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Scattering by two staggered semi-infinite cracks on square lattice: an application of asymptotic Wiener-Hopf factorization

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Pith's one-line read The paper derives an approximate solution for two staggered semi-infinite cracks on a square lattice by first-order asymptotic Wiener-Hopf factorization, yielding scattered-field integrals and far-field formulas that match numerical…

desk verdict New staggered-crack lattice scattering problem, but the far-field formulas are undermined by a sign error in the incident pole that must be fixed before the numerics can be trusted. read the letter →

arxiv 1908.01952 v1 pith:XZ4GTRZ3 submitted 2019-08-06 math-ph math.MP

Gaurav Maurya , Basant Lal Sharma This is my paper · ORCID
classification math-phmath.MP MSC 45E1074J20
keywords latticewavesWiener-Hopffactorizationstaggeredcrackssquarescatteringasymptoticstationaryphasemethodanti-planedisplacement
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

An incident time-harmonic lattice wave hits two parallel, semi-infinite cracks on a square lattice whose tips are staggered by $M$ lattice spacings. The paper reduces the discrete scattering problem to a coupled $2\times2$ Wiener-Hopf equation whose kernel cannot be factorized by standard methods, and it adapts an asymptotic factorization technique to the unit circle to get a first-order approximate factorization in the small-offset parameter. Using that factorization, the Wiener-Hopf technique yields an integral representation of the scattered displacement field and stationary-phase far-field formulas. The paper shows numerically that these formulas agree with a direct lattice simulation when the offset is small, that agreement worsens as $M$ grows, and that the low-frequency numerical solution reproduces the known continuum solution for two staggered plates.

What carries the argument

The load-bearing object is the $2\times2$ Wiener-Hopf kernel $K(z)=L(z)G_M(z)$, with $L=h/r$ the scalar factor familiar from the single semi-infinite crack and $G_M$ the symmetric matrix $\begin{pmatrix}1 & z^{-M}\lambda^N \\ z^M\lambda^N & 1\end{pmatrix}$ that carries the stagger. A similarity transformation by $R_M=\mathrm{diag}(z^{-M/2},z^{M/2})$ recasts $G_M$ as a matrix $F$ with entries $1$ and $\lambda^N$, which is exactly factorized through its eigenvalues $1\pm\lambda^N$. The remaining part is written as $I+N_M$ and split additively into factors analytic inside and outside the unit circle, giving the approximate product $G_M \approx (I+N_{1M-})(I+N_{1M+})$. This approximate factorization is what permits the Liouville step that produces the displacement formulas.

What would settle it

Compute the unit-circle norm of the difference between $G_M$ and the first-order product $(I+N_{1M-})(I+N_{1M+})$ for $M=1,2,3,\ldots$ at fixed $N$ and frequency; if the normalized residual does not shrink with the offset parameter, the first-order factorization fails. A direct check is to compare the far-field angle dependence from (113)-(114) with an independent lattice simulation at $M=3$, $N=4$, $\omega=0.35$, where the paper's figures already show visible deviation.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the matrix kernel $K(z)=L(z)G_M(z)$ of the coupled Wiener-Hopf equations can be multiplicatively factorized to first order as $K_-K_+$, where $L$ is the scalar factor from the single-crack problem, $G_M$ is the symmetric matrix encoding the stagger, and the perturbation enters through a matrix $N_M$ whose size is controlled by $\epsilon = \lambda^N \sin(\xi M/2)$. Once this factorization is made, the standard Wiener-Hopf argument gives explicit expressions for the half-range transforms, hence the contour integrals (93)-(94) for the displacement and the stationary-phase far-field estimates (113)-(114). The paper presents these formulas as the approximate solution of the two-staggered-crack lattice problem for small offset, and it reports graphical agreement with numerical lattice solutions in that regime.

Load-bearing premise

The load-bearing premise is that the offset between the two crack tips is small enough that keeping only the first-order correction to the zero-offset kernel is accurate; the paper supplies no error bound and notes that agreement with numerics gets worse as the offset increases.

