The reflection and transmission of elastic waves through a plane of spheres in periodic arrangement

The work presented in this paper focuses on the reflection and transmission coefficients of an incident plane wave which impinges obliquely a plane of identical spheres arranged periodically in a homogeneous host with infinite extension. The Bloch theorem of periodic structure and the addition theorem of spherical wave functions are used to obtain the total scattering wave from all spherical scatterers periodically arranged in a plane. The total scattering wave in series form of spherical wave functions is then transformed into plane wave form in order to derive the reflection and transmission coefficients. Some numerical examples are given for different size, material constants and array patterns of spherical scatterers, and their influences on the reflection and transmission coefficients of a plane of spheres are discussed based on the numerical results. This study implies that a plane of spheres can be elaborately designed to serve as a sound barrier at a certain frequency range.     

Resonance scattering of elastic waves by an elastic inclusion in an elastic matrix. Numerical calculations

The scattering of an incident plane ultrasonic (longitudinal) wave by an elastic spherical inhomogeneity contained within an elastic matrix is studied. The emphasis is on the computation and analysis of basic multiple cases that result when different material behaviours are present in the matrix and in the inclusion. These calculations are useful in underwater acoustic applications. The behaviors are dominated by the soft or rigid backgrounds of the resonance scattering theory (RST). The first three multiple coefficients appearing in the expansions for the total elastodynamic fields developed around the inhomogeneity during the scattering process have been calculated in suitable frequency bands in all the cases considered. The examination of modulus, the real parts, and the imaginary parts of these (complex) coefficients under the RST approach allows the quantitative assessment of the conditions under which monopole or dipole resonances will occur and their relative magnitudes. The decomposition of the multiple coefficients into their resonance and background portions shows that it is the upward frequency shift of the background curves that controls the dominance of either radial (monopole) or translation (dipole) oscillations of the inclusion. This has an effect on the dispersion curves of the composite, which develop optical as well as acoustical branches. The real and imaginary parts of the multiple coefficients are respectively proportional to the attenuation and the effective wave speed in this simple inhomogeneous composite.     

Generalized theory of resonance excitation by sound scattering from an elastic spherical shell in a nonviscous fluid

This work presents the general theory of resonance scattering (GTRS) by an elastic spherical shell immersed in a nonviscous fluid and placed arbitrarily in an acoustic beam. The GTRS formulation is valid for a spherical shell of any size and material regardless of its location relative to the incident beam. It is shown here that the scattering coefficients derived for a spherical shell immersed in water and placed in an arbitrary beam equal those obtained for plane wave incidence. Numerical examples for an elastic shell placed in the field of acoustical Bessel beams of different types, namely, a zero-order Bessel beam and first-order Bessel vortex and trigonometric (nonvortex) beams are provided. The scattered pressure is expressed using a generalized partial-wave series expansion involving the beam-shape coefficients (BSCs), the scattering coefficients of the spherical shell, and the half-cone angle of the beam. The BSCs are evaluated using the numerical discrete spherical harmonics transform (DSHT). The far-field acoustic resonance scattering directivity diagrams are calculated for an albuminoidal shell immersed in water and filled with perfluoropropane gas, by subtracting an appropriate background from the total far-field form function. The properties related to the arbitrary scattering are analyzed and discussed. The results are of particular importance in acoustical scattering applications involving imaging and beam-forming for transducer design. Moreover, the GTRS method can be applied to investigate the scattering of any beam of arbitrary shape that satisfies the source-free Helmholtz equation, and the method can be readily adapted to viscoelastic spherical shells or spheres.     

The scattering of electro-elastic waves by a spherical piezoelectric particle in a polymer matrix

This paper is devoted to the study of scattering of plane harmonic waves by a piezoelectric sphere with spherical isotropy embedded in an unbounded isotropic polymer matrix. The scattered displacement field and the electric potential in the matrix are expressed in terms of spherical vector wave functions and spherical harmonic functions, respectively. For the field points inside the inhomogeneity, new displacement functions are introduced. Expansion of the new displacement functions and the electric potential in terms of spherical harmonic functions, the equations of motion and electrostatic lead to four second order ordinary differential equations (odes), where three of them are coupled. The coupled system of odes is solved by the generalized Frobenius series. This approach is readily used to handle low and high frequencies. Three different types of piezoelectric inhomogeneities, PZT-4, PZT-5H, and BaTiO₃ are considered and the associated piezoelectric effects on the electro-mechanical fields, differential and total scattering cross-sections are addressed.     

