Wavenumber Components Research Articles

Wavenumber-aware diffusion sampling to regularize multi-parameter elastic full waveform inversion

Summary Multi-parameter elastic full waveform inversion (EFWI) provides a more realistic depiction of the subsurface models than the standard acoustic approximation. In practice, however, the significant additional cost and interdependency between the unknown parameters (cross-talks) hinder the application of such algorithms. Diffusion model-based regularization can be used to improve the inversion results while simultaneously injecting prior information into the solution. The main challenge here is how to inject such priors into the EFWI iterations that can better complement the solution’s evolution. To address this challenge, we incorporate a model wavenumber continuation process into a diffusion model-based regularization contribution to multi-parameter EFWI. To do so, we promote a sampling strategy such that at the early iteration, the proposed regularization updates account for the low wavenumber component more and increase progressively with the iteration. We first train the diffusion model on elastic moduli images in an unsupervised manner and incorporate the trained model during the EFWI inversion. We deliberately use single-component measurements, which is the most common acquisition scenario, during the inversion to demonstrate the effectiveness of our regularization. At the inference stage, the proposed framework provides more accurate solutions with negligible additional computational cost compared to several conventional regularization algorithms.

Imaging Moveout in Acoustic Factorized Orthorhombic Media

Summary In velocity model building using full waveform inversion, diving waves are a source of the low-frequency information needed to update the low- to intermediate wavenumber components in the velocity model. Because full waveform inversion is a highly non-linear optimization problem attempting to fit the information from the whole wavefield, it is prone to get stuck in local minimums and experience cycle skipping if the initial model does not contain accurate enough information about the low-wavenumbers to constrain the kinematics. Such initial velocity models will typically be estimated using alternative inversion methods that only apply information from parts of the wavefield in the forward modelling and recorded data. In migration velocity analysis, the unfocusing of the image, caused by inconsistency in the kinematics of reflected waves, is measured and is used to update the velocity model by applying differential semblance optimization. The same concept is applicable when using information from diving waves. For diving waves, little is known about the focusing behaviour, also known as imaging moveout, especially in the presence of azimuthal anisotropy, such as orthorhombic anisotropy, which is a good approximation for volcanic plains and sedimentary basins. Obtaining more knowledge about the focusing behaviour of diving waves for this type of anisotropy may help when choosing parameters to invert during inversion that yield a low trade-off between the parameters. Here, we derive approximate series-based equations describing the imaging moveout of diving qP-waves in acoustic factorized orthorhombic anisotropic media. Due to the complexity of the analytical solutions for diving wave properties in such models, we find a vertical transverse isotropic type of simplification that manages to conserve the curvature and horizontal velocity of the diving qP-wave. The accuracy of the simplified expressions is compared to the exact form and found to yield small errors. To verify the validity of the derived equations for imaging moveout, we compare the generated surfaces with images containing the shape of the unfocusing obtained using reverse time migration, showing a good fit. Of the anisotropic parameters defining the acoustic factorized orthorhombic model, the ε1 and ε2 parameters have the largest influence on the shape of the surface. The anelliptical parameters are shown only to induce minimal changes. Our method to obtain the series expressions describing the imaging moveout in the acoustic factorized orthorhombic model is also applicable to higher symmetry special cases of orthorhombic anisotropy by setting appropriate conditions. Here, this is demonstrated by additionally finding an approximate series expression describing the imaging moveout in acoustic factorized vertical transversely isotropic media, yielding a good match with the image obtained from reverse time migration.

SuperDARN Radar Wind Observations of Eastward-Propagating Planetary Waves

An array of SuperDARN meteor radars at northern high latitudes was used to investigate the sources and characteristics of eastward-propagating planetary waves (EPWs) at 95 km, with a focus on wintertime. The nine radars provided the daily mean meridional winds and their anomalies over 180 degrees of longitude, and these anomalies were separated into eastward and westward waves using a fast Fourier transform (FFT) method to extract the planetary wave components of zonal wavenumbers 1 and 2. Years when a sudden stratospheric warming event with an elevated stratopause (ES-SSW) occurred during the winter were contrasted with years without such events and composited through superposed epoch analysis. The results show that EPWs are a ubiquitous—and unexpected—feature of meridional wind variability near 95 km. Present even in non-ES-SSW years, they display a regular annual cycle peaking in January or February, depending on the zonal wavenumber. In years when an ES-SSW occurred, the EPWs were highly variable but enhanced before and after the onset.