Editorial extensions

If this is right

  • For small tip offset, the scattered field anywhere outside the crack region can be evaluated from one-dimensional contour integrals rather than by solving the full two-dimensional lattice problem.
  • The factorization separates the geometry: the single-crack scalar factor $L_\pm$ and the zero-offset matrix factors are universal, while the crack separation $N$ enters through $\lambda^N$ and explicit Chebyshev product factors.
  • In the aligned-tip limit $M=0$ the perturbation matrix vanishes and the formulas reduce to the exact zero-offset solution, giving a built-in consistency check.
  • The same factorization can be applied to a wave incident from the waveguide between the cracks; only the right-hand side of the Wiener-Hopf equation changes.
  • Because agreement with numerics degrades as $M$ increases, the derived formulas are reliable only for small-to-moderate stagger relative to the crack separation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The residual $G_M-(I+N_{1M-})(I+N_{1M+})$ could be measured numerically on the unit circle; if it scales like $|\epsilon|$ over a wider parameter range, the first-order factorization may be promoted to a rigorous asymptotic expansion by iterating the same correction.
  • The Chebyshev-polynomial product forms for the scalar factors suggest that for even $N$ the contour integrals defining the factors can be evaluated in closed form, removing the numerical contour integration the paper identifies as a practical difficulty.
  • A similar stagger matrix arises for cracks on triangular or honeycomb lattices, so the same $R_M F R_M^{-1}$ decomposition may yield first-order factorization for those geometries as well.
  • An analytic estimate of the growth of $K_-(z)K_-^{-1}(z_P)$ near $z=0$ would replace the numerical inspection used to justify the Liouville step and would tell whether the approximation remains uniform in the observation angle.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper formulates scattering of a time-harmonic plane wave on a square lattice by two parallel semi-infinite cracks with staggered edges as a 2x2 matrix Wiener-Hopf equation (36). Because the matrix kernel is not exactly factorable, the authors adapt an asymptotic factorization (61) based on the small parameter epsilon = lambda^N sin(xi M/2), solve the Wiener-Hopf equation formally, and give integral representations of the scattered field in (93)-(94). The far field is then approximated by stationary phase, leading to (113)-(114), and these formulas are compared with direct numerical lattice simulations and with the low-frequency continuum solution of Abrahams and Wickham.

Significance. If correct, the paper would provide the first lattice analogue of the two staggered semi-infinite crack/plate diffraction problem and would demonstrate that asymptotic Wiener-Hopf factorization can be adapted to a circular contour for a discrete matrix kernel. The work has genuine strengths: the Wiener-Hopf setup is coherent, the zero-offset limit is tied to an exact solution in [50], the low-frequency comparison with the continuum Abrahams-Wickham solution is physically meaningful, and the numerical comparisons involve no fitted free parameters. However, the central displayed far-field formulas currently contain an internal sign/pole inconsistency, and several load-bearing steps (the first-order truncation, the Liouville argument, and the dropped pole contribution) lack analytic or quantitative control. These issues must be addressed before the claims can be accepted.

