pith. sign in

arxiv: 2606.18359 · v1 · pith:7OZA6E4Wnew · submitted 2026-06-16 · 🧮 math.DS · math-ph· math.MP

Time and Frequency domain analysis of Love waves generated by Gaussian, Ricker and double couple seismic sources in a memory dependent fractured poroviscoelastic layer on a heterogeneous viscoelastic half-space

Anisha Kumari , Subhajyoti Sarkar , Santimoy Kundu This is my paper

Pith reviewed 2026-06-26 22:07 UTC · model grok-4.3

classification 🧮 math.DS math-phmath.MP
keywords Love wavesporoviscoelasticfractional derivativesseismic sourcesdispersion relationheterogeneous mediumsynthetic seismogramsviscoelastic half-space
0
0 comments X

The pith

Distributed seismic sources are formulated for Love wave analysis in memory-dependent fractured poroviscoelastic layers using fractional calculus.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes a mathematical formulation for the propagation of Love waves in a fractured poroviscoelastic layer overlying a heterogeneous viscoelastic half-space. Fractional derivatives model the memory effects in the material's response to three types of distributed seismic sources: Gaussian, Ricker, and double-couple. Fourier transforms and Green's functions produce a complex dispersion relation whose roots are found iteratively. Numerical results examine how heterogeneity, viscoelasticity, and porosity affect wave characteristics, with synthetic seismograms confirming physical consistency. A simple oscillator model assesses the surface motion induced by these sources.

Core claim

The authors claim that incorporating distributed sources such as Gaussian, Ricker, and double-couple into the analysis of Love waves within a framework that uses Riemann-Liouville fractional derivatives for constitutive relations in a stratified poroviscoelastic medium has not been done previously, enabling detailed time and frequency domain studies of wave propagation influenced by material heterogeneity and porosity.

What carries the argument

The complex dispersion relation derived from Fourier transform techniques combined with Green's function methodology, solved using a hybrid Newton-Raphson algorithm.

If this is right

  • The effects of heterogeneity, fractional viscoelasticity, and porosity can be quantified through numerical simulations of wave propagation characteristics.
  • Synthetic seismograms verify that the solutions are physically consistent and meaningful.
  • The surface response to different seismic sources can be evaluated using a single degree of freedom oscillator model.
  • Parameters exerting the most significant influence on the system response are identified through the analysis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This formulation could allow for better modeling of seismic wave behavior in real geological structures with fractures and porosity.
  • Extending the approach to other source types or wave modes might reveal additional insights into earthquake dynamics.
  • Validation against field data would be necessary to confirm the accuracy of the fractional model in practical applications.

Load-bearing premise

The assumption that Riemann-Liouville fractional derivatives in the constitutive relations accurately capture the memory-dependent mechanical behavior of the fractured poroviscoelastic material without requiring empirical calibration or validation against observed data.

What would settle it

Observation of Love wave dispersion or attenuation in a real fractured poroviscoelastic geological setting that significantly deviates from the numerically computed complex roots would falsify the model's predictive power.

Figures

Figures reproduced from arXiv: 2606.18359 by Anisha Kumari, Santimoy Kundu, Subhajyoti Sarkar.