Estimation of the depth-variant seismic wavelet based on the modified unscaled S-transform

Seismic inversion is one of the key techniques used for reservoir characterization. Depth-domain seismic inversion eliminates the cumulative errors associated with depth-to-time and time-to-depth conversions, thus providing geologists and reservoir engineers with an intuitive basis for geological interpretation. The method has received increasing attention in the field of reservoir characterization. Extracting accurate depth-domain seismic wavelets is a prerequisite for successful depth-domain seismic inversion. However, the depth-domain wavelet is velocity-dependent and exhibits significant non-stationarity, which leads to the failure of seismic wavelet estimation methods based on the stationary convolutional model. To this end, we propose a modified wavenumber-domain unscaled S-transform (MWUST) method to accomplish accurate estimation of depth-domain seismic wavelets. The proposed method enhances the accuracy of wavenumber components by removing the linear wavenumber-dependent term from the S-transform. Furthermore, it introduces slope and intercept parameters to improve the depth resolution at low wavenumbers, thereby yielding a more reliable depth–wavenumber spectrum. Subsequently, the relationship between the non-stationary depth-domain seismic wavelet and the depth–wavenumber spectrum is established, allowing for the accurate extraction of non-stationary wavelets under the assumption that the depth-domain reflectivity is a random sequence. Synthetic and real data applications have been used to verify the effectiveness of the proposed method.

Detection of delamination using laser-generated Lamb waves with improved wavenumber analysis

Abstract Delamination is a typical failure mode in multilayered materials, which may potentially threaten the safety and reliability of the materials if not detected promptly. This paper proposes a novel quantitative detection method for delamination in multilayered materials based on laser-generated Lamb waves with improved wavenumber analysis. First, a three-dimensional(3D) finite element model is established to simulate the interaction between delamination and laser-generated Lamb waves. The propagation characteristics of Lamb waves in multilayered materials without defects and with delamination of three different shapes, i.e., rectangular, circular and triangular are analyzed. And the mechanism that delamination can lead to the appearance of scattered wave and new wavenumber components is investigated, facilitating the quantitative detection of delamination. Then, an improved wavenumber analysis method for quantitative detection of delamination is proposed. The new wavenumber components caused by delamination are extracted by broadband adaptive wavenumber filtering to reduce imaging interference and artifacts caused by irrelevant components, and the spatial wavenumber spectrum is obtained by fast broadband local wavenumber estimation algorithm to enhance the delamination imaging accuracy and spatial resolution. The experimental results indicate that, across various experimental scenarios, the proposed method achieves superior imaging precision compared to traditional wavenumber filtering or local wavenumber estimation methods, and can quantitatively identify delamination defects of various shapes and sizes.

A Self-adaptive numerical dispersion suppressed method for shallow seismic simulation

Near-surface in seismic exploration generally refers to the low deceleration zone and a section of stratum below it, which can be described in geophysical language as “low velocity”, “low Q”, and “Free surface”, etc. Due to the presence of low-velocity layers in near-surface, shallow seismic simulation is plagued by numerical dispersion. Numerical dispersion affects the accuracy of seismic wave simulation, especially severe in shallow areas containing low-velocity layers. To improve the simulated quality, denser grids or higher-order difference schemes are used, but both of which would seriously reduce computational efficiency, especially for frequency-domain numerical modeling. Differentiation is replaced by difference in wave field simulation, which would inevitably result in numerical errors. These numerical errors are manifested as the difference between the phase velocity of the discretized wave field and the medium velocity. The waves with different wavenumber components have different phase velocities, and the larger the wavenumber, the more that its phase velocity lags the group velocity. Based on this theoretical analysis, a regularization factor is added to the frequency-domain acoustic wave equation to correct the phase velocity of high wavenumber components. According to the Von Neumann stability requirements, a regularization factor that adaptively changes with simulation parameters is derived. Compared to the fixed regularization factor, adaptive regularization factor can better match the velocity field and protect the effective wave field to the maximum extent while suppressing numerical dispersion. The improved acoustic wave equation can effectively suppress numerical dispersion without increasing the number of grids. Different numerical models demonstrate the effectiveness and efficiency of the improved acoustic equation with the adaptive regularization factor for suppressing numerical dispersion.

