Love Waves in a Viscoelastic Stratum: Explicit Expression for Complex Velocity as a Function of Frequency

2021 ◽  
Author(s):  
Mohan D. Sharma
Keyword(s):  
Phase Velocity ◽  
Love Wave ◽  
Complex Analysis ◽  
Love Waves ◽  
Complex Velocity ◽  
Complex Phase ◽  
Real Frequency ◽  
Analysis Technique ◽  
Layered Stratum ◽  
Complex Dispersion

ABSTRACT Propagation of Love wave is considered in a two-layered stratum of isotropic viscoelastic solids. The complex dispersion equation for this wave is solved through a complex analysis technique. This gets an analytical expression for complex velocity, as a function of real frequency rather than the complex wavenumber. This complex (phase) velocity is used further to calculate the (complex) group velocity. Numerical example is solved to analyze the dispersion in speed and attenuation of the viscoelastic Love waves.

Love-Wave Phase-Velocity Estimation from Array-Based Rotational Motion Microtremor

2020 ◽  
Author(s):  
Kunikazu Yoshida ◽  
Hirotoshi Uebayashi
Keyword(s):  
Phase Velocity ◽  
Rayleigh Waves ◽  
Love Wave ◽  
Velocity Estimation ◽  
Love Waves ◽  
Phase Velocities ◽  
Wave Phase ◽  
Wave Phase Velocity ◽  
Spatial Derivatives ◽  
Rotational Motions

ABSTRACT The most popular array-based microtremor survey methods estimate velocity structures from the phase velocities of Rayleigh waves. Using the phase velocity of Love waves improves the resolution of inverted velocity models. In this study, we present a method to estimate the phase velocity of Love waves using rotational array data derived from the horizontal component of microtremors observed using an ordinal nested triangular array. We obtained discretized spatial derivatives from a first-order Taylor series expansion to calculate rotational motions from observed array seismograms. Rotational motions were obtained from a triangular subarray consisting of three receivers using discretized spatial derivatives. Four rotational-motion time histories were calculated from different triangular subarrays in the nested triangular arrays. Phase velocities were estimated from the array of the four rotational motions. We applied the proposed Love-wave phase-velocity estimation technique to observed array microtremor data obtained using a nested triangular array with radii of 25 and 50 m located at the Institute for Integrated Radiation and Nuclear Science, Kyoto University. The phase velocities of rotational and vertical motions were estimated from the observed data, and results showed that the former were smaller than those of the latter. The observed phase velocities obtained from vertical and rotational components agreed well with the theoretical Rayleigh- and Love-wave phase velocities calculated from the velocity structure model derived from nearby PS logs. To show the ability of the rotation to obtain Love wave, we estimated apparent phase velocities from north–south or east–west components. The apparent velocities resulted in between the theoretical velocities of Rayleigh and Love waves. This result indicates that the calculated rotation effectively derived the Love waves from a combination of Love and Rayleigh waves.

scholarly journals The Effect of Micro-Inertia and Flexoelectricity on Love Wave Propagation in Layered Piezoelectric Structures

Nanomaterials ◽  
2021 ◽  
Vol 11 (9) ◽  
pp. 2270
Author(s):  
Olha Hrytsyna ◽  
Jan Sladek ◽  
Vladimir Sladek
Keyword(s):  
Wave Propagation ◽  
Phase Velocity ◽  
Love Wave ◽  
Short Circuit ◽  
Open Circuit ◽  
Love Waves ◽  
Inertia Effect ◽  
Wave Phase Velocity ◽  
Piezoelectric Structure ◽  
Piezoelectric Structures

The non-classical linear governing equations of strain gradient piezoelectricity with micro-inertia effect are used to investigate Love wave propagation in a layered piezoelectric structure. The influence of flexoelectricity and micro-inertia effect on the phase wave velocity in a thin homogeneous flexoelectric layer deposited on a piezoelectric substrate is investigated. The dispersion relation for Love waves is obtained. The phase velocity is numerically calculated and graphically illustrated for the electric open-circuit and short-circuit conditions and for distinct material properties of the layer and substrate. The influence of direct flexoelectricity, micro-inertia effect, as well as the layer thickness on Love wave propagation is studied individually. It is found that flexoelectricity increases the Love-wave phase velocity, while the micro-inertia effect reduces its value. These effects become more significant for Love waves with shorter wavelengths and small guiding layer thicknesses.

