Spherical Wave Field Research Articles

Reflection and transmission coefficients of spherical waves at an interface separating two dissimilar viscoelastic solids

SUMMARY Spherical wave reflection and transmission (R/T) coefficients at an interface are not only of theoretical significance but also play an important role in the amplitude variation with offset (AVO) analysis of wide-angle reflection seismic data and cross-borehole surveys. For sources close to the interface the resulting wavefields cannot be adequately described in terms of a single incident plane wave. Rather, the spherical waves must be viewed as the superposition of an infinite number of plane waves. Moreover, the R/T coefficients for each individual plane wave in viscoelastic media have proven to be more complicated than expected due to the difficulty in selecting the correct vertical slowness. In such attenuating media the R/T coefficients cannot be properly determined by simply replacing the real elastic parameters with their complex viscoelastic counterparts. In this study, the reflection and transmission coefficients of spherical waves at a plane interface separating two dissimilar viscoelastic solids are rigorously investigated. The difficulty in selecting the vertical slowness is shown to be circumvented if the spherical wavefields are calculated from the plane wavefields using the Sommerfeld integral appropriate for the dissipative materials. However, some resulting phase curves of the complex spherical wave R/T coefficients tend to be of opposite sign to the corresponding phase curves of plane waves due to non-uniqueness of the latter for post-critical wave incidence. In this contribution we propose a new definition of spherical wave R/T coefficients for viscoelastic media which differs from the conventional one. Its advantages are that it is not explicitly expressed as a function of the R/T angles, it is valid for both P and S waves, yet it is consistent with the existing definitions of spherical wave R/T coefficients but is more robust. By way of examples we compute both spherical wave reflection coefficients (SWRC) and spherical wave transmission coefficients (SWTC) for two different viscoelastic models. Unlike plane waves, both the SWRC and the SWTC of converted PS waves are found to be non-zero at vertical incidence and may be drastically affected by the existence of longitudinal PS waves which are confirmed by full waveform calculations for the converted PS waves.

Method of theoretically calculating spherical stress wave field in linear-hardening materials under impact load

Based on the linear hardening plastic constitutive model, the theoretical solution of the elastic-plastic spherical stress wave field under impact load is established. Firstly, the influence of the impact load unloading rate on the propagation of the spherical stress wave is analyzed, and three different types of propagation images are obtained. On this basis, the method of theoretically calculating the spherical wave equation in elastic stage, plastic loading stage and unloading stage is established separately, and the calculation scheme of particle displacement, particle velocity, stress and strain is given. Compared with the existing theoretical methods, this method takes into account the different propagation patterns of the stress waves under different unloading rates, and shows how to calculate the stress wave parameters in unloading stage, which is more applicable. This method is used to calculate the elastic-plastic spherical stress wave field under constant shock load and exponential attenuation shock load. The calculated results are in good agreement with those from the existing theoretical method and numerical simulation results in the elastic stage and also in the plastic loading stage. In the unloading stage, the existing theoretical method is no longer applicable, while the results obtained in this paper are in good agreement with the numerical simulation results, which verifies the correctness of the theoretical method.

Adaptation of the Standard Off-Axis Digital Holographic Microscope to Achieve Variable Magnification

Traditional microscopy provides only for a small set of magnifications using a finite set of microscope objectives. Here, a novel architecture is proposed for quantitative phase microscopy that requires only a simple adaptation of the traditional off-axis digital holographic microscope. The architecture has the key advantage of continuously variable magnification, resolution, and Field-of-View, by simply moving the sample. The method is based on combining the principles of traditional off-axis digital holographic microscopy and Gabor microscopy, which uses a diverging spherical wavefield for magnification. We present a proof-of-concept implementation and ray-tracing is used to model the magnification, Numerical Aperture, and Field-of-View as a function of sample position. Experimental results are presented using a micro-lens array and shortcomings of the method are highlighted for future work; in particular, the problem of aberration is highlighted, which results from imaging far from the focal plane of the infinity corrected microscope objective.