Figure 1
Figure 1. Figure 1: Configuration of the Problem Under these assumptions, the resulting motion represents a two-dimensional shear-horizontal wave field that is guided within the surface layer and decays into the underlying half-space, which is a defining char￾acteristic of Love-wave propagation in layered viscoelastic and poroelastic media. 3. Mechanical Framework and Governing Differential Equations 3.1. Particle dynamics wi… view at source ↗
Figure 2
Figure 2. Figure 2: Ricker source functions for varying spatial standard deviations. Subfigures (a)–(c) depict the symmetric case [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Gaussian-regularized double-couple source distributions for different spatial configurations. Subfigure (a) shows the [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Validation graphs: (a) for dispersion (b) for attenuation [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) Synthetic wavefield traces showing receiver-offset variation with time. (b) Snapshot representation of wave [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Time-domain response v1(t) under different source mechanisms at x = 0 and z = 0: (a) classical Love wave propagation, (b) point source excitation, (c) Gaussian source excitation with excitation force equal to the point source, (d) Gaussian source excitation using peak normalization, (e) Ricker source excitation and (f) double-couple source excitation. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Space-domain response v1(x) under different source mechanisms at t = 1 and z = 0: (a) classical Love wave propagation, (b) point source excitation, (c) Gaussian source excitation with excitation force equal to the point source, (d) Gaussian source excitation using peak normalization, (e) Ricker source excitation and (f) double-couple source excitation. 6.3. Physical implications of Heterogeneity, fractiona… view at source ↗
Figure 8
Figure 8. Figure 8: Influence of heterogeneity parameter ξ1 on wave characteristics: (a) Phase and group velocity profiles using (Re(c)) (b) Attenuation (Im(c)) (c) Spatial Attenuation. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Influence of heterogeneity parameter b1 on wave characteristics: (a) Phase and group velocity profiles using (Re(c)) (b) Attenuation (Im(c)) (c) Spatial Attenuation. Next, the attenuation characteristics are examined to understand how the governing material parameters influence the dissipation of wave energy during propagation. The parameters ξ1, b1, α1 and α2 collectively govern the viscoelastic character… view at source ↗
Figure 10
Figure 10. Figure 10: Effect of scaling parameter a on wave characteristics: (a)Phase and group velocity profiles using (Re(c)) ; (b) Attenuation (Im(c)); (c) Spatial Attenuation. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Influence of heterogeneity parameter b2 on wave characteristics: (a) Phase and group velocity profiles using (Re(c)), (b) Attenuation (Im(c)), (c) Spatial Attenuation. Having discussed the influence of the governing parameters on the dispersion and attenuation character￾istics of Love waves, it is also instructive to examine the relationship between the phase and group velocities. The phase velocity repre… view at source ↗
Figure 12
Figure 12. Figure 12: Influence of Fractional parameter α1 on wave characteristics: (a) Phase and group velocity profiles (Re(c)) ; (b) Attenuation (Im(c)); (c) Spatial Attenuation. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Influence of Fractional parameter α2 on wave characteristics: (a) Phase and group velocity profiles using (Re(c)) ; (b) Attenuation (Im(c)); (c) Spatial Attenuation. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Influence of volume fraction parameter ν1 on wave characteristics: (a) Phase and group velocity profiles using (Re(c)) ; (b) Attenuation (Im(c)); (c) Spatial Attenuation. 6.4. Integrated Quantitative Sensitivity and Cutoff Frequency Analysis of Wave Propagation Characteristics To understand the conditions under which Love waves can exist in the considered medium, the cutoff frequency is evaluated for diff… view at source ↗
Figure 15
Figure 15. Figure 15: Heat Maps: (a) for dispersion (b) for attenuation [PITH_FULL_IMAGE:figures/full_fig_p031_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Angular frequency dependence of the percentage change in phase velocity under varying heterogeneity parameters: [PITH_FULL_IMAGE:figures/full_fig_p033_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Angular frequency dependence of the percentage change in phase velocity under varying fractional parameters: (a) [PITH_FULL_IMAGE:figures/full_fig_p033_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Angular frequency dependence of the percentage change in phase velocity under varying volume fraction parameter [PITH_FULL_IMAGE:figures/full_fig_p033_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: 2D schematic illustration of a Single-degree-of-freedom (SDOF) system undergoing Love wave induced ground [PITH_FULL_IMAGE:figures/full_fig_p034_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Influence of different source mechanisms on the frequency response characteristics of the SDOF system. (a) Point [PITH_FULL_IMAGE:figures/full_fig_p036_20.png] view at source ↗
read the original abstract