major comments (4)
  1. [§3.1, Eqs. (57)-(61)] The first-order factorization G1,M(z) ≈ (I + N1M-(z))(I + N1M+(z)) is adopted without any error bound or convergence estimate. The parameter epsilon = lambda^N sin(xi M/2) is asserted to be small, and the paper itself notes that agreement with numerics degrades as M increases, but there is no quantitative criterion for when the truncated product is a valid replacement for G_M. Since (61) is the basis of the entire solution, this requires either a rigorous remainder estimate or a systematic numerical study of the truncation error.
  2. [§3.2, Eq. (77) and Figs. 2-3] The Liouville step J(z)=0 is justified only by numerical inspection of plots. The solution formulas (78)-(79) depend on the entire function J being identically zero; this requires analytic control of the growth/decay of the two sides of (77), not visual inspection of a few computed curves. Without such control, the derivation of (78)-(79) is incomplete.
  3. [§4, Eqs. (93)-(94), (109), (113)-(114)] There is an internal sign/pole inconsistency in the inversion and far-field formulas. Equation (31) sets zP = e^{ik cos Theta}, and equations (91)-(92) place the pole at z = zP. However, the printed inversion integrals (93)-(94) contain K_-^{-1}(zP^{-1}) and denominator z - zP^{-1}; with the mapping z = e^{-i xi} this places the pole at xi = k cos Theta, as stated in (109), whereas (31)-(32) require xi = -k cos Theta. The stationary-phase amplitudes (113)-(114) inherit the wrong evaluation point. Unless a reciprocal convention is introduced, the displayed far-field formulas are not the asymptotics of the field defined by (91)-(92). This must be corrected and the numerical comparisons rerun with the corrected sign.
  4. [§4, after Eq. (112) and note following (114)] The far-field asymptotics are stated 'modulo the contribution of pole', and the pole contribution is then dropped without an estimate. In scattering problems, the pole typically gives the reflected or plane-wave component, which can be of the same order as the saddle-point contribution. As written, (113)-(114) cannot be asserted as asymptotic equivalents unless the pole contribution is shown to be negligible in the relevant angular sector; otherwise it must be included in the comparison with numerical data.
minor comments (4)
  1. [§1, Eq. (16)] The phase in vi_2x is written as e^{ik(N sin Theta + M sin Theta)}, but the subsequent shifted Fourier transform (32) and the Wiener-Hopf forcing (37c) use M cos Theta; this should be reconciled.
  2. [§3.2, Eq. (76)] The sentence 'C+ and C+ are analytic' should presumably read 'C+ and C- are analytic'; as printed it is a typographical error.
  3. [§3.2, near Eq. (77)] The Liouville argument refers to 'Examining (135), (136), and (59)' with equations from Appendix A that assume N is even, but the main solution is stated for general N; the scope of the numerical justification should be clarified.
  4. [§4, Figs. 4-5] The claimed agreement between the semi-analytical and numerical far fields is only visual; the paper would be substantially strengthened by a quantitative error metric (e.g., relative L2 or pointwise error as a function of M and omega).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the analysis is self-contained, with no fitted parameters and no load-bearing self-citation; the central staggered-offset result is derived from the stated equations.

full rationale

The paper's derivation chain is not circular. The matrix Wiener-Hopf equation (36) with kernel K = L G_M is solved using an approximate factorization developed in the paper itself: G_M is rewritten via R_M and F, then expanded as I + N_M and factorized to first order as (I + N1M-)(I + N1M+) (Eqs. 38-61). The only imported factorization is that of the scalar function L(z), taken from the first author's earlier work [44], but Eq. (66) gives the explicit closed-form factors L_±(z) = C_L sqrt((1-z_h z^{-1})/(1-z_r z^{-1})). This is a parameter-free, published result whose assumptions do not include the staggered-offset target, so it constitutes independent support rather than circularity. The zero-offset exact solution [50] by the same authors is used only as a validation check, not as an ingredient in the staggered factorization. The low-frequency comparison to the continuum solution [4,5] is an external benchmark. No parameter is fitted: omega, Theta, N, and M are physical inputs, and the far-field formulas (93)-(94) and (113)-(114) follow by stationary phase from the derived integral representation; the Cauchy-projector coefficients are computed numerically, not tuned to match the plotted outputs. The Liouville step J(z) = 0 is justified by numerical inspection of Figs. 2-3 rather than by an analytic estimate; this is a rigor gap, not circularity. Similarly, the apparent mismatch between the pole at z_P in (84)-(85) and at z_P^{-1} in (93)-(94) is an internal consistency or correctness issue, not a reduction of the output to the input. Therefore the paper receives a circularity score of 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No invented entities and no fitted free parameters. The load-bearing imported items are the factorization of L(z) from [44], the assumed smallness of the perturbation N_M, and the numerical justification of the Liouville step; each is listed above.

