Temporal focusing by use of composite X waves

It is shown that highly focused pulses can be shaped by exciting a finite aperture with a spread-out pulse train of X waves. The basis of the proposed scheme is that the peaks of X waves, characterized by different apex angles, travel at different velocities. This property allows one to vary the temporal starting points of the initial excitations of a sequence of X waves so that all their peaks meet at a chosen focusing point. It is demonstrated that this simple criterion can be effective in producing a highly focused, composite X-wave pulse that exhibits a slower decay behavior than the individual X-wave components used in synthesizing it.     

An X wave transform

Limited diffraction beams such as X waves can propagate to an infinite distance without spreading if they are produced with an infinite aperture and energy. In practice, when the aperture and energy are finite, these beams have a large depth of field with only limited diffraction. Because of this property, limited diffraction beams could have applications in medical imaging, tissue characterization, blood flow velocity vector imaging, nondestructive evaluation of materials, communications, and other areas such as optics and electromagnetics. In this paper, a new transform, called X wave transform, is developed. In the transform, any well behaved solutions to the isotropic-homogeneous wave equation or limited diffraction beams can he expanded using X waves as basis functions. The coefficients of the expansions can be calculated with the properties that X waves are orthogonal. Examples are given to demonstrate the efficacy of the X wave transform. The X wave transform reveals an intrinsic relationship between any well behaved solutions to the wave equation and X waves, including limited diffraction beams. This provides a theoretical foundation to develop new limited diffraction beams or solutions to the wave equation that may have practical usefulness.     

Experimental verification of nondiffracting X waves

The propagation of acoustic waves in isotropic/homogeneous media and electromagnetic waves in free space is governed by the isotropic/homogeneous (or free space) scalar wave equation. A zeroth-order acoustic X wave (axially symmetric) was experimentally produced with an acoustic annular array transducer. The generalized expression includes a term for the frequency response of the system and parameters for varying depth of field versus beam width of the resulting family of beams. Excellent agreement between theoretical predictions and experiment was obtained. An X wave of finite aperture driven with realizable (causal, finite energy) pulses is found to travel with a large depth of field (nondiffracting length).     

New X-wave solutions of free-space scalar wave equation and their finite size realization

Based on the method proposed by Donnelly and Ziolkowski [1], [2], a new general solution has been obtained for the isotropic/homogeneous scalar wave equation in cylindrical coordinates. It is shown that well-known limited diffraction beams such as Durnin's Bessel beams [4], Lu and Greenleaf's X-wave [15], localized waves of Donnelly and Ziolkowski [1], [2], and limited-diffraction, band-limited waves of Li and Bharath [19], [20] can be obtained from this generic solution as particular cases. In addition, we have obtained new X-wave solutions and have calculated the field characteristics for one of them using a finite aperture realization. It is shown that with a proper choice of the free parameter values, well-behaved X-waves with narrow beamwidths and large depths of field can be achieved. For similar source spectra, the results are compared with Lu and Greenleaf's zeroth-order X-wave, and it is shown that the depth of field and beamwidth are very comparable.     

Nonlinear waves in the Pitaevskii-Gross equation

Nonlinear waves, solitary and periodic, are studied exactly in the Pitaevskii-Gross equation for the wave function of the condensate of a superfluid. We also study the relationship between these two waves and Bogoliubov's phonon, and the energies associated with these waves. The creation energy of a solitary wave with amplitudeA is proportional toA ^3/2. Solitary waves show interesting behavior on their collision due to their localized character. The effect of collision on solitary waves can be described by the phase shift. We give a formula of the phase shift on a collision of two solitary waves. We further discuss the decay of an arbitrary initial disturbance into solitary waves.On leave from Department of Physics, Kyushu University, Fukuoka, Japan.     

Formation of cylindrical shock waves in radiation-magneto gas dynamics

The propagation of acceleration waves in radiation-magneto gas dynamic flow which is induced by the motion of a piston advancing with finite acceleration into a constant state of rest has been studied along with the characteristic path by using the characteristics of the governing quasilinear system as the reference coordinate system. It is shown that a linear solution in the characteristic plane can exhibit nonlinear behaviour in the physical plane. A differential equation governing the growth and decay of an acceleration wave is derived. The critical time is obtained when all the characteristics will pile up at the wave front to form a shock wave. It is found that the effect of magnetic field on compressive waves which owe their origin to radiation is to cause an early shock formation, while on those of expansion waves which owe their origin to radiation the effect of magnetic field is to decrease the decay rate. However, the effect of coupling of radiation and magnetic field on waves (compressive) which are emanating from piston movement is to slow down the motion of a breakdown point and thus increase the cylindrical shock formation time, while on those of expansion waves, the effect is to enhance the decay rate.     