Magnetoelastic wave tunneling via a gap between ferroelectric crystals with relative longitudinal displacement

The tunneling of a plane monochromatic acoustic wave via a gap between two ferromagnetic crystals exhibiting relative longitudinal displacement has been studied. It is shown that, at a gap width comparable with the wavelength, the acoustic wave can exhibit complete transmission at the Damon-Eshbach mode frequency. If the gap width is much smaller than the wavelength, the complete transmission takes place at two resonance frequencies. Allowance for the longitudinal displacement of one crystal leads in all cases to violation of the resonance conditions, which results in a significant decrease in the transmission coefficient. The greater the velocity of crystal displacement, the stronger the decrease in the acoustic wave transmission though the gap between ferromagnetic crystals.     

Mie resonances and Bragg-like multiple scattering in opacity of two-dimensional photonic crystals

The lowest (main) and high-order Mie resonances and the Bragg-like multiple scattering of electromagnetic (EM) waves are determined as three mechanisms of formation and frequency position of two opaque bands, with narrow peaks in one of the bands in the transmission spectra of 2D photonic crystals composed of dielectric cylinders arranged parallel to the EM wave's electric vector in the square lattice. The main Mie resonance in a single cylinder defines the frequency position of the main gap whose formation results from the Bragg-like scattering. An additional gap with narrow transmission peaks opens in the spectrum of a cylinder layer and becomes pronounced with the number of layers. It is argued that higher-order Mie resonances are responsible for the transmission peaks within the additional band of a perfect crystal. It is shown that 2D photonic crystals with a filling factor ranging from 3% to 20% at a fixed crystal period may be a good zero approximation to study wave transmission through a localizing 2D dense random medium slab.     

焦区附近球形粒子对高斯光束的弹性散射

本文应用复源球面波理论,将高斯光束场按矢量球波函数展开,对高斯光束入射到单个球形粒子上时的弹性散射问题进行了理论分折。并对球形粒子在波束传播轴上时的远场散射光强角分布进行了数值计算,同时还与平面波散射的结果进行了比较。     

Scattering of elastic waves by spherical inclusions with applications to low frequency wave propagation in composites

Scattering of plane elastic waves by a spherical inclusion is considered. A unified method of solution is presented which treats compressional and shear incidence on a similar basis. Explicit results are given for Rayleigh scattering. We apply the results of the single scattering problem to the propagation of low frequency waves in a composite containing a dilute concentration of spherical inclusions. Explicit formulae are given for the effective wave speeds and attenuations when the inclusions are voids. Both the compressional and shear wave speeds decrease initially as a function of frequency.     

临界折射纵波(LCR波)传播机理的研究

文章旨在用有限元法模拟LCR波的传播过程，初步探讨LCR波的传播机理。作者用有限元法模拟了LCR波在钢轨中传播的声场分布，并且对钢轨中LCR波的传播作了实验。用有限元法得到的接收楔块斜面所接收到的能量对时间的曲线与实验中接收楔块接收到的信号有较一致的结果：波形曲线趋势相同，在第一个波峰后都有第二个峰值。从有限元分析和实验结果得出结论：LCR波是纵波以第一临界角入射时产生的特殊模式，是一种特殊的头波，这与激发超声波的换能器和被测材料是有限尺寸的介质而不是无限大介质有关。     

The coated thermoelastic body within a low-frequency elastodynamic field

A superposition of a longitudinal and a transverse plane elastic wave excites a small body which is embedded in an infinite elastic medium. The interior of the body exhibits thermoelastic behaviour of the Biot type and it contains a core which is also thermoelastic but with different thermal and elastic parameters. Integral representations for the near as well as the far-field are obtained which involve volume integrals over the shell and the core and surface integrals over the surface of the scatterer and the core–shell interface. Complete low-frequency expansions are provided and the scattering problem is reduced to a sequence of transmission problems for the determination of the coefficients of these expansions. It is shown that the thermal character of the interior media is observed in the low-frequency approximations of order higher or equal to three, when we are close to the scatterer and higher or equal to five, when we are far away from it. Furthermore, the thermoelastic behaviour of the scatterer affects only the radial scattering amplitude, which is of the longitudinal type, while the tangential scattering amplitudes, which are of the transverse type, coincide with the corresponding expressions for scattering by an elastic body with a penetrable elastic core.     

Acoustic diffraction by two concentric coaxial soft spherical caps

Summary The subject of this paper is the problem of diffraction of a time-harmonic axially symmetric acoustic wave by two concentric coaxial soft spherical caps. An integral equation technique is employed to solve such a boundary value problem involving two concentric coaxial spherical caps. Approximate expressions are derived for the far field amplitude as well as the scattering cross section for this problem when the incident wave is a low frequency axially symmetric plane wave travelling along the common axis of the two caps. By taking appropriate limits, the formulae for scattering cross section for the corresponding problems for a soft spherical cap, a soft sphere and a soft sphere bounded by a concentric soft spherical cap are also derived. Furthermore, the total electrostatic charge required to raise the two concentric coaxial spherical caps to unit potentials in a free space is readily evaluated from the analysis of this paper.     