Bandgap formation in topological metamaterials with spatially modulated resonators

Within the field of elastic metamaterials, topological metamaterials have recently received much attention due to their ability to host topologically robust edge states. Introducing local resonators to these metamaterials also opens the door for many applications such as energy harvesting and reconfigurable metamaterials. However, the interactions between phenomena from local resonance and modulation patterning are currently unknown. This work fills that gap by studying multiple cases of spatially modulated metamaterials with local resonators to reveal the mechanisms behind bandgap formation. Their dispersion relations are determined analytically for infinite chains and validated numerically using eigenvalue analysis. The inverse method is used to determine the imaginary wavenumber components from which each bandgap is characterized by its formation mechanism. The topological nature of the bandgaps is also explored through calculating the Chern number and integrated density of states. The band structures are obtained for various sources of modulation as well as multiple resonator parameters to illustrate how both local resonance and modulation patterning interact together to influence the band structure. Other unique features of these metamaterials are further demonstrated through the mode shapes obtained using the eigenvectors. The results reveal a complex band structure that is highly tunable, and the observations given here can be used to guide designers in choosing resonator parameters and patterning to fit a variety of applications.

A closure mechanism for screech coupling in rectangular twin jets

The twin-jet configuration allows two different scenarios to close the screech feedback. For each jet, there is one loop involving disturbances which originate in that jet and arrive at its own receptivity point in phase (self-excitation). The other loop is associated with free-stream acoustic waves that radiate from the other jet, reinforcing the self-excited screech (cross-excitation). In this work, the role of the free-stream acoustic mode and the guided-jet mode as a closure mechanism for twin rectangular jet screech is explored by identifying eligible points of return for each path, where upstream waves propagating from such a point arrive at the receptivity location with an appropriate phase relation. Screech tones generated by these jets are found to be intermittent with an out-of-phase coupling as a dominant coupling mode. The instantaneous phase difference between the twin jets computed by the Hilbert transform suggests that a competition between out-of-phase and in-phase coupling is responsible for the intermittency. To model wave components of the screech feedback while ensuring perfect phase locking, an ensemble average of leading spectral proper orthogonal decomposition modes is obtained from several segments of large-eddy simulation data that correspond to periods of invariant phase difference between the two jets. Each mode is then extracted by retaining relevant wavenumber components produced via a streamwise Fourier transform. Spatial cross-correlation analysis of the resulting modes shows that most of the identified points of return for the cross-excitation are synchronised with the guided jet mode self-excitation, supporting that it is preferred in closing rectangular twin-jet screech coupling.

Investigating Zonal Asymmetries in Stratospheric Ozone Trends From Satellite Limb Observations and a Chemical Transport Model

AbstractThis study investigates the origin of a zonal asymmetry in stratospheric ozone trends at northern high latitudes, identified in satellite limb observations over the past two decades. We use a merged data set consisting of ozone profiles retrieved at the University of Bremen from SCIAMACHY and OMPS‐LP measurements to derive ozone trends. We also use TOMCAT chemical transport model (CTM) simulations, forced by ERA5 reanalyses, to investigate the factors that drive the asymmetry observed in the long‐term changes. By studying seasonally and longitudinally resolved observation‐based ozone trends, we find, especially during spring, a well‐pronounced asymmetry at polar latitudes with values up to +6 % per decade over Greenland and −5 % per decade over western Russia. The control CTM simulation agrees well with these observed trends, whereas sensitivity simulations indicate that chemical mechanisms involved in the production and removal of ozone, or their changes, are unlikely to explain the observed behavior. The decomposition of TOMCAT ozone time series and ERA5 geopotential height into the first two wavenumber components shows a clear correlation between the two variables in the middle stratosphere and demonstrates a weakening and a shift in the wavenumber‐1 planetary wave activity over the past two decades. Finally, the analysis of the polar vortex position and strength points to a decadal oscillation with a reversal pattern at the beginning of the century. The same is found in the ozone trend asymmetry. This further stresses the link between changes in the polar vortex position and the identified ozone trend pattern.