Rheological model of Love wave propagation in viscoelastic layered media under gravity

2015 ◽  
Vol 11 (3) ◽  
pp. 424-436
Author(s):  
Rajneesh Kakar
Keyword(s):  
Internal Friction ◽  
Phase Velocity ◽  
Love Wave ◽  
Velocity Ratio ◽  
Love Waves ◽  
Wave Number ◽  
Viscoelastic Layer ◽  
Content Type ◽  
Surface Plot ◽  
Inhomogeneity Factor

Purpose – The purpose of this paper is to deal with the propagation of Love waves in inhomogeneous viscoelastic layer overlying a gravitational half-space. It has been observed velocity of Love waves depends on viscosity, gravity, inhomogeneity and initial stress of the layer. Design/methodology/approach – The dispersion relation for the Love wave in closed form is obtained with Whitaker’s function. Findings – The effect of various non-dimensional inhomogeneity factors, gravity factor and internal friction on the non-dimensional Love wave velocity has been shown graphically. The authors observed that the dispersion curve of Love wave increases as the inhomogeneity factor increases. It is seen that increment in gravity, inhomogeneity and internal friction decreases the damping phase velocity of Love waves but it is more prominent in case of internal friction. Originality/value – Surface plot of Love wave reveals that the velocity ratio increases with the increase of non-dimensional phase velocity and non-dimensional wave number. The above results may attract seismologists and geologists.

scholarly journals Influence of Poroelasticity of the Surface Layer on the Surface Love Wave Propagation

2018 ◽  
Vol 85 (5) ◽  
Author(s):  
Adil El Baroudi
Keyword(s):  
Wave Propagation ◽  
Surface Layer ◽  
Phase Velocity ◽  
Dispersion Equation ◽  
Love Wave ◽  
Wave Attenuation ◽  
Love Waves ◽  
Velocity Variation ◽  
Fluid Viscosity ◽  
Benchmark Solutions

This work presents a theoretical method for surface love waves in poroelastic media loaded with a viscous fluid. A complex analytic form of the dispersion equation of surface love waves has been developed using an original resolution based on pressure–displacement formulation. The obtained complex dispersion equation was separated in real and imaginary parts. mathematica software was used to solve the resulting nonlinear system of equations. The effects of surface layer porosity and fluid viscosity on the phase velocity and the wave attenuation dispersion curves are inspected. The numerical solutions show that the wave attenuation and phase velocity variation strongly depend on the fluid viscosity, surface layer porosity, and wave frequency. To validate the original theoretical resolution, the results in literature in the case of an homogeneous isotropic surface layer are used. The results of various investigations on love wave propagation can serve as benchmark solutions in design of fluid viscosity sensors, in nondestructive testing (NDT) and geophysics.

scholarly journals Upper-mantle shear velocities beneath southern California determined from long-period surface waves

1997 ◽  
Vol 87 (1) ◽  
pp. 200-209
Author(s):  
J. Polet ◽  
H. Kanamori
Keyword(s):  
Phase Velocity ◽  
Rayleigh Wave ◽  
Upper Mantle ◽  
Love Wave ◽  
Southern California ◽  
Velocity Structure ◽  
Velocity Model ◽  
Love Waves ◽  
Long Period ◽  
Mantle Velocity

Abstract We used long-period surface waves from teleseismic earthquakes recorded by the TERRAscope network to determine phase velocity dispersion of Rayleigh waves up to periods of about 170 sec and of Love waves up to about 150 sec. This enabled us to investigate the upper-mantle velocity structure beneath southern California to a depth of about 250 km. Ten and five earthquakes were used for Rayleigh and Love waves, respectively. The observed surface-wave dispersion shows a clear Love/Rayleigh-wave discrepancy that cannot be accounted for by a simple isotropic velocity model with smooth variations of velocity with depth. Separate isotropic inversions for Love- and Rayleigh-wave data yield velocity models that show up to 10% anisotropy (transverse isotropy). However, tests with synthetic Love waves suggest that the relatively high Love-wave phase velocity could be at least partly due to interference of higher-mode Love waves with the fundamental mode. Even after this interference effect is removed, about 4% anisotropy remains in the top 250 km of the mantle. This anisotropy could be due to intrinsic anisotropy of olivine crystals or due to a laminated structure with alternating high- and low-velocity layers. Other possibilities include the following: upper-mantle heterogeneity in southern California (such as the Transverse Range anomaly) may affect Love- and Rayleigh-wave velocities differently so that it yields the apparent anisotropy; higher-mode Love-wave interference has a stronger effect than suggested by our numerical experiments using model 1066A. If the high Love-wave velocity is due to causes other than anisotropy, the Rayleigh-wave velocity model would represent the southern California upper-mantle velocity structure. The shear velocity in the upper mantle (Moho to 250 km) of this structure is, on average, 3 to 4% slower than that of the TNA model determined for western North America.