Graphical simulation of superluminal acoustic localized wave pulses

This paper investigates, graphically, a simulation of the launching of recently obtained superluminal localized wave solutions of the homogeneous wave equation. These solutions represent focused interference patterns that travel with speed greater than c in the direction of propagation. Graphical simulation indicates that approximations to these solutions may be launched from a finite array of point sources, and the superluminal nature of the launched pulses is maintained in the near field of the array     

Light bullets in coupled nonlinear Schrödinger equations with spatially modulated coefficients and Bessel trapping potential

We discuss three-dimensional (3D) light bullets (LBs) in a system of coupled nonlinear Schrödinger equations with spatially modulated diffraction and nonlinearity coefficients, under the action of a Bessel trapping potential. Exact spatiotemporal vector solitary waves, or LBs, are obtained using the method of separation of variables and the Hirota’s bilinear method. An inverse solution procedure is introduced, in which the desired localized solutions of equations are proposed first and then the corresponding diffraction and nonlinearity coefficients determined. New 3D wave packets are built with the help of spherical harmonics in the form of multipole, necklace, and toroidal solitary pulses. Numerical solution of the full system of equations indicates that an initial wave in the form of such 3D wave packets is longlived but slowly changing along the propagation direction.     

不同应力波穿过多条非线性变形节理时的透射特性

基于节理非线性位移不连续模型,分析不同弹性纵波正向入射多条节理时的透射规律。利用半解析半数值解法得到透射初至波质点振动速度一维数值差分解,进而研究不同类型、不同振幅的入射纵波穿过多条节理时,初至波质点速度幅值、能量、频谱及时间延迟的变化特性。研究表明,透射能力由大到小的顺序为:矩形波、半正弦波、三角形波。速度透射系数、能量透射系数、透射波中高谐波频域幅值均随入射脉冲幅值增大而增大;延迟时间随入射波振幅的增大而减小,且三角波入射时的时间延迟大于半正弦波。随着节理条数的增加,速度和能量透射系数逐渐降低,且下降速度逐渐减慢;高谐波频域幅值先增大后下降,且下降速度逐渐减慢;时间延迟增大的速度逐渐加快。     

Shear horizontal wave propagation in periodically layeredcomposites

The transverse resonance approach to guided wave analysis is applied to shear horizontal (SH) wave propagation in periodically layered composites. It is found for SH waves that at high values of the guided wavevector β, the wave energy is trapped in the slower of the two media and propagates accordingly at the slower wavespeed. At low values of β, however, the modes demonstrate a clustering behavior, indicative of the underlying Floquet wave structure. The number of modes in a cluster is observed to correlate with the number of unit cells in the layered plate. New physical insights into the behavior of these systems are obtained by analyzing the partial waves of the guided SH modes in terms of Floquet waves. We show that the fast and slow shear waves in the periodically layered composite play an analogous role to the longitudinal and shear partial waves comprising Lamb waves in a homogeneous plate     

Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence

The calculation of the aperture-averaged angle-of-arrival variance, observed with a telescope with a circular aperture, of a plane or spherical wave propagating through homogeneous and isotropic turbulence is one of the classical problems in the theory of wave propagation through random media. We present and discuss approximate closed-form solutions on the basis of the Rytov approximation. For both plane and spherical waves, the accuracy of the approximations is better than 0.25% for all ratios of aperture diameter and Fresnel length.     