assumptions (5)
  • ad hoc to paper The first-order perturbation factorization (61) is valid when epsilon = lambda^N sin(xi M/2) is small; no error bound is supplied.
    Adopted in Section 3.1 to replace G_M by (I+N1M-)(I+N1M+); the paper concedes agreement with numerics worsens as M grows.
  • ad hoc to paper The function J(z) in (77) has decay and growth behavior sufficient for Liouville's theorem.
    Justified by numerical plots in Figures 2 and 3, with no analytic proof.
  • domain assumption Existence and uniqueness of the discrete scattering solution hold as in the single-crack analysis [46].
    The Introduction states this is anticipated but not shown.
  • standard math The factors L+(z) and L-(z) from [44] apply to the present matrix kernel.
    Relied on in equations (62)-(66) and cited to Sharma [44]; the zero-offset numerical comparison and continuum limit provide partial checks.
  • ad hoc to paper R_M(z) in (39) is analytic, bounded, and locally Holder-continuous on the unit circle.
    Stated without proof in Section 3.1.

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Pith. "Pith review of Scattering by two staggered semi-infinite cracks on square lattice: an application of asymptotic Wiener-Hopf factorization." pith.science (2026). https://pith.science/paper/XZ4GTRZ3

@misc{pith2026190801952,
  author       = {Pith},
  title        = {Pith review of: Scattering by two staggered semi-infinite cracks on square lattice: an application of asymptotic Wiener-Hopf factorization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XZ4GTRZ3}},
  note         = {Machine review of arXiv:1908.01952}
}
read the original abstract

Scattering of time-harmonic plane wave by two parallel semi-infinite rows, but with staggered edges, is considered on square lattice. The condition imposed on the semi-infinite rows is a discrete analogue of Neumann boundary condition. A physical interpretation assuming an out-of-plane displacement for the particles arranged in the form of a square lattice and interacting with nearest-neighbours, associates the scattering problem to lattice wave scattering due to the presence of two staggered but parallel crack tips. The discrete scattering problem is reduced to the study of a pair of Wiener-Hopf equation on an annulus in complex plane, using Fourier transforms. Due to the offset between the crack edges, the Wiener-Hopf kernel, a 2x2 matrix, is not amenable to factorization in a desirable form and an asymptotic method is adapted. Further, an approximation in the far field is carried out using the stationary phase method. A graphical comparison between the far-field approximation based on asymptotic Wiener-Hopf method and that obtained by a numerical solution is provided. Also included is a graphical illustration of the low frequency approximation, where it has been found that the numerical solution of the scattering problem coincides with the well known formidable solution in the continuum framework.

Figures

Figures reproduced from arXiv: 1908.01952 by the authors.

Figure 1
Figure 1. Schematic of a pair of staggered, parallel cracks on square lattice (with [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Determinant of Kk−(z) as z → 0, with z = z(x) = x + 0.005i, x on the horizontal axis. The black curve shows the determinant while the grey plot shows the slope of the determinant. The parameters chosen for plotting purpose are shown in the plot label. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. The behaviour of the elements of (A) C−(z), (B) C+(1/z) and (C, D) K+(1/z) as z → 0, with z = z(x) = x + 0.005i, x on the horizontal axis. The function δD+(zz−1 P ) has a simple pole at z = zP, which lies outside the annulus of analyticity A but inside the unit circle in complex plane. The vector function C can be additively factorised as C(z) = C+(z) + C−(z); (75) the factors are given by C+(z) = (−K+(z) + K−1 − (z… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Modulus of the diffracted field |u| against the observation angle θ (100) for angle of incidence Θ = 45◦ , N = 4 and (A) M = 0, (B) M = 1, (C) M = 2 and (D) N = 6 and M = 2. The numerical results are shown in black while the semi-analytical results are shown in blue. T…
Figure 5
Figure 5. Figure 5: Modulus of the diffracted field |φ| against the observation angle θ for angle of incidence Θ = 45◦ , kr = 30, kh = π/2, ka = 0.1, 2, 4 (from left to right). The parameters are chosen as given in [5]. The thick black curve shows the exact solution obtained by [4, 5] whi…

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Works this paper leans on

54 extracted references · 41 canonical work pages

  1. [50]