The present study develops a detailed theoretical and mathematical formulation to analyze the time and frequency domain propagation characteristics of Love waves in a stratified fractured poroviscoelastic continuum.The top stratum is modeled as a fractured poroviscoelastic material,whereas the lower semi infinite region exhibits heterogeneity and a gradual transition from viscoelastic behavior near the interface to purely elastic response at greater depths.Fractional order constitutive relations are incorporated to capture the memory-dependent mechanical behavior of the medium using Riemann Liouville fractional derivatives. Three distributed source models, namely Gaussian, Ricker and double-couple sources, are considered. To the best of our knowledge, the mathematical formulation of these distributed sources within the present framework has not been established in earlier studies, where the excitation is typically modeled using an idealized point source. By applying Fourier transform techniques in conjunction with Greens function methodology, the complex dispersion relation is obtained. Since the resulting dispersion equation yields complex roots, a hybrid Newton Raphson iterative algorithm is employedto compute these roots efficiently. Synthetic seismograms are generated to verify that the obtained solutions remain physically consistent and meaningful. Numerical simulations are then performed to investigate the effects of heterogeneity, fractional viscoelasticity and porosity on wave propagation characteristics, thereby identifying the parameters that exert the most significant influence on the system response. Furthermore, to examine the structural implications of the propagated waves, a single degree of freedom SDOF oscillator model is employed to evaluate the surface response corresponding to different types of seismic sources.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a theoretical formulation for Love-wave propagation in a fractured poroviscoelastic layer overlying a heterogeneous viscoelastic half-space, employing Riemann-Liouville fractional derivatives to model memory effects. It introduces mathematical expressions for three distributed seismic sources (Gaussian, Ricker, and double-couple), derives the complex dispersion relation via Fourier transforms combined with Green's functions, solves the resulting equation with a hybrid Newton-Raphson algorithm, generates synthetic seismograms, and performs numerical studies on the influence of heterogeneity, fractional order, and porosity, together with an SDOF oscillator analysis of surface response.

Significance. If the source-term insertions and root-finding procedure are correctly executed, the work supplies a concrete extension of prior point-source models to distributed sources within a fractional poroviscoelastic setting. The parameter-sensitivity results and seismogram generation constitute falsifiable outputs that could be compared with field data; the explicit use of Green's functions for the inhomogeneous problem is a standard but useful technical step.

major comments (2)
  1. [Source-term formulation and Green's-function construction] The central novelty claim concerns the insertion of the three distributed sources into the governing system. The dispersion relation itself is extracted from the homogeneous boundary-value problem and is therefore independent of the particular source; the manuscript should therefore make explicit (with the relevant transformed equations) how the Gaussian, Ricker, and double-couple terms appear only in the particular solution while still constituting a new formulation within this constitutive framework.
  2. [Numerical root-finding algorithm] The hybrid Newton-Raphson procedure is stated to locate complex roots of the dispersion equation, yet no convergence analysis, initial-guess strategy, or verification against limiting cases (e.g., vanishing fractional order or homogeneous half-space) is supplied. Because the physical consistency of the subsequent seismograms rests on these roots, such verification is load-bearing.
minor comments (2)
  1. [Abstract] The abstract contains typographical errors ("employ edto", missing apostrophe in "Green's", inconsistent hyphenation of "double-couple").
  2. [Constitutive relations and model parameters] Notation for the fractional-order parameters and the heterogeneity functions should be introduced once, with a single table of symbols, to avoid repeated re-definition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We address each major comment below, agreeing where clarification or additional verification is warranted, and indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: [Source-term formulation and Green's-function construction] The central novelty claim concerns the insertion of the three distributed sources into the governing system. The dispersion relation itself is extracted from the homogeneous boundary-value problem and is therefore independent of the particular source; the manuscript should therefore make explicit (with the relevant transformed equations) how the Gaussian, Ricker, and double-couple terms appear only in the particular solution while still constituting a new formulation within this constitutive framework.

    Authors: We agree that the complex dispersion relation is obtained from the homogeneous boundary-value problem and is independent of the source terms. The contribution of the work lies in the formulation of the three distributed sources (Gaussian, Ricker, and double-couple) within the memory-dependent fractured poroviscoelastic constitutive framework and their incorporation via Green's functions into the particular solution of the inhomogeneous problem. In the revised manuscript we will add the relevant transformed governing equations that explicitly isolate the source terms in the particular solution, thereby clarifying how the present formulation extends earlier point-source treatments while preserving the source-independent dispersion relation. revision: yes

  2. Referee: [Numerical root-finding algorithm] The hybrid Newton-Raphson procedure is stated to locate complex roots of the dispersion equation, yet no convergence analysis, initial-guess strategy, or verification against limiting cases (e.g., vanishing fractional order or homogeneous half-space) is supplied. Because the physical consistency of the subsequent seismograms rests on these roots, such verification is load-bearing.