A Study on the Theoretical Calculation Method for Spherical Stress Wave Field in Linear Hardening Materials under Impact Load

基于线性硬化塑性本构模型，建立了冲击载荷作用下弹塑性球面应力波场的理论求解方法。首先，分析了冲击载荷卸载速率对球面应力波传播的影响，得到了三种不同类型的应力波传播图像；在此基础上，建立了弹性阶段、塑性加载阶段以及卸载阶段球面波动方程的理论求解方法，给出了质点位移、质点速度、应力和应变等物理量的计算方案。与已有理论方法相比，该方法考虑了不同载荷卸载速率条件下应力波的不同传播情况，并且给出了卸载阶段应力波参量计算方法，具有更广的适用范围。利用上述方法计算了恒定冲击载荷和不同指数衰减冲击载荷作用下弹塑性球面应力波场参量，在弹性阶段和塑性加载阶段，理论计算得到的物理量与已有理论方法和数值模拟结果基本吻合，在卸载阶段，已有理论方法不再适用，而本文理论计算得到的物理量与数值模拟结果基本吻合，验证了该理论方法的正确性。

3D Display System Based on Spherical Wave Field Synthesis

We present a novel concept and first experimental results of a new type of 3D display, which is based on the synthesis of spherical waves. The setup comprises a lens array (LA) with apertures in the millimeter range and a liquid crystal display (LCD) panel. Each pixel of the LCD creates a spherical wave cutout that propagates towards the observer. During the displaying process, the curvature of the spherical waves is dynamically changed by either changing the distance between LA and LCD or by adapting the focal lengths of the lenses. Since the system, similar to holography, seeks to approximate the wavefront of a natural scene, it provides true depth information to the observer and therefore avoids any vergence–accommodation conflict (VAC).

Application of optical interferometry in focused acoustic field measurement

Optical interferometry has been successfully applied in measuring acoustic pressures in plane-wave fields and spherical-wave fields. In this paper, the “effective” refractive index for focused acoustic fields was developed, through numerical simulation and experiments, the feasibility of the optical method in measuring acoustic fields of focused transducers was proved. Compared with the results from a membrane hydrophone, it was concluded that the optical method has good spatial resolution and is suitable for detecting focused fields with fluctuant distributions. The influences of a few factors (the generated lamb wave, laser beam directivity, etc.) were analyzed, and corresponding suggestions were proposed for effective application of this technology.

A note on X-ray spherical wavefields in the Laue case for perfect crystals.

Spherical wavefields in the Laue case are obtained when the real part of the crystal structure factor is zero or small compared with the imaginary part. The results are the same as in the conventional case where the real part is large. Through this work, it is shown that virtual wavefields on the vacuum side should be taken into account to explain the obtained results. It is also shown that, to obtain the virtual wavefields, a modification of the conventional spherical wave theory is necessary.

Optical Image Encryption With Divergent Illumination and Asymmetric Keys

A flexible optical encryption system based on divergent illumination and asymmetric encoding is proposed. The input image is encrypted by use of two random phase masks (RPMs) located at the input and the conjugate plane in a diverging spherical wave field. Compared with the counterparts using planar illumination and symmetric keys, a significant difference is that continuous change of positions of optical elements applied for encryption is allowed, resulting in decryption keys that are different from the encryption keys (or their conjugates) and variable size display of encrypted/decrypted images. A detailed mathematical description and calculation results with different bandwidths of the system support our proposal.

Amplitude and phase versus angle for elastic wide-angle reflections in the τ‐p domain

Near- and postcritical spherical-wave reflections contain amplitude and phase variations with incident angle that are not predicted by plane-wave solutions. However, if a spherical wavefield is decomposed into plane waves by a time-intercept-slowness ([Formula: see text]) transform, then plane-wave reflection coefficients (the Zoeppritz) can be used as the basis of amplitude/phase versus angle analysis. The spherical-wave effects on reflection coefficients near the critical angle (in the time-offset domain) were decomposed by [Formula: see text] transformation into plane waves. Kinematic ray tracing linked the reflection angle at the target reflector and the apparent slowness at the surface receiver, which enabled extracting the amplitude/phase versus angle data at the reflector from the surface [Formula: see text] data. The most reliable inversion results were obtained by combining the extracted amplitudes and phases in a composite inversion for the elastic parameters below the target reflector.