FREQUENCY-INDEPENDENT INFINITE ELEMENTS FOR ANALYSING SEMI-INFINITE PROBLEMS

Two drawbacks exist with the infinite elements used for simulating the unbounded domains of semi-infinite problems. The first is the lack of an adequate measure for calculating the decay parameter. The second is the frequency-dependent characteristic of the finite/infinite element mesh used for deriving the impedance matrices. Based on the properties of wave propagation, a scheme is proposed in this paper for evaluating the decay parameter. In addition, it is shown that by the method of dynamic condensation, the far-field impedance matrices for waves of lower frequencies can be obtained repetitively from the one for waves of the highest frequency, using exactly the same finite/infinite element mesh. Such an approach ensures that accuracy of the same degree can be maintained for waves of all frequencies within the range of consideration. Effectiveness of the proposed method is demonstrated in the numerical examples through comparison with previous results.     

一种量测松散介质对应力波衰减效应的实验方法及其在珊瑚砂中的应用

砂土等松散介质对在其中传播的应力波有非常明显的衰减作用,因此,松散介质常常作为爆炸波消波材料被广泛应用于防护工程中。为了准确地量测松散介质对应力波的衰减效应,基于并改进了传统SHPB装置,提出了一种定量研究应力波在砂土介质中衰减规律的新方法。该文方法适用于所有在冲击荷载的应变率范围内（约1~102 s-1）应变率效应不明显的松散介质。方法基于拟合的透射系数T₂,通过杆中的三波（入射波、反射波和透射波）计算得到试件两端真实的峰值应力,还可以计算试件的弹性波速、峰值应力速度、试件端部应力波的前沿升时等关键参数,简单实用,可操作性强。采用提出的方法,对干燥珊瑚砂进行了应力波衰减实验,得出了应力波荷载峰值随传播距离的衰减规律。经对比实验与参数讨论发现,拟合透射系数引起的结果误差不超过2.83%,具有很好的可靠性与实用性。     

Optical generation of focus wave modes

Localized wave solutions of free-space wave equation can be used in numerous applications where the localized transmission of electromagnetic energy is of major importance. However, an optical implementation of localized wave fields has not been accomplished yet, except for an ultrashort version of the Bessel beams or the so called Bessel-X pulses. We propose an approach to constructing realizable optical schemes for generation of localized wave fields. We show that wavelength dispersion of the cone angle of axicons and circular diffraction gratings can be used to generate good approximation to focus wave modes.     

Practical Shear Wave Lens Design for Improved Resolution with Acoustic Microscope

This paper presents a practical new lens design for acoustic microscopy. The new lens provides a factor-of-2 higher resolution than currently available commercial lenses for acoustic microscopy, and a reduction in the influence of surface roughness on the image formation. Analysis, computer simulations, and demonstration examples provide convincing evidence that new lens design works efficiently. Whereas most current lens designs emphasize the use of longitudinal waves, the designs presented here focus on the use of transverse or shear waves. In the present study, two kinds of lens designs have been developed: One is a ``center-sealed' acoustic lens used at the center frequency of 400 MHz and 1 GHz for use with acoustic tone bursts, and the other is a ``high-NA acoustic lens' used in the center frequency of 30 MHz for use with short pulses. The center-sealed acoustic lens has its center area aperture sealed to prevent longitudinal waves from traveling into the sample so that the acoustic image is substantially composed of shear wave components. The ``high-NA' acoustic lens has an aperture with a large aperture angle for exciting shear waves in the object. In this study, the mechanisms of image formation with both of these lenses are described and their features are evaluated.     

Numerical sampling rules for paraxial regime pulse diffraction calculations

Sampling rules for numerically calculating ultrashort pulse fields are discussed. Such pulses are not monochromatic but rather have a finite spectral distribution about some central (temporal) frequency. Accordingly, the diffraction pattern for many spectral components must be considered. From a numerical implementation viewpoint, one may ask how many of these spectral components are needed to accurately calculate the pulse field. Using an analytical expression for the Fresnel diffraction from a 1-D slit, we examine this question by varying the number of contributing spectral components. We show how undersampling the spectral profile produces erroneous numerical artifacts (aliasing) in the spatial-temporal domain. A guideline, based on graphical considerations, is proposed that determines appropriate sampling conditions. We show that there is a relationship between this sampling rule and a diffraction wave that emerges from the aperture edge; comparisons are drawn with boundary diffraction waves. Numerical results for 2-D square and circular apertures are presented and discussed, and a potentially time-saving calculation technique that relates pulse distributions in different z planes is described.     