A hybrid wave and finite element/boundary element method for predicting the vibroacoustic characteristics of finite-width complex structures

This paper presents a hybrid wave and finite element/boundary element method for predicting the vibroacoustic characteristics of complex panels characterised by an irregular cross-sectional shape with a non-flat upper surface. In this approach, the panels are treated as waveguides and only a small segment of the panel is modelled using finite elements. The mass and stiffness matrices of the segment are then post-processed to project the motion onto its cross-sectional boundaries interfacing with the surrounding fluids. The acoustic pressures in the fluids are modelled by applying Fourier transforms to the Helmholtz boundary integral equations in the wavenumber domain. For each wavenumber component, the pressures acting on the segment are transformed into external nodal forces by evaluating the virtual work performed by the surface pressure on the segment. The wave and finite element formulations are subsequently coupled to the discretised boundary integral equations to determine the response of the panel to acoustic wave excitation. The hybrid method for predicting dispersion curves and sound transmission is validated through comparisons with predictions obtained from both an analytical model and a wave-based method found in the literature. To demonstrate the practical utility of the approach, the hybrid method is employed to predict sound transmission through a corrugated sandwich panel, where developing an analytical model is difficult. The method provides accurate predictions at a low computational cost.

Strong diffraction of nonlinear surface acoustic waves in crystals

Nonlinear distortion and shock formation in planar surface acoustic waves in anisotropic crystals have been modeled without [Hamilton et al., JASA (1999)] and with [Cormack et al., JASA 2022] piezoelectricity. Weak diffraction has been included in the paraxial approximation for nonlinear surface wave beams in isotropic solids [Shull et al., JASA (1995)]. Here, a procedure for including strong diffraction, i.e., without the paraxial approximation, in the model for nonlinear surface waves in crystals is presented, which incorporates the angular spectrum approach described by Kharusi and Farnell (JASA 1970). Anisotropy is defined by expressing the phase speed of a plane wave, and therefore, the magnitude of the corresponding wavenumber, as a function of the direction of propagation. Next, an explicit expression relating the two wavenumber components in the planar surface along which the wave propagates must be obtained as a function of direction, requiring iterative solution of a transcendental relation. The beam is then propagated incrementally away from the source, advancing the angular spectrum in k-space and including nonlinear interactions in the spatial domain to characterize the combined effects of diffraction and nonlinearity. Preliminary results are presented for converging nonlinear surface waves in crystals. [Work supported by IR&D at ARL:UT.]

Pre-resonance effects in deep UV Raman spectra of normal and deuterated water.

We have investigated the shape of the OH/OD stretching Raman band of water as a function of the excitation wavelength in the deep UV region (200-266 nm). By analyzing the spectral profiles, we highlighted selective pre-resonance effects in the high wavenumber component of the OH/OD stretching band, associated to distorted H-bonded water configurations. A van't Hoff treatment of the temperature-dependent Raman spectra provides an estimate of the thermal energy associated to the change from ordered (ice-like) to disordered configurations that agrees with values obtained by related methods based on a two-state model of water. These results open the possibility of exploiting the observed pre-resonance deep-UV signal enhancement to investigate H-bonding properties in aqueous media.

Q-compensated wavefield depth extrapolation-based migration using a viscoacoustic wave equation

In a viscoacoustic medium, intrinsic attenuation causes seismic wavefields to attenuate in amplitude and become dispersed in phase, leading to distorted structural imaging and inaccurate migrated amplitudes. To address this problem, viscoacoustic reverse time migration corrects for dispersion and amplitude attenuation effects in wavefield propagation in forward and backward directions. To provide a parallel alternative approach, the time fractional derivative viscoacoustic wave equation in the depth domain is solved and a Q-compensated wavefield depth extrapolation scheme is established to compensate for viscoacoustic effects during recursive wavefield depth extrapolation. This approach decouples the amplitude attenuation and phase dispersion effects from the viscoacoustic vertical wavenumber. To suppress high wavenumber components and address the problem of exponential amplitude growth during wavefield depth extrapolation, we limit the imaginary part of the vertical wavenumber in the frequency wavenumber domain and use an adaptive stabilization solution. Numerical experiments of impulse responses in a viscoacoustic isotropic medium demonstrate that our proposed scheme can observe calculated wavefields exhibiting amplitude attenuation and phase dispersion effects in comparison to wavefields of the acoustic medium, indicating good agreement with theoretical expectations. We also demonstrate the capability of our proposed scheme to recover imaging amplitudes through a variety of numerical experiments, such as imaging of three-layer, Marmousi, and BP gas reservoir models. The results indicate that our proposed scheme can recover amplitude attenuation and phase dispersion effects more accurately than acoustic migration. We also use marine seismic data featuring natural gas hydrates to indicate that our proposed Q-compensated scheme can generate an enhanced imaging result, especially for the bottom simulating reflectors, compared with the conventional imaging algorithm without Q-compensation.