A random physical model of seismic ground motion field on local engineering site

This paper presents a random physical model of seismic ground motion field on a specific local engineering site. With this model, artificial ground motions which are consistent with realistic records at SMART-1 array on spatial correlation are synthesized. A two-scale modeling method of seismic random field is proposed. In large scale, the seismic ground motion field on bedrock surface is simplified to a two-dimensional spherical wave field based on the seismic point source and homogeneous isotropic media model. In small scale, the seismic ground motion field on the engineering site has a plane waveform. By introducing the physical models of seismic source, path and local site and considering the randomness of the basic physical parameters, the random model of seismic ground motion field is completed in a random functional form. This model is applied to simulation of the acceleration records at SMART-1 array by using the superposition method of wave group.

A Second-Order Asymptotic Approximation for the Sommerfeld Half-Space Problem

Investigation is made on the Sommerfeld half-space problem that concerns the spherical-wave fields on the sparser side of a planar interface involving both permittivity and permeability contrasts. For a point current source placed at the interface, the fields are described by a second-order approximation that properly evaluates the asymptotic component having the inverse-square dependence on the distance from the source. The validity of this approximation is demonstrated in the time domain by comparisons with the waveforms of the simpler approximations and the wavenumber-synthetic waveforms.

Study on Blasting Propagation at Great Depth

The elasticity cavity swelling is investigated, and based on equivalent loading and elastic wave theory, blasting propagation model at great depth is established. Using the potential functions which is equal to different boundary conditions, a transformation rule between velocity field and stress field is given. The result shows that for the same stress condition, the particle’s velocity is smaller, the reason for which is that the factors influencing the magnitude of this difference are the propagation distance and speed. Through the stress field attenuation analyses under shock wave loading, variable attenuation of spherical wave field is r-1at least. At the beginning of the medium’s movement, the circumferential stress is tensile stress, and then turns into compressive stress. The tensile stress may cause radial cracks. The shearing stress is the decreasing function of r/R0in the area from seismic source to elastic wave front.

Near‐field seismic effects in a homogeneous medium and their removal in vertical seismic profile attenuation estimates

ABSTRACTTo better understand (and correct for) the factors affecting the estimation of attenuation (Q), we simulate subsurface wave propagation with the Weyl/Sommerfeld integral. The complete spherical wavefield emanating from a P‐wave point source surrounded by a homogeneous, isotropic and attenuative medium is thus computed. In a resulting synthetic vertical seismic profile, we observe near‐field and far‐field responses and a 90° phase rotation between them. Depth dependence of the magnitude spectra in these two depth regions is distinctly different. The logarithm of the magnitude spectra shows a linear dependence on frequency in the far‐field but not in those depth regions where the near‐field becomes significant. Near‐field effects are one possible explanation for large positive and even negative Q‐factors in the shallow section that may be estimated from real vertical seismic profile data when applying the spectral ratio method. We outline a near‐field compensation technique that can reduce errors in the resultant Q estimates.

Comparison of three measurement techniques for the normal absorption coefficient of sound absorbing materials in the free field

Three different techniques for evaluating the absorption coefficient of sound absorbing materials in free field conditions are discussed. One technique measures the acoustic impedance at one point nearby a specimen, the other two techniques evaluate the impedance from the transfer function of two sound pressures and two particle velocities at two points. These are called "PU-method," "PP-method," and "UU-method," respectively. An iterative algorithm to estimate the acoustic impedance of the locally reactive specimen in the spherical wave field is also applied. First, the effect of receiver positions, specimen areas, and source heights to the measured normal absorption coefficient is investigated by the boundary element method. According to these investigations, the PU-method is most stable against the effect of specimen area, and the UU-method is easily affected by that effect. Closer source to the specimen distance is advantageous for the signal to noise ratio of these measurement techniques, but correction for the effect of the spherical wave field has to be applied. As a finding, the iterative algorithm works for all of three techniques. Finally, the PU-method is applied experimentally with a pressure-velocity sensor and a loudspeaker in a hemi-anechoic room. As a result, the calculated results have been verified.

