of a temperature growing linearly with depth. Using a perturbation method, the dispersion relation is derived in the form of Hill's determinant. Numerical solutions of this equation lead to the following conclusions: (a) a weak random field (with σ_eff?=?0.05) essentially affects long waves which experience attenuation and a frequency reduction; (b) for a stronger random field (with σ_eff?=?0.1), high-order sound modes behave as sound waves as they are attenuated and their frequencies are increased; (c) for a sufficiently strong random field (with σ_eff?=?0.2), mode coupling occurs, as a result of which the dispersive curves cross each other, the sound modes loose their identities, and some modes are amplified. Here σ_eff denotes the effective strength of a random field.     

Dispersion of circumferential waves in cylindrically anisotropic layered pipes in plane strain

Dispersion spectra of circumferential waves along the periphery of circular pipes made of layered anisotropic materials do not seem to be available in literature. This note attempts to partially fill this gap by providing the dispersion spectra in two and three layered cylindrically anisotropic pipes in plane strain motion. The spectra for pipes executing time harmonic vibrations in plane strain condition are obtained as roots of a numerical characteristic equation derived extending a weighted residual method of solution of the governing equations for a single layer pipe [Towfighi et al., J. Appl. Mech. 69, 283-291 (2002)] to a general N layered pipe. The anisotropic elastic coefficients are considered to be independent of position coordinates and the bond condition at interfaces of the layers is assumed to be perfect. Numerical illustrations are presented for two and three layered pipes with anisotropy directions differing in adjacent layers. Increase in curvature of the pipe and inclination of the fiber orientation in the outermost layers to propagation direction are factors that seem to influence the mode number and pattern within the limited examples worked out.     

Rigorous solutions for sound radiation from circular ducts with hyperbolic horns or infinite plane baffle

A rigorous treatment is presented of sound radiation from circular ducts with either a hyperbolic horn or an infinite plane baffle. In the analysis hyperboloidal wave functions are used, which are defined here, for the first time, as a class of eigensolutions of the wave equation for oblate spheroidal co-ordinates. The numerical results include the complex conversion (or reflection) coefficients and the radiation directivity for various incident wave modes, spinning modes as well as axisymmetric modes. The solutions are valid for the whole frequency range including frequencies above and below the cut-off frequencies of the duct modes involved.     

Natural frequencies of free finite-length circular cylinders

Sets of tables are given for the natural frequencies of the first five symmetric and first five antisymmetric modes of a hollow or solid cylinder for circumferential wave numbers n = 0, 1, 2. Contour graphs are given for the lowest frequencies as functions of (length)/(mean radius) and (thickness)/(mean radius).     

Free vibration of a pneumatic tire-wheel unit using a ring on an elastic foundation and a finite element model

Natural frequencies and mode shapes of a pneumatic tire without suspension are investigated using a 12-d.o.f., geometrically non-linear, doubly curved, thin shell finite element of revolution with laminate composite materials. The wheel is assumed to be free to move within its own plane. The results of the free vibration analysis indicate that only the radial modes of n = 1 are affected by the wheel's freedom to move. To evaluate the finite element modeling, a simplified elastic ring-spring model is studied. The tire is modeled as a circular, elastic ring supported by distributed spring in both radial and circumferential directions. The wheel is modeled as a rigid mass to which the disturbed spring is attached. The two models are found to agree and complement each other. While the simplified ring-spring model is easy and practical to use to obtain preliminary results, the complex finite element model can give more detailed and accurate results for both free vibration and dynamic response analyses.     

Locally nonequilibrium magnetic excitations in insulators

The equation of motion for the magnetic moment vector M in a locally nonequilibrium medium is derived. The dispersion and attenuation of the coupled modes of the magnetic vector potential and magnetization are determined. It is shown that the continuous spectrum contains frequencies corresponding to undamped waves or constant-phase damped waves.     

Characterization of circumferential guided waves in a cylindrical cortical bone-mimicking phantom

The femoral neck cortical shell was recently demonstrated to act like a waveguide for circumferential waves. Femoral neck assessment with ultrasound could be enhanced by guided waves measurement. In this study, the decomposition of the time reversal operator (DORT) method is used to measure the phase velocities of circumferential guided modes in a circular tube with dimensions characteristic of femoral neck. The tube is made of a bone-mimicking material. Five guided modes are obtained and compared to theoretical predictions. The work substantiates the feasibility of measuring guided waves in a relatively thick tube of attenuating material with the DORT method.