    Authors: We acknowledge that the manuscript does not presently supply a convergence study, an explicit description of the initial-guess strategy, or direct comparisons with limiting cases. In the revised version we will include a brief convergence analysis of the hybrid Newton-Raphson algorithm, document the initial-guess procedure, and present verification results for the limiting cases of vanishing fractional order and a homogeneous half-space. These additions will confirm that the roots used to generate the synthetic seismograms are physically consistent. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation proceeds from standard constitutive relations (Riemann-Liouville fractional derivatives for the poroviscoelastic layer, heterogeneity in the half-space) to the governing PDEs, then applies Fourier transforms and Green's functions to obtain the complex dispersion relation from the homogeneous boundary-value problem. Source terms (Gaussian, Ricker, double-couple) enter only the particular solution and do not appear in the characteristic equation. No self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior work are present in the provided text. The numerical root-finding and seismogram generation are downstream applications of the independently derived dispersion equation. The model is therefore self-contained against its own stated assumptions.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Based on abstract only: the model rests on fractional-order constitutive relations, specific heterogeneity profiles, and the validity of the three source representations; no independent evidence or external benchmarks are described.

free parameters (2)
  • fractional order parameters
    Order of Riemann-Liouville derivatives chosen to capture memory effects; values not specified in abstract but required for the dispersion relation.
  • heterogeneity and porosity parameters
    Gradual transition functions and porosity values that affect wave propagation; fitted or chosen to define the half-space model.
axioms (2)
  • domain assumption Riemann-Liouville fractional derivatives correctly represent memory-dependent viscoelastic behavior in the poroviscoelastic layer
    Invoked in the constitutive relations section implied by the abstract.
  • domain assumption The lower half-space transitions from viscoelastic near the interface to elastic at depth
    Stated as part of the medium model.

pith-pipeline@v0.9.1-grok · 5823 in / 1314 out tokens · 26616 ms · 2026-06-26T22:07:57.454545+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

43 extracted references · 32 canonical work pages

  1. [1]

    Love, A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, Cam- bridge, 1920

    A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, Cam- bridge, 1920

    1920

  2. [2]

    Biot, Generalized theory of acoustic propagation in porous dissipative media, J

    M.A. Biot, Generalized theory of acoustic propagation in porous dissipative media, J. Acoust. Soc. Am. 34 (1962) 1254–1264. https://doi.org/10.1121/1.1918315. 41

    work page doi:10.1121/1.1918315 1962

  3. [3]

    Frenkel, Theory of seismic and seismoelectric phenomena in wet soil, Izv

    Y.I. Frenkel, Theory of seismic and seismoelectric phenomena in wet soil, Izv. Akad. Nauk SSSR, Ser. Geogr. Geofiz. 8 (1944) 133–150

    1944

  4. [4]

    Berryman, H.F

    J.G. Berryman, H.F. Wang, The elastic coefficients of double-porosity models for fluid transport in jointed rock, J. Geophys. Res. Solid Earth 100 (1995) 24611–24627. https://doi.org/10.1029/95JB02161

    work page doi:10.1029/95jb02161 1995

  5. [5]

    Berryman, H.F

    J.G. Berryman, H.F. Wang, Elastic wave propagation and attenuation in a double-porosity dual- permeability medium, Int. J. Rock Mech. Min. Sci. 37 (2000) 63–78. https://doi.org/10.1016/S1365- 1609(99)00092-1

    work page doi:10.1016/s1365- 2000

  6. [6]

    Dai, Z.-B

    Z.-J. Dai, Z.-B. Kuang, Love waves in double porosity media, J. Sound Vib. 296 (2006) 1000–1012. https://doi.org/10.1016/j.jsv.2006.03.029

    work page doi:10.1016/j.jsv.2006.03.029 2006

  7. [7]

    Beskos, I

    D.E. Beskos, I. Vgenopoulou, Wave propagation in saturated, fissured, poroelastic rocks, in: Structural Dynamics — Vol. 1, Routledge, 2022, pp. 267–272

    2022

  8. [8]