    Discrete scattering by a pair of parallel defects

    B. L. Sharma and G. Maurya. “Discrete scattering by a pair of parallel defects”. In: Philosophical Transactions of the Royal Society A: Mathe- matical, Physical and Engineering Sciences accepted (Aug. 2019), pp. 1–

    work page 2019

  2. [1]

    M. J. Ablowitz and A. S. Fokas. Complex variables: introduction and applications. Second. Cambridge Texts in Applied Mathematics. Cam- bridge University Press, Cambridge, 2003, pp. xii+647. doi: 10 . 1017 / CBO9780511791246. url: https://doi.org/10.1017/CBO9780511791246

    work page doi:10.1017/cbo9780511791246 2003

  3. [2]

    Acoustic scattering by two parallel slightly staggered rigid plates

    I. D. Abrahams and G. R. Wickham. “Acoustic scattering by two parallel slightly staggered rigid plates”. In: Wave Motion 12.3 (1990), pp. 281–

    work page 1990

  4. [3]

    General Wiener-Hopf factorization of matrix kernels with exponential phase factors

    I. D. Abrahams and G. R. Wickham. “General Wiener-Hopf factorization of matrix kernels with exponential phase factors”. In:SIAM J. Appl. Math. 50.3 (1990), pp. 819–838. doi: 10.1137/0150047 . url: https://doi. org/10.1137/0150047

    work page doi:10.1137/0150047 1990

  5. [4]

    On the scattering of sound by two semi-infinite parallel staggered plates. I. Explicit matrix Wiener-Hopf fac- torization

    I. D. Abrahams and G. R. Wickham. “On the scattering of sound by two semi-infinite parallel staggered plates. I. Explicit matrix Wiener-Hopf fac- torization”. In: Proc. Roy. Soc. London Ser. A 420.1858 (1988), pp. 131– 156

    work page 1988

  6. [5]

    The scattering of sound by two semi- infinite parallel staggered plates. II. Evaluation of the velocity potential for an incident plane wave and an incident duct mode

    I. D. Abrahams and G. R. Wickham. “The scattering of sound by two semi- infinite parallel staggered plates. II. Evaluation of the velocity potential for an incident plane wave and an incident duct mode”. In: Proc. Roy. Soc. London Ser. A 427.1872 (1990), pp. 139–171

    work page 1990

  7. [6]

    On the application of the Wiener-Hopf technique to problems in dynamic elasticity

    I. D. Abrahams. “On the application of the Wiener-Hopf technique to problems in dynamic elasticity”. In: Wave Motion 36.4 (2002). Dedicated to Jan D. Achenbach on the occasion of his 65th birthday, pp. 311–333. doi: 10.1016/S0165- 2125(02)00027- 6 . url: https://doi.org/10. 1016/S0165-2125(02)00027-6

    doi:10.1016/s0165- 2002

  8. [7]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun, eds. Handbook of mathematical func- tions with formulas, graphs, and mathematical tables . Reprint of the 1972 edition. Dover Publications, Inc., New York, 1992, pp. xiv+1046

    work page 1972

Show all 54 references
  1. [8]

    J. D. Achenbach. Wave propagation in elastic solids . first. Vol. 16. North- Holland Series in Applied Mathematics and Mechanics. North-Holland Publishing Co., Amsterdam, 1976, front matter+425

    1976

  2. [9]

    B¨ ottcher and B

    A. B¨ ottcher and B. Silbermann. Analysis of Toeplitz operators . Second. Springer Monographs in Mathematics. Prepared jointly with Alexei Karlovich. Springer-Verlag, Berlin, 2006, pp. xiv+665. 22

    2006

  3. [10]

    Diffraction theory

    C. J. Bouwkamp. “Diffraction theory”. In: Reports on Progress in Physics 17 (1954), pp. 35–100

    1954

  4. [11]

    Brillouin

    L. Brillouin. Wave Propagation in Periodic Structures. Electric Filters and Crystal Lattices. McGraw-Hill Book Company, Inc., New York, 1946, pp. xii+247