A simulation and diagnosis of tire radiation noise

Due to the recent pass-by noise regulation, reduction of tire noise has been being required. In order to meet the requirements of the regulation, it is important to simulate and diagnose tire radiation noise for efficiently finding a quieter tire structure. In this paper, an accurate simulation model is employed. With the model tire radiation physics are diagnosed by patch contribution and wavenumber contribution. The diagnosis showed that radiation was dominantly generated at near contact patch and lower wavenumber component dominantly contributed to radiation. In order to further diagnose about what behavior contributes, an approach that calculates radiation efficiency of wavetypes is proposed. The proposed approach finds high radiation efficiency wavetypes at each frequencies so that behavior of highly contributing to radiation at frequencies of interest can efficiently be found, which helps designers to think about design modification to reduce tire radiation noise.

Theoretical and numerical investigation into the suppressing mechanism of radiated noise in an ABH-cavity system

The problem of radiated noise of a coupling system between acoustic space and flexible structure is very common in practical applications. In this paper, the acoustic black hole (ABHs) structure is used for reducing the radiated noise of the plate-cavity system as an efficient passive control technology. The suppression mechanism of the ABH-cavity system is investigated. Theoretical analysis shows that enhanced radiated noise suppressing effect can be obtained by rearranging the numbers and locations of ABHs compared to its uniform counterpart at cavity modal frequencies. Due to this suppression mechanism, ABH structure exhibits a reduced radiation efficiency of the panel, which can be explained by the wavenumber analysis that lots of high wavenumber components are generated and effective wavenumbers in the radiation region are reduced.

Effects of bulk viscosity, heat capacity ratio, and Prandtl number on the dispersion relationship of compressible flows

Here, we present the variation of the dispersion characteristics of the three-dimensional (3D) linearized compressible Navier–Stokes equation (NSE) to bulk viscosity ratio, specific heat ratio (γ), and Prandtl number (Pr). The 3D compressible NSE supports five types of waves, two vortical, one entropic, and two acoustic modes. While the vortical and entropic modes are non-dispersive, the acoustic modes are dispersive only up to a specific bifurcation wavenumber. We illustrate the characteristics and variation of relative (with respect to the vortical mode) diffusion coefficient for entropic and acoustic modes and a specially designed dispersion function for acoustic modes with depressed wavenumber η=KM/Re, the bulk viscosity ratio, γ, and Prandtl number Pr of the flow. Here, K, M, and Re denote the absolute wavenumber of disturbances, Mach number, and Reynolds number of the flow, respectively. At lower wavenumber components, the deviation of the dispersion function from the inviscid and adiabatic case is proportional to η2 at the leading order, and the relative diffusion coefficients increase linearly with bulk viscosity ratio and γ while varying inversely with Pr. With the increase in the bulk viscosity ratio, the shape and extent of the dispersion function alter significantly, and the change is more substantial for higher wavenumber components. The relative diffusion coefficient for entropic and acoustic modes shows contrasting variation with wavenumber depending upon bulk viscosity ratio, γ, and Pr. We also show by solving linearized compressible NSE that relatively significant evolution and radiation of acoustic and entropic disturbances occur when the bulk viscosity ratio is close to the corresponding critical value of maximum bifurcation wavenumber. Based on this criterion, we have presented an empirical relation for estimating bulk viscosity ratio depending upon γ and Pr, giving the corresponding range for obtaining relatively significant disturbance evolution.

Efficient source wavefield reconstruction for full-waveform inversion

Full-waveform inversion (FWI) is a useful tool for estimating the properties of subsurface media, but it has the disadvantage of incurring considerable computing costs. Therefore, we develop a method for reducing the computing memory requirement in the time domain of FWI that uses a new approach with the discrete cosine transform (DCT) and the Nyquist sampling theorem. Our method enables accurate reconstruction of a wavefield with several optimal DCT coefficients derived by considering the maximum wavenumber component of the virtual source wavefield and the maximum frequency of the source wavelet. Consequently, virtual source wavefields can be stored in a computer’s memory rather than on a disk. Our FWI algorithm is compared with existing compression methods that use DCT and wavelet transform in a numerical example. The results of this comparison demonstrate that our algorithm provides significantly improved computational efficiency.