    Gupta, R

    S. Gupta, R. Dutta, S. Das, Love-type wave propagation in an inhomogeneous cracked porous medium loaded by heterogeneous viscous liquid layer, J. Vib. Eng. Technol. 9 (2021) 433–448. https://doi.org/10.1007/s42417-020-00237-y

    work page doi:10.1007/s42417-020-00237-y 2021

  9. [9]

    Rajak, S

    B.P. Rajak, S. Kundu, S. Gupta, D. Kumar, Love wave propagation characteristics in a fluid- saturated cracked double porous layered structure, Mech. Adv. Mater. Struct. 31 (2024) 2349–2361. https://doi.org/10.1080/15376494.2022.2155333

    work page doi:10.1080/15376494.2022.2155333 2024

  10. [10]

    Pranav, A.K

    P. Pranav, A.K. Rahul, Love-type wave propagation in a heterogeneous medium between a liquid and a fractured porous half-space: A WKB and variable separation approach, Eur. J. Mech. A Solids (2025) 105763. https://doi.org/10.1016/j.euromechsol.2025.105763

    work page doi:10.1016/j.euromechsol.2025.105763 2025

  11. [11]

    Gajroiya, J.S

    K. Gajroiya, J.S. Sikka, Dynamics of Love-type wave propagation in composite transversely isotropic porous structures, Appl. Math. Model. 137 (2025) 115723. https://doi.org/10.1016/j.apm.2024.115723

    work page doi:10.1016/j.apm.2024.115723 2025

  12. [12]

    Kumhar, S

    R. Kumhar, S. Kundu, S. Gupta, Modelling of Love waves in fluid saturated porous viscoelastic medium resting over an exponentially graded inhomogeneous half-space influenced by gravity, J. Appl. Comput. Mech. 6 (2020) 517–530. https://doi.org/10.22055/JACM.2019.29246.1575

    work page doi:10.22055/jacm.2019.29246.1575 2020

  13. [13]

    Negi, A.K

    A. Negi, A.K. Singh, S. Koley, On the scattering of Love waves in a layered transversely isotropic irregular poro-viscoelastic composite rock structure, J. Earthq. Eng. 27 (2023) 1900–1919. https://doi.org/10.1080/13632469.2022.2089406

    work page doi:10.1080/13632469.2022.2089406 2023

  14. [14]

    M. Xu, W. Tan, Intermediate processes and critical phenomena: Theory, method and progress of fractional operators and their applications to modern mechanics, Sci. China Ser. G 49 (2006) 257–272. https://doi.org/10.1007/s11433-006-0257-2

    work page doi:10.1007/s11433-006-0257-2 2006

  15. [15]

    Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, Singapore, 2022

    F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, Singapore, 2022

    2022

  16. [16]

    Meral, T.J

    F.C. Meral, T.J. Royston, R. Magin, Fractional calculus in viscoelasticity: an experimental study, Com- mun. Nonlinear Sci. Numer. Simul. 15 (4) (2010) 939–945. https://doi.org/10.1016/j.cnsns.2009.05.004

    work page doi:10.1016/j.cnsns.2009.05.004 2010

  17. [17]

    Farno, J.-C

    E. Farno, J.-C. Baudez, N. Eshtiaghi, Comparison between classical Kelvin–Voigt and fractional deriva- tive Kelvin–Voigt models in prediction of linear viscoelastic behaviour of waste activated sludge, Sci. Total Environ. 613 (2018) 1031–1036. https://doi.org/10.1016/j.scitotenv.2017.09.206

    work page doi:10.1016/j.scitotenv.2017.09.206 2018

  18. [18]

    Bagley, On the equivalence of the Riemann–Liouville and the Caputo fractional order derivatives in modeling of linear viscoelastic materials, Fract

    R. Bagley, On the equivalence of the Riemann–Liouville and the Caputo fractional order derivatives in modeling of linear viscoelastic materials, Fract. Calc. Appl. Anal. 10 (2007) 123–126

    2007

  19. [19]

    J. Long, R. Xiao, W. Chen, Fractional viscoelastic models with non-singular kernels, Mech. Mater. 127 (2018) 55–64. https://doi.org/10.1016/j.mechmat.2018.07.012. 42

    work page doi:10.1016/j.mechmat.2018.07.012 2018

  20. [20]