    1946

  5. [12]

    Systems of Integral Equations on a Half Line with Kernels Depending on the Difference of Arguments

    I. C. Gohberg and M. G. Krein. “Systems of Integral Equations on a Half Line with Kernels Depending on the Difference of Arguments”. In: Am. Math. Soc.Transl. 14 (1960), pp. 217–287

    1960

  6. [13]

    L. Collatz. The numerical treatment of differential equations. 3d ed. Trans- lated from a supplemented version of the 2d German edition by P. G. Williams. Die Grundlehren der mathematischen Wissenschaften, Bd. 60. Springer-Verlag, Berlin-G¨ ottingen-Heidelberg, 1960, xv+568 pp....

    1960

  7. [14]

    On the solution of two coupled Wiener–Hopf equations

    V. Daniele. “On the solution of two coupled Wiener–Hopf equations”. In: SIAM Journal on Applied Mathematics 44.4 (1984), pp. 667–680

    1984

  8. [15]

    S. Elaydi. An introduction to difference equations . Third. Undergraduate Texts in Mathematics. Springer, New York, 2005, pp. xxii+539

    2005

  9. [16]

    L. C. Evans. Partial differential equations. Second. Vol. 19. Graduate Stud- ies in Mathematics. American Mathematical Society, Providence, RI, 2010, pp. xxii+749. url: https://doi.org/10.1090/gsm/019

    2010 doi

  10. [17]

    L. B. Felsen and N. Marcuvitz. Radiation and scattering of waves. Prentice- Hall Microwaves and Fields Series. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973, pp. xxxii+888

    1973

  11. [18]

    F. D. Gakhov. Boundary value problems . Translated from the Russian, Reprint of the 1966 translation. Dover Publications, Inc., New York, 1990, pp. xxii+561

    1966

  12. [19]

    Gilbarg and N

    D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order . Classics in Mathematics. Reprint of the 1998 edition. Springer-Verlag, Berlin, 2001, pp. xiv+517

    1998

  13. [20]

    Gohberg and M

    I. Gohberg and M. A. Kaashoek, eds. Constructive methods of Wiener- Hopf factorization. Vol. 21. Operator Theory: Advances and Applications. Birkh¨ auser Verlag, Basel, 1986, pp. xii+409.doi: 10.1007/978-3-0348- 7418-2. url: https://doi.org/10.1007/978-3-0348-7418-2

    1986 doi

  14. [21]

    An Overview of Ma- trix Factorization Theory and Operator Applications

    I. Gohberg, M. A. Kaashoek, and I. M. Spitkovsky. “An Overview of Ma- trix Factorization Theory and Operator Applications”. In: Factorization and Integrable Systems. Ed. by I. Gohberg, N. Manojlovic, and A. F. dos Santos. Basel: Birkh¨ auser Basel, 2003, pp. 1–102

    2003

  15. [22]

    J. G. Harris. Linear elastic waves . Vol. 26. Cambridge University Press, 2001

    2001

  16. [23]

    The radiation and transmission properties of a pair of semi- infinite parallel plates. I

    A. E. Heins. “The radiation and transmission properties of a pair of semi- infinite parallel plates. I”. In: 6 (1948), pp. 157–166

    1948

  17. [24]

    The radiation and transmission properties of a pair of semi- infinite parallel plates. II

    A. E. Heins. “The radiation and transmission properties of a pair of semi- infinite parallel plates. II”. In: 6 (1948), pp. 215–220. 23

    1948

  18. [25]

    The scope and limitations of the method of Wiener and Hopf

    A. E. Heins. “The scope and limitations of the method of Wiener and Hopf”. In: IX (1956), pp. 447–466

    1956

  19. [26]

    Factorization of a Wiener-Hopf matrix

    D. S. Jones. “Factorization of a Wiener-Hopf matrix”. In: IMA J. Appl. Math. 32.1-3 (1984), pp. 211–220. doi: 10.1093/imamat/32.1- 3.211 . url: https://doi.org/10.1093/imamat/32.1-3.211

