Skip to main content
SpringerLink
Log in

On PC Ansatzs

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

The paper is devoted to a detailed consideration of an ansatz known from the seventies:

$$e^i \,^{kl(x)} [AD_p (\sqrt k e^{ - \frac{\pi }{4}} m(x))] + k^{ - \frac{1}{2}} e^{\frac{\pi }{4}} BD'_p (\sqrt k e)^{ - \frac{\pi }{4}} m(x))],$$

where

$$A = \sum\limits_{s = 0}^\infty {\frac{{A_s (x)}}{{(ik)^s }},\quad B} = \sum\limits_{s = 0}^\infty {\frac{{B_s (x)}}{{(ik)^s }}.}$$

Here the Dp are parabolic-cylinder functions. Analytic expressions in the first approximation for the wave field in the penumbra of the wave reflected by an impedance or transparent cone are obtained. Bibliography: 11 titles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. A. L. Brodskaya, A. V. Popov, and S. A. Khozioskii, “Asymptotics of the wave reflected by a cone in penumbra,” in: 6th All-Union Symposium on the Diffraction and Propagation of Waves, Moscow-Erevan (1976), pp. 227–231.

  2. F. Olver, Asymptotics and Special Functions [Russian translation], Nauka, Moscow (1990).

    Google Scholar 

  3. Yu. A. Kravtsov, “On a modification of the method of geometric optics,” Izv. Vuzov, Radiofizika, 7, No.4, 664–673 (1964).

    Google Scholar 

  4. N. V. Tsepelev, “Some special solutions of the Helmholtz equation,” Zap. Nauchn. Semin. LOMI, 51, 197–202 (1975).

    MATH  Google Scholar 

  5. I. S. Gradstein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], Moscow (1951).

  6. V. A. Borovikov and B. E. Kinber, Geometric Theory of Diffraction [in Russian], Moscow (1978).

  7. V. A. Fock, Problems in the Diffraction and Propagation of Electromagnetic Waves [in Russian], Moscow (1970).

  8. V. P. Smyshlyaev, “On the diffraction by a cone at high frequencies,” LOMI Preprint E-9-89, Leningrad (1989).

  9. N. Bleistein, “Uniform asymptotic expansion of integrals with stationary points near algebraic singularities,” Comm. Pure Appl. Math., 19, No.4, 353–370 (1966).

    MATH  MathSciNet  Google Scholar 

  10. V. A. Borovikov, Diffraction on Polygons and Polyhedra [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  11. V. M. Babich, D. B. Dement'ev, B. A. Samokish, and V. P. Smyshlyaev, “Scattering of a high-frequency wave on the vertex of an arbitrary cone. Singular directions,” Zap. Nauchn. Semin. POMI, 264, 7–21 (2000).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg Department, Steklov Mathematical Institute, St. Petersburg, Russia

    V. M. Babich

Authors
  1. V. M. Babich
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to P. V. Krauklis on the occasion of his seventieth birthday

__________

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 9–22.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Babich, V.M. On PC Ansatzs. J Math Sci 132, 2–10 (2006). https://doi.org/10.1007/s10958-005-0470-y

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-005-0470-y

Keywords

Search

Navigation