    Zhang, X

    Q. Zhang, X. Gu, Q. Dong, J. Liang, Modified fractional–Zener model—Numerical applica- tion in modeling the behavior of asphalt mixtures, Constr. Build. Mater. 388 (2023) 131690. https://doi.org/10.1016/j.conbuildmat.2023.131690

    work page doi:10.1016/j.conbuildmat.2023.131690 2023

  21. [21]

    Kundu, R

    S. Kundu, R. Kumhar, M. Maity, S. Gupta, Influence of point source on Love-type waves in anisotropic layer overlying viscoelastic FGM half-space: Green’s function approach, Int. J. Geomech. 20 (2020) 04019141. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001531

    work page doi:10.1061/(asce)gm.1943-5622.0001531 2020

  22. [22]

    Sadab, G

    M. Sadab, G. Saikia, S. Kundu, Effects of parameters on the Green’s function problem in the presence of an impulsive point source in highly heterogeneous viscoelastic media, ZAMM Z. Angew. Math. Mech. 105 (2025) e70060. https://doi.org/10.1002/zamm.70060

    work page doi:10.1002/zamm.70060 2025

  23. [23]

    Pradhan, S

    N. Pradhan, S. Manna, S.K. Samal, SH-type wave motion in a geophysical model with monoclinic and heterogeneous media due to a point source at the interface, Arch. Appl. Mech. 93 (2023) 2613–2629. https://doi.org/10.1007/s00419-023-02399-z

    work page doi:10.1007/s00419-023-02399-z 2023

  24. [24]

    Pramanik, S

    D. Pramanik, S. Manna, O. Şahin, Love-type wave propagation in a coated fluid-saturated fractured poro-viscoelastic layer with sliding contacts and point source effect, Appl. Math. Model. 125 (2024) 424–444. https://doi.org/10.1016/j.apm.2023.08.045

    work page doi:10.1016/j.apm.2023.08.045 2024

  25. [25]

    Pramanik, S

    D. Pramanik, S. Manna, Love-like wave fields at the interface of sliding contact with non-local elas- tic heterogeneous fluid-saturated fractured poro-viscoelastic layer, Eur. J. Mech. A/Solids 107 (2024) 105350. https://doi.org/10.1016/j.euromechsol.2024.105350

    work page doi:10.1016/j.euromechsol.2024.105350 2024

  26. [26]

    Sarkar, S

    S. Sarkar, S. Kundu, Modeling of Love-type wave propagation in a fractional poro-viscoelastic layer over a viscoelastic-to-elastic half-space, induced by a Gaussian source, Appl. Math. Model. (2025) 116441. https://doi.org/10.1016/j.apm.2025.116441

    work page doi:10.1016/j.apm.2025.116441 2025

  27. [27]

    Gholamy, V

    A. Gholamy, V. Kreinovich, Why Ricker wavelets are successful in processing seismic data: Towards a theoretical explanation, in: Proc. 2014 IEEE Symp. Comput. Intell. Eng. Solut. (CIES), IEEE, 2014, pp. 11–16. https://doi.org/10.1109/CIES.2014.7011824

    work page doi:10.1109/cies.2014.7011824 2014

  28. [28]

    Wang, Frequencies of the Ricker wavelet, Geophysics 80 (2015) A31–A37

    Y. Wang, Frequencies of the Ricker wavelet, Geophysics 80 (2015) A31–A37. https://doi.org/10.1190/geo2014-0441.1

    work page doi:10.1190/geo2014-0441.1 2015

  29. [29]

    Okal, Earthquake, focal mechanism, in: Encyclopedia of Solid Earth Geophysics, Springer, 2020, pp

    E.A. Okal, Earthquake, focal mechanism, in: Encyclopedia of Solid Earth Geophysics, Springer, 2020, pp. 1–6. https://doi.org/10.1007/978-3-030-58631-7_158

    work page doi:10.1007/978-3-030-58631-7_158 2020

  30. [30]

    Y. Gao, H. Hu, Seismoelectromagnetic waves radiated by a double couple source in a saturated porous medium, Geophys. J. Int. 181 (2010) 873–896. https://doi.org/10.1111/j.1365-246X.2010.04526.x

    work page doi:10.1111/j.1365-246x.2010.04526.x 2010

  31. [31]