基于快速多极子基本解方法(FMM-MFS)的弹性波二维散射模拟研究

针对弹性波二维散射问题,发展一种新的快速多极子基本解方法（FMM-MFS）。方法基于单层位势理论,通过在虚边界上设置膨胀波线源和剪切波线源以构造散射波场,从而避免了奇异性的处理和边界单元离散;结合快速多极子展开技术（FMM）,大幅度降低了计算量和存储量,突破了传统方法难以处理大规模散射问题的瓶颈。以全空间孔洞对P、SV波的二维散射为例,给出了具体求解步骤,并在个人计算机上实现了上百万自由度问题的快速精确计算。在方法效率和精度检验基础上,分别以单孔洞和随机孔洞群对平面波（P、SV波）的散射为例进行计算模拟,揭示了孔洞（群）周围弹性波散射的若干重要规律。     

Effects of fiber ellipticity and orientation on dynamic stress concentrations in porous fiber-reinforced composites

Interaction of time harmonic fast longitudinal and shear incident plane waves with an elliptical fiber embedded in a porous elastic matrix is studied. The novel features of Biot dynamic theory of poroelasticity along with the classical method of eigen-function expansion and the pertinent boundary conditions are employed to develop a closed form series solution involving Mathieu and modified Mathieu functions of complex arguments. The complications arising due to the non-orthogonality of angular Mathieu functions corresponding to distinct wave numbers in addition to the problems associated with appearance of additional angular dependent terms in the boundary conditions are all avoided by expansion of the angular Mathieu functions in terms of transcendental functions and subsequent integration, leading to a linear set of independent equations in terms of the unknown scattering coefficients. A MATHEMATICA code is developed for computing the Mathieu functions in terms of complex Fourier coefficients which are themselves calculated by numerically solving appropriate sets of eigen-systems. The analytical results are illustrated with numerical examples in which an elastic fiber of elliptic cross section is insonified by a plane fast compressional or shear wave at normal incidence. The effects of fiber cross sectional ellipticity, angle of incidence (fiber two-dimensional orientation), and incident wave polarization (P, SV, SH) on dynamic stress concentrations are studied in a relatively wide frequency range. Limiting cases are considered and fair agreements with well-known solutions are established.     

Low-frequency scattering of coated spherical obstacles

This work deals with the scattering of a plane harmonic elastic wave by a penetrable spherical scatterer with a concentric spherical penetrable inclusion. We evaluate the zeroth and first-order approximations of the Rayleigh expansion of the displacement fields. The major line of applications belongs to the science of the particulate composite material. So, as an application of the method, a typical particulate composite material is examined and the behaviour of the scattering cross section with respect to the elastic properties of the medium is presented.     

Optimized reflector stacks for solidly mounted bulk acoustic wave resonators

The quality factor (Q) of a solidly mounted bulk acoustic wave resonator (SMR) is limited by substrate losses, because the acoustic mirror is traditionally optimized to reflect longitudinal waves only. We propose two different design approaches derived from optics to tailor the acoustic mirror for effective reflection of both longitudinal and shear waves. The first one employs the stopband theory in optics; the second one takes advantage of the periodic nature of reflection spectra in a Bragg reflector: the diffraction grating design approach. The optimized design using stopband theory reaches a calculated minimum transmission of 25 dB and 20 dB at resonance frequency for longitudinal and shear waves, respectively, for various practical reflector material combinations. Using the diffraction grating approach, a near quarter-wave performance is maintained for longitudinal waves, whereas shear waves reach minimum transmission below 26 dB. However, this design does necessitate relatively thick layers. The experimental results show good agreement with finite element models (FEM). The extracted 1-D Q for the realized shear optimized devices was increased to around 3300.     

Computer-generated holograms of three-dimensional objects synthesized from their multiple angular viewpoints

Synthesizing computer-generated holograms (CGHs) of a general three-dimensional (3D) object is usually a heavy computational task. We propose and demonstrate a new algorithm for computing CGHs of 3D objects. In our scheme, many different angular projections of computer-designed 3D objects are numerically processed to yield a single two-dimensional complex matrix. This matrix is equivalent to the complex amplitude of a wave front on the rear focal plane of a spherical lens when the object is located near the front focal point and illuminated by a plane wave. Therefore the computed matrix can be used as a CGH after it is encoded to a real positive-valued transparency. When such CGH is illuminated by a plane wave, a 3D real image of the objects is constructed. The number of computer operations are equivalent to those of a two-dimensional Fourier CGH. Computer and optical constructions of 3D objects, both of which show the feasibility of the proposed approach, are described.     