Determination of optimum tuber planting density in the potato varieties Pentland Squire, Cara, Estima, Maris Piper and King Edward

1992 ◽  
Vol 119 (1) ◽  
pp. 35-44 ◽  
Author(s):  
D. C. E. Wurr ◽  
J. R. Fellows ◽  
E. J. Allen
Keyword(s):  
Complex Analysis ◽  
Seed Tuber ◽  
Tuber Size ◽  
Planting Density ◽  
Potato Seed ◽  
Stem Density ◽  
Seed Tubers ◽  
Net Returns ◽  
Analysis Technique ◽  
The Cost

SummaryThirty-two experiments examining the effects of the weight and within-row spacing of potato seed tubers on graded tuber yields of five varieties were conducted on eight sites from 1980 to 1985. A complex analysis technique was used to combine these data and estimate the optimum tuber planting densities for different ratios of seed cost to small (40–60 mm) and large (60–80 mm) ware value. The same technique could be applied to any other combination of seed cost, ware size and ware value.The optimum tuber planting density decreased with increasing seed-tuber weight. Differences in optimum planting density between varieties were much greater with small (35 g) than with large (105 g) seed tubers and decreased as the cost of seed increased relative to the value of ware. As large ware became worth more than small ware the influence of increasing seed cost on the optimum density was reduced. As the value of large ware increased, net returns increased and the effect of seed cost on net returns was reduced. Mean tuber size decreased with increasing stem density at harvest and at the same stem density was lower in varieties producing more daughter tubers/stem. Changes of mean tuber size (μ) and the spread of yield across size grades (σ) with time were well described by parallel curves in different varieties. It is suggested that in future it may not be necessary to determine optimum tuber planting densities by complex experiments involving several seed-tuber weights and spacings. Instead μ and σ could be estimated from simple experiments and tuber spacings determined by comparison with control varieties.

Finite-Difference Modeling and Characteristics Analysis of Love Waves in Anisotropic-Viscoelastic Media

2021 ◽  
Author(s):  
Shichuan Yuan ◽  
Zhenguo Zhang ◽  
Hengxin Ren ◽  
Wei Zhang ◽  
Xianhai Song ◽  
...  
Keyword(s):  
Finite Difference ◽  
Phase Velocity ◽  
Velocity Dispersion ◽  
Love Waves ◽  
Velocity Anisotropy ◽  
Viscoelastic Media ◽  
Phase Velocities ◽  
Phase Velocity Dispersion ◽  
Anisotropic Elastic ◽  
Attenuation Anisotropy

ABSTRACT In this study, the characteristics of Love waves in viscoelastic vertical transversely isotropic layered media are investigated by finite-difference numerical modeling. The accuracy of the modeling scheme is tested against the theoretical seismograms of isotropic-elastic and isotropic-viscoelastic media. The correctness of the modeling results is verified by the theoretical phase-velocity dispersion curves of Love waves in isotropic or anisotropic elastic or viscoelastic media. In two-layer half-space models, the effects of velocity anisotropy, viscoelasticity, and attenuation anisotropy of media on Love waves are studied in detail by comparing the modeling results obtained for anisotropic-elastic, isotropic-viscoelastic, and anisotropic-viscoelastic media with those obtained for isotropic-elastic media. Then, Love waves in three typical four-layer half-space models are simulated to further analyze the characteristics of Love waves in anisotropic-viscoelastic layered media. The results show that Love waves propagating in anisotropic-viscoelastic media are affected by both the anisotropy and viscoelasticity of media. The velocity anisotropy of media causes substantial changes in the values and distribution range of phase velocities of Love waves. The viscoelasticity of media leads to the amplitude attenuation and phase velocity dispersion of Love waves, and these effects increase with decreasing quality factors. The attenuation anisotropy of media indicates that the viscoelasticity degree of media is direction dependent. Comparisons of phase velocity ratios suggest that the change degree of Love-wave phase velocities due to viscoelasticity is much less than that caused by velocity anisotropy.