Intensity diffraction tomography with fixed detector plane

Intensity diffraction tomography (IDT) is an imaging method for reconstructing the complex refractive index distribution of a weakly scattering three-dimensional object. Unlike classic diffraction tomography, which requires measurement of the transmitted wave-field phase, IDT accomplished this reconstruction from knowledge of wave-field intensity measurements on parallel detector planes at each tomographic view angle. In this work, novel scanning protocols and reconstruction algorithms for IDT are proposed that require two intensity measurements on a single detector plane positioned at a fixed distance from the object. Each measurement corresponds to a probing spherical wave field that possesses a distinct curvature. Accordingly, the form of the incident wavefield, rather than the detector position, represents the degree of freedom in the imaging system that is varied for acquisition of the necessary measurement data. Computer simulation studies are conducted to investigate the developed data acquisition and image reconstruction algorithms.

Spatial sound reproduction with wave field synthesis

Wave field synthesis is a reproduction technique developed at TU Delft, that enables the generation of high-quality three-dimensional spatial sound fields. The benefit of the method is that spatial impressions are highly independent of the position of the listeners within a large listening area. In short, the method uses a limited number of audio channels that are reproduced by generating plane and spherical wave fields with arrays of loudspeakers that surround the listening place. Applications include spatial sound reproduction in the home and in cinemas, sound reinforcement in theaters, teleconferencing with large video screens, and variable acoustics.

A method of interpolating the diffractive information of the sphere-baffled microphone in the sound field of the spherical wave

The sphere-baffled microphone (SBM) has been used for estimating DOA of progressive plane waves and their extractions, but it is now planned to construct the diffractive information directory (DID) of the SBM with DOA entry and source—distance entry in the sound field of spherical wave. This paper deals with the interpolation methods of getting information associated with entries not appearing in DID. Numerical computation of the diffractive information is done for the 2-element SBM (right and left microphones at both ends of a diameter of 17.68 cm) at frequency points corresponding to the sampling frequency 22.05 kHz and 1024 point DFT. Two data sets of DOA interval of 6° [0,6,12,...] and [3,9,15,...] are used for interpolation and evaluation of errors. Interpolations and their error evaluations are done in the regions of both the log magnitude versus frequency and the group delay versus frequency. Both the bilinear interpolation and cubic interpolation method result in significant interpolation errors up to 10 dB and 0.14 ms for the shadow region. Finally, a new method is presented with high accuracy of interpolation based on the landmarks-loci of the diffractive information.

WAVE PROPAGATION THROUGH SOILS IN CENTRIFUGE TESTING

Wave propagation phenomena in soils can be experimentally simulated using centrifuge scale models. An original excitation device (drop-ball arrangement) is proposed to generate short wave trains. Wave reflections on model boundaries are taken into account and removed by homomorphic filtering. Propagation is investigated through dispersion laws. For drop-ball experiments, spherical wave field analysis assuming linear viscoelasticity leads to a complete analytical description of wave propagation. Damping phenomena are examined and evaluated using this description.

Wave propagation through soils in centrifuge testing

Wave propagation phenomena in soils can be experimentally simulated using centrifuge scale models. An original excitation device (drop-ball arrangement) is proposed to generate short wave trains. Wave reflections on model boundaries are taken into account and removed by homomorphic filtering. Propagation is investigated through dispersion laws. For drop-ball experiments, spherical wave field analysis assuming linear viscoelasticity leads to a complete analytical description of wave propagation. Damping phenomena are examined and evaluated using this description.

Estimation of locations and strengths of broadband planar and spherical noise sources using coherent signal subspace

This paper presents the estimation methods of the planar and spherical wave sources with broadband frequency by using two types of coherent signal-subspace processing; one uses frequency correction of the unwrapped phase of the measured sound pressures at the microphone array in space domain, and the other uses the focusing procedures by using the focusing matrix. The simulations show that the method based on frequency correction has a rather simple procedure compared to that of the methods based on focusing procedures. The performances of these methods have been also investigated in terms of the frequency band, and the distance between sources. The MUSIC power spectrum by the former gives broader peaks than that of the latter but the latter makes some spurious peaks in the MUSIC power spectrum when the bearing angle between the sources and/or the frequency band of interest increases, provided that one assumes a planar wave field. For the spherical wave field, these phenomena were less sensitive than those of the planar wave field.