    1984 doi

  20. [27]

    E. I. Jury. Theory and Application of the z-Transform Method . Wiley, 1964

    1964

  21. [28]

    An Iterative Wiener–Hopf method for triangular matrix func- tions with exponential factors

    A. V. Kisil. “An Iterative Wiener–Hopf method for triangular matrix func- tions with exponential factors”. In:SIAM Journal on Applied Mathematics 78.1 (2018), pp. 45–62

    2018

  22. [29]

    Levy and F

    H. Levy and F. Lessman. Finite difference equations. Reprint of the 1961 edition. Dover Publications, Inc., New York, 1992, pp. viii+278

    1961

  23. [30]

    J. C. Mason and D. C. Handscomb. Chebyshev polynomials. Chapman & Hall/CRC, Boca Raton, FL, 2003, pp. xiv+341

    2003

  24. [31]

    On some problems involving multiple scattering due to edges

    G. Maurya. “On some problems involving multiple scattering due to edges”. PhD thesis. Indian Institute of Technology Kanpur, 2018

    2018

  25. [32]

    Elastodynamical scattering by N parallel half-planes in R3

    E. Meister and K. Rottbrand. “Elastodynamical scattering by N parallel half-planes in R3”. In: Math. Nachr. 177 (1996), pp. 189–232. doi: 10. 1002/mana.19961770112 . url: http://dx.doi.org/10.1002/mana. 19961770112

    1996 doi

  26. [33]

    Elastodynamical scattering by N parallel half-planes in R3. II. Explicit solutions for N = 2 by explicit symbol factorization

    E. Meister and K. Rottbrand. “Elastodynamical scattering by N parallel half-planes in R3. II. Explicit solutions for N = 2 by explicit symbol factorization”. In: Integral Equations Operator Theory29.1 (1997), pp. 70–

    1997

  27. [34]

    Wiener-Hopf equations for waves scattered by a system of parallel Sommerfeld half-planes

    E. Meister, K. Rottbrand, and F.-O. Speck. “Wiener-Hopf equations for waves scattered by a system of parallel Sommerfeld half-planes”. In:Math. Methods Appl. Sci. 14.8 (1991), pp. 525–552.doi: 10.1002/mma.1670140802. url: http://dx.doi.org/10.1002/mma.1670140802

    1991 doi

  28. [35]

    Wiener–Hopf Factorization of Certain Non- Rational Matrix Functions in Mathematical Physics

    E. Meister and F.-O. Speck. “Wiener–Hopf Factorization of Certain Non- Rational Matrix Functions in Mathematical Physics”. In: The Gohberg Anniversary Collection. Ed. by H. Dym et al. Vol. 41. Operator Theory: Advances and Applications. Birkhauser Basel, 1989, pp. 385–394. doi...

    1989 doi

  29. [36]

    Miklowitz

    J. Miklowitz. The theory of elastic waves and waveguides . Vol. 22. North- Holland Series in Applied Mathematics and Mechanics. North-Holland Publishing Co., Amsterdam-New York, 1978, pp. xvi+618

    1978

  30. [37]

    An asymptotic method of factorization of a class of matrix functions

    G. Mishuris and S. Rogosin. “An asymptotic method of factorization of a class of matrix functions”. In: Proc. R. Soc. A . Vol. 470. The Royal Society. 2014, p. 20140109. 24

    2014

  31. [38]

    Factorization of a class of matrix-functions with stable partial indices

    G. Mishuris and S. Rogosin. “Factorization of a class of matrix-functions with stable partial indices”. In: Mathematical Methods in the Applied Sci- ences 39.13 (2016), pp. 3791–3807

    2016

  32. [39]

    Regular approximate factorization of a class of matrix-function with an unstable set of partial indices

    G. Mishuris and S. Rogosin. “Regular approximate factorization of a class of matrix-function with an unstable set of partial indices”. In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474.2209 (2018), p. 20170279

    2018

  33. [40]