scholarly journals Combinatorial-Semantic Analysis of Collocations as a Method of Linguistic Research (on the Example of Colour Collocations Formed by an Adjective Type)

2020 ◽  
pp. 40-57
Author(s):  
M. V. Vlavatskaya
Keyword(s):  
Detailed Analysis ◽  
Complex Analysis ◽  
Semantic Analysis ◽  
Natural Languages ◽  
Linear Relations ◽  
Linguistic Research ◽  
Analysis Technique ◽  
Mechanisms Of Formation

The article is devoted to the application of the method of complex semantic-syntagmatic analysis of collocations, or combinatorially determined lexical and phraseological units, characterized by structural-semantic integrity and having functional, idio-ethnic and other limitations. The relevance of the work is due to the importance of their study as universal units inherent in all natural languages. The purpose of the article is to substantiate the advisability of using the method of complex analysis of collocations and with its help to identify the semantic, functional, syntagmatic mechanisms of their formation, as well as to discover the universal and specific characteristics of these units. The method is developed in the framework of combinatorial lexicology, studying the linear relations of words and their combinatorial potential. The novelty of the study is in the development of a detailed analysis technique, with the help of which it will be possible to detect the mechanisms of formation of various collocations. The material for study was colour collocations, which are created by the adjective type. An algorithm for performing complex analysis is presented, and the procedure for its implementation is demonstrated. Particular attention is paid to the semantic and combinatorial-syntagmatic blocks as the most important aspects of the study of the compatibility of lexemes in the framework of the combinatorial science of words.

scholarly journals Sensitivities of surface wave velocities to the medium parameters in a radially anisotropic spherical Earth and inversion strategies

2015 ◽  
Vol 58 (5) ◽  
Author(s):  
Sankar N. Bhattacharya
Keyword(s):  
Surface Wave ◽  
Phase Velocity ◽  
Group Velocity ◽  
Wave Velocity ◽  
Love Waves ◽  
P Wave ◽  
Model Parameters ◽  
Nonlinear Inversion ◽  
Wave Velocities ◽  
Spherical Earth

<p>Sensitivity kernels or partial derivatives of phase velocity (<em>c</em>) and group velocity (<em>U</em>) with respect to medium parameters are useful to interpret a given set of observed surface wave velocity data. In addition to phase velocities, group velocities are also being observed to find the radial anisotropy of the crust and mantle. However, sensitivities of group velocity for a radially anisotropic Earth have rarely been studied. Here we show sensitivities of group velocity along with those of phase velocity to the medium parameters <em>V<sub>SV</sub>, V<sub>SH </sub>, V<sub>PV</sub>, V<sub>PH , </sub></em><em>h</em><em> </em>and density in a radially anisotropic spherical Earth. The peak sensitivities for <em>U</em> are generally twice of those for <em>c</em>; thus <em>U</em> is more efficient than <em>c</em> to explore anisotropic nature of the medium. Love waves mainly depends on <em>V<sub>SH</sub></em> while Rayleigh waves is nearly independent of <em>V<sub>SH</sub></em> . The sensitivities show that there are trade-offs among these parameters during inversion and there is a need to reduce the number of parameters to be evaluated independently. It is suggested to use a nonlinear inversion jointly for Rayleigh and Love waves; in such a nonlinear inversion best solutions are obtained among the model parameters within prescribed limits for each parameter. We first choose <em>V<sub>SH</sub></em>, <em>V<sub>SV </sub></em>and <em>V<sub>PH</sub></em> within their corresponding limits; <em>V<sub>PV</sub></em> and <em>h</em> can be evaluated from empirical relations among the parameters. The density has small effect on surface wave velocities and it can be considered from other studies or from empirical relation of density to average P-wave velocity.</p>

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