    Ten Kroode, S

    F. Ten Kroode, S. Bergler, C. Corsten, J.W. de Maag, F. Strijbos, H. Tijhof, Broad- band seismic data—The importance of low frequencies, Geophysics 78 (2) (2013) WA3–WA14. https://doi.org/10.1190/geo2012-0294.1

    work page doi:10.1190/geo2012-0294.1 2013

  32. [32]

    X. Yin, J. Xia, C. Shen, H. Xu, Comparative analysis on penetrating depth of high-frequency Rayleigh and Love waves, J. Appl. Geophys. 111 (2014) 86–94. https://doi.org/10.1016/j.jappgeo.2014.09.022

    work page doi:10.1016/j.jappgeo.2014.09.022 2014

  33. [33]

    K. Aki, P. Richards, Quantitative Seismology, MIT Press, Sausalito, CA, 2002

    2002

  34. [34]

    Chattopadhyay, S

    A. Chattopadhyay, S. Gupta, P. Kumari, V.K. Sharma, Effect of point source and heterogeneity on the propagation of SH-waves in a viscoelastic layer over a viscoelastic half space, Acta Geophys. 60 (2012) 119–139. https://doi.org/10.2478/s11600-011-0059-4

    work page doi:10.2478/s11600-011-0059-4 2012

  35. [35]

    Medina, J.A

    C.R. Medina, J.A. Rupp, D.A. Barnes, Effects of reduction in porosity and permeability with depth on storage capacity and injectivity in deep saline aquifers: A case study from the Mount Simon Sandstone aquifer, Int. J. Greenh. Gas Control 5 (2011) 146–156. 43

    2011

  36. [36]

    Athy, Density, porosity and compaction of sedimentary rocks, AAPG Bull

    L.F. Athy, Density, porosity and compaction of sedimentary rocks, AAPG Bull. 14 (1930) 1–24

    1930

  37. [37]

    X. Fang, L. Wang, J. Lou, H. Fan, J. Du, Flexoelectricity and size effects on SH surface waves in a nanoscale piezoelectric semiconductor layered medium, Appl. Math. Model. 141 (2025) 115928. https://doi.org/10.1016/j.apm.2025.115928

    work page doi:10.1016/j.apm.2025.115928 2025

  38. [38]

    Balachandra Rao, Numerical Methods: With Programs in BASIC, FORTRAN, Pascal and C++, Universities Press (India), Hyderabad, 2004

    S. Balachandra Rao, Numerical Methods: With Programs in BASIC, FORTRAN, Pascal and C++, Universities Press (India), Hyderabad, 2004

    2004

  39. [39]

    Kiełczyński, Direct Sturm–Liouville problem for surface Love waves propagating in layered viscoelas- tic waveguides, Appl

    P. Kiełczyński, Direct Sturm–Liouville problem for surface Love waves propagating in layered viscoelas- tic waveguides, Appl. Math. Model. 53 (2018) 419–432. https://doi.org/10.1016/j.apm.2017.09.013

    work page doi:10.1016/j.apm.2017.09.013 2018

  40. [40]

    Stein, M

    S. Stein, M. Wysession, An Introduction to Seismology, Earthquakes and Earth Structure, John Wiley & Sons, Hoboken, NJ, 2009

    2009

  41. [41]

    B. Mi, J. Xia, J.H. Bradford, C. Shen, Estimating near-surface shear-wave-velocity structures via multichannel analysis of Rayleigh and Love waves: An experiment at the Boise Hydrogeophysical Research Site, Surv. Geophys. 41 (2020) 323–341. https://doi.org/10.1007/s10712-019-09582-4

    work page doi:10.1007/s10712-019-09582-4 2020

  42. [42]

    K. D. Hjelmstad, Fundamentals of Structural Dynamics, Springer, 2022

    2022

  43. [43]

    Mimura, A

    T. Mimura, A. Mita, Automatic estimation of natural frequencies and damping ratios of building structures, Procedia Engineering 188 (2017) 163–169. https://doi.org/10.1016/j.proeng.2017.04.470. 44

    work page doi:10.1016/j.proeng.2017.04.470 2017