    Mitra and S

    R. Mitra and S. Lee. Analytical techniques in the theory of guided waves . Macmillan, 1971

    1971

  34. [41]

    B. Noble. Methods based on the Wiener-Hopf technique for the solution of partial differential equations. International Series of Monographs on Pure and Applied Mathematics. Vol. 7. Pergamon Press, New York-London- Paris-Los Angeles, 1958, pp. x+246

    1958

  35. [42]

    Constructive methods for factorization of matrix-functions

    S. Rogosin and G. Mishuris. “Constructive methods for factorization of matrix-functions”. In: IMA Journal of Applied Mathematics 81.2 (2015), pp. 365–391

    2015

  36. [43]

    Continuum limit of discrete Sommerfeld problems on square lattice

    B. L. Sharma. “Continuum limit of discrete Sommerfeld problems on square lattice”. In: S¯ adhan¯ a42.5 (2017), pp. 713–728

    2017

  37. [44]

    Diffraction of waves on square lattice by semi-infinite crack

    B. L. Sharma. “Diffraction of waves on square lattice by semi-infinite crack”. In: SIAM J. Appl. Math. 75.3 (2015), pp. 1171–1192. doi: 10 . 1137/140985093. url: https://doi.org/10.1137/140985093

    2015 doi

  38. [45]

    Diffraction of waves on square lattice by semi-infinite rigid constraint

    B. L. Sharma. “Diffraction of waves on square lattice by semi-infinite rigid constraint”. In: Wave Motion 59 (2015), pp. 52–68. doi: 10 . 1016 / j . wavemoti.2015.07.008. url: https://doi.org/10.1016/j.wavemoti. 2015.07.008

    2015 doi

  39. [46]

    Near-tip field for diffraction on square lattice by crack

    B. L. Sharma. “Near-tip field for diffraction on square lattice by crack”. In: SIAM J. Appl. Math. 75.4 (2015), pp. 1915–1940. doi: 10 . 1137 / 15M1010646. url: https://doi.org/10.1137/15M1010646

    2015 doi

  40. [47]

    Near-tip field for diffraction on square lattice by rigid constraint

    B. L. Sharma. “Near-tip field for diffraction on square lattice by rigid constraint”. In: Z. Angew. Math. Phys. 66.5 (2015), pp. 2719–2740. doi: 10 . 1007 / s00033 - 015 - 0508 - z. url: https : / / doi . org / 10 . 1007 / s00033-015-0508-z

    2015

  41. [48]

    On linear waveguides of square and triangular lattice strips: an application of Chebyshev polynomials

    B. L. Sharma. “On linear waveguides of square and triangular lattice strips: an application of Chebyshev polynomials”. In:S¯ adhan¯ a42.6 (2017), pp. 901–927

    2017

  42. [49]

    Wave propagation in bifurcated waveguides of square lat- tice strips

    B. L. Sharma. “Wave propagation in bifurcated waveguides of square lat- tice strips”. In: SIAM J. Appl. Math. 76.4 (2016), pp. 1355–1381. doi: 10.1137/15M1051464. url: https://doi.org/10.1137/15M1051464

    2016 doi

  43. [51]

    L. I. Slepyan. Models and phenomena in fracture mechanics . Foundations of Engineering Mechanics. Springer-Verlag, Berlin, 2002, pp. xviii+576. doi: 10.1007/978-3-540-48010-5 . url: https://doi.org/10.1007/ 978-3-540-48010-5 . A Further simplification and factorization of G1 an...

    2002 doi

  44. [53]

    eprint: http://arxiv.org/abs/ 1906.11404

    doi: 10.1098/rsta.2019.0102 . eprint: http://arxiv.org/abs/ 1906.11404. 25

    2019

  45. [109]

    url: http://dx.doi.org/10.1007/ BF01191481

    doi: 10.1007/BF01191481 . url: http://dx.doi.org/10.1007/ BF01191481

    doi

  46. [297]

    url: https://doi.org/10

    doi: 10.1016/0165-2125(90)90044-5. url: https://doi.org/10. 1016/0165-2125(90)90044-5

    doi

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.