Multiple scattering of electromagnetic waves by an aggregate of uniaxial anisotropic spheres

An exact analytical solution is obtained for the scattering of electromagnetic waves from a plane wave with arbitrary directions of propagation and polarization by an aggregate of interacting homogeneous uniaxial anisotropic spheres with parallel primary optical axes. The expansion coefficients of a plane wave with arbitrary directions of propagation and polarization, for both TM and TE modes, are derived in terms of spherical vector wave functions. The effects of the incident angle α and the polarization angle β on the radar cross sections (RCSs) of several types of collective uniaxial anisotropic spheres are numerically analyzed in detail. The characteristics of the forward and backward RCSs in relation to the incident wavelength are also numerically studied. Selected results on the forward and backward RCSs of several types of square arrays of SiO? spheres illuminated by a plane wave with different incident angles are described. The accuracy of the expansion coefficients of the incident fields is verified by comparing them with the results obtained from references when the plane wave is degenerated to a z-propagating and x- or y-polarized plane wave. The validity of the theory is also confirmed by comparing the numerical results with those provided by a CST simulation.     

A self-consistent approach to dynamic effective properties of a composite reinforced by distributed spherical particles

Summary A self-consistent approach to dynamic effective properties of a composite reinforced by randomly distributed spherical inclusions is studied. The coherent plane waves propagating through the particle-reinforced composite are of attenuation nature. It implies that there is an analogy between the particle-reinforced composite and the effective medium with complex-valued elastic constants from the viewpoints of wave propagation. A composite sphere consisting of the inclusion, the matrix and the interphase between them is assumed embedded in the effective medium. The effective wavenumbers of the coherent plane waves propagating through the particle-reinforced composite are obtained by the dynamic self-consistent conditions which require that the forward scattering amplitudes of such a composite sphere embedded in the effective medium are equal to zero. The dynamic effective properties (effective phase velocity, effective attenuation and effective elastic constants) obtained by the present dynamic self-consistent approach for SiC-Al composites are compared numerically with that obtained by the effective field approach at various volume concentrations. It is found that there is a good agreement between the two approaches at a relatively low frequency and low volume concentration but the numerical results deviate from each other at a relatively high frequency and high volume concentration.     

板状粘接结构中对称和反对称纵波与界面的相互作用

为了激发粘接结构中的导波或界面波,通常需要将声波从两半无限介质同相位或反相位地同时入射多层系统。针对此问题,基于矩阵方法,推导了界面处于理想连接的情况下,对称或反对称纵波从上下半无限空间入射时,三层板状粘接结构中纵、横波的反射与透射系数表达式。分析了入射角度、粘接层厚度以及基体材料等对声波反射(或透射)特性的影响。结果表明,对称或反对称纵波垂直入射时不发生波型转换。粘接结构中声波的反射(或透射)特性与入射角度、频率以及粘接层厚度等参数密切相关。在相同的粘接层厚度(或频率)范围内,随着声波频率(或粘接层厚度)的增加,谐振频率曲线向低频漂移。该方法可作为粘接结构中体波或导波传播特性研究的重要理论基础。     

Potential and limits of holographic reconstruction algorithms

For the purpose of ultrasonic nondestructive testing of materials, holography in connection with digital reconstruction algorithms has been proposed as a modern tool to extract crack sizes from ultrasonic scattering data. Defining the typical holographic reconstruction algorithm as the application of the scalar Kirchhoff diffraction theory to backward wave propagation, we demonstrate its general incapability of reconstructing equivalent sources, and hence, geometries of scattering bodies. Only the special case of a planar measurement recording surface, that is to say, a hologram plane, and a planar crack with perfectly rigid boundary conditions parallel to the hologram plane and perpendicular to the incident field yields a nearly perfect correlation between crack size and reconstructed image; the reconstruction algorithm is then referred to as the Rayleigh-Sommerfeld formula; it therefore represents the optimal case matched to that special geometrical situation and, hence, may be interpreted as a quasi-matched spatial filter. Using integral equation theory and physical optics, we compute synthetic holographic data for a linear cracklike scatterer for both plane and spherical wave incidence, the latter case simulating a synthetic aperture impulse echo situation, thus illustrating how the Rayleigh-Sommerfeld algorithm or its Fresnel approximation increasingly fail for cracks inclined to the hologram plane and excited nonperpendicularly. Furthermore, we point out how the physical data recording process may additionally influence the reconstruction accuracy, and, finally, guidelines for a careful and serious application of these holographic reconstruction algorithms are given. The theoretical results are supported by measurements.