Asymptotic multi-layer analysis of wind over unsteady monochromatic surface waves

Asymptotic multi-layer (AML) analyses and computation of solutions for turbulent flows over steady and unsteady monochromatic surface waves are reviewed in the limits of low-turbulence stresses and small wave amplitude. The structure of the flow is defined in terms of asymptotically matched thin layers, namely the surface layer and a critical layer (CL), whether it is ‘elevated’ or ‘immersed’, corresponding to its location above or within the surface layer. The results particularly demonstrate the physical importance of the singular flow features and physical implications of the elevated CL in the limit of the unsteadiness tending to zero. These agree with the variational mathematical solution of Miles (J Fluid Mech, 3:185–204, 1957) for a small but finite growth rate, but they are not consistent physically or mathematically with his analysis in the limit of a growth rate tending to zero. As this and other studies conclude, in the limit of zero growth rate, the effect of the elevated CL is eliminated by finite turbulent diffusivity, so that the perturbed flow and the drag force are determined by the asymmetric or sheltering flow in the surface shear layer and its matched interaction with the upper region. But for groups of waves, in which the individual waves grow and decay, there is a net contribution of the elevated CL to the wave growth. CLs, whether elevated or immersed, affect this asymmetric sheltering mechanism, but in quite a different way from their effect on growing waves. These AML methods lead to physical insights and suggest approximate methods for analysing higher-amplitude and more complex flows, such as flow over wave groups.     

Elastic stress transmission in cellular systems—Analysis of wave propagation

A closely packed array of thin-walled rings constitutes an idealisation of a cellular structure. Elastic waves propagating through such structures must do so via the ring (cell) walls. A theoretical investigation into the propagation of elastic stresses in thin-walled circular rings is undertaken to examine the nature of wave transmission. Three modes of motion, corresponding to shear, extensional and flexural waves, are established and their respective velocities defined by a cubic characteristic equation. The results show that all three waves are dispersive. By neglecting extension of the centroidal axis and rotary inertia, explicit approximate solutions can be obtained for flexural waves. Employment of Love's approach for extensional waves [Love AEH. A treatise on the mathematical theory of elasticity, 4th ed. New York: Dover Publications; 1944. p. 452–3] enables approximate solutions for shear waves to be derived. The three resulting approximate solutions exhibit good agreement with the exact solutions of the characteristic equation over a wide range of wavelengths. The effects of material property, ring wall thickness and ring diameter on the three wave modes are discussed, and the results point to flexural waves as the dominant means of elastic energy transmission in such cellular structures. Wave velocities corresponding to different frequency components determined from experimental results are compared with theoretical predictions of group velocity for flexural waves and good correlation between experimental data and theory affirms this conclusion.     

Designing limited diffraction beams

Theoretically, limited diffraction beams can only be produced with an infinite aperture. In practice, they can be closely approximated with a finite aperture over a large depth of field. Because of this property, these beams could have applications in medical imaging, tissue characterization, Doppler velocity estimation, and nondestructive evaluation (NDE) of materials, as well as other physics-related areas such as electromagnetics and optics. In this paper, a new method is developed to design limited diffraction beams of desired beam shapes within a finite aperture of interest. It uses previously discovered limited diffraction beams such as Bessel beams and X waves as basis functions, and constructs new beams with linear superpositions of the bases. To construct a new beam of a desired shape, coefficients of the basis functions in the linear superposition are chosen so that the difference between the new beam and a desired beam is minimized under the criterion of least-squares error within the aperture. This procedure is implemented by digitizing both the basis beams and desired beams in the aperture and solving a system of linear equations from its normal equation. The method is applied to several desired beams that are limited diffraction beams known previously. Results show that the designed beams and the desired beams are virtually identical. If the desired beams are not solutions to the wave equation, the designed beams are new limited diffraction beams that are similar in shapes to the desired beams. This suggests that the method may be a powerful and practical tool for developing new limited diffraction beams of desired properties.     

Waves in thermo-viscoelastic rods

Summary This paper deals with the propagation and decay of shock and acceleration waves in nonhomogeneous linear thermo-viscoelastic rods. The growth-decay laws governing the propagation of such waves is obtained. These clearly exhibit the effects of nonhomogeneity, viscous damping and thermal damping on wave amplitude. An example is treated which brings into play thermo-mechanical coupling in bending.