3D frequency-domain acoustic full-waveform inversion using diffraction-angle filtering

The success of seismic imaging techniques such as full-waveform inversion (FWI) depends on using a good starting velocity model or a background velocity model, which can be effectively built by long-wavelength components in the early stages of FWI. This study proposes an optimal strategy for performing 3D frequency-domain FWI with diffraction-angle filtering (DAF). This approach can perform optimized single-frequency FWI with lower computational costs and build a long-wavelength velocity structure by separating low-wavenumber components with DAF. Prior to applying this strategy, we confirm the performance of 3D frequency-domain FWI with DAF in the radiation pattern analysis. DAF effectively separates the diffraction energy of partial derivative wavefields according to the diffraction angle with a negligible increase in computational costs. We then apply our 3D FWI strategy with DAF on synthetic data generated from the 3D SEG/EAGE overthrust model. The wavenumber components are effectively separated, and a long-wavelength velocity model is estimated with DAF compared to conventional FWI. Finally, we apply our strategy to 3D ocean-bottom seismic data from the North Sea. As a result, we observe that 3D FWI with the DAF method successfully reconstructs a long-wavelength velocity structure on the real data using only a single-frequency component.

Anisotropy and dispersion of elastic waves in periodically multilayered anisotropic media

The propagation characteristics of elastic waves in general periodically multilayered media formed by alternate arrangement of two or more arbitrarily-anisotropic materials are comprehensively studied. Combining with the state-space formalism for obtaining the wave solutions to the constituent layers, the Floquet-Bloch theorem for providing the periodic relations between neighboring unit cells, the method of reverberation-ray matrix (MRRM) is employed to derive the general dispersion equations of the periodically multilayered anisotropic media. By solving the dispersion equations in cases of all possible sorts of wavenumber components, the complete anisotropic and dispersion characteristics of elastic waves can be provided. By numerical examples, the slowness and the phase velocity curves are provided to summarize the innovative anisotropic characteristics of elastic waves in three coordinate planes. The effects of the anisotropic parameters on the slowness/phase velocity curves have also been discussed. Moreover, all kinds of dispersion curves including the frequency-related and phase velocity-related curves along the directions perpendicular, parallel, and oblique to the layering are provided to reflect the general dispersion characteristics of elastic waves from various viewpoints. It is found that the innovative anisotropy characteristics of elastic waves in the periodically multilayered anisotropic media include that the slowness curves have periodicity in the thickness direction and the least positive period decreases with the increasing of frequency. As the frequency becomes big enough, the slowness/phase velocity curves show interactions between the modes within the overlapping neighboring periods. The common dispersion characteristics of elastic waves along various directions include the symmetry of wavenumber, the asymptotic of the wavelength and the phase velocity curves to the frequency lines corresponding to zero wavenumber, as well as the cutoff property of the phase velocities of the lowest three basis modes. The waves in oblique directions are special in that the mode transition and repulsion due to the zone folding effect of periodicity along thickness as well as the cutoff property of higher-than-third-order overtones along layering are simultaneously in effective.

Study of SH-wave in a pre-stressed anisotropic magnetoelastic layer sandwich by heterogeneous semi-infinite media

A brief study of SH-wave has been encountered in a pre-stressed anisotropic magnetoelastic layer loaded by heterogeneous half-spaces. The semi-infinite media are heterogeneous with different heterogeneity parameters and the upper half-space is taken exponentially varying with depth while the lower medium varying linearly. The solutions are more apparent with analytical methods clubbed with Mathematica-7 to get the numerical results for a particular model with available data. The study involves obtaining the closed-form expression for reflection and transmission coefficients with the continuous boundary conditions at interfaces. The terms for reflection and transmission coefficients manifest the notable dependencies of material coefficients with inhomogeneity parameters, layer thickness, and the known components of wave propagation, i.e., phase velocity, wave number, and incidence angle. The two special cases have been considered, i.e., one when the layer vanishes means only two heterogeneous half-spaces and the other two isotropic half-spaces that lead to validation of results with well-known reference. The numerical results are shown by graphs which make the result transpicuous to interpret physically.