Skip to main content
SpringerLink
Log in

Computing numerically the Green’s function of the half-plane Helmholtz operator with impedance boundary conditions

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

In this article we compute numerically the Green’s function of the half-plane Helmholtz operator with impedance boundary conditions. A compactly perturbed half-plane Helmholtz problem is used to motivate this calculation, by treating it through integral equation techniques. These require the knowledge of the calculated Green’s function, and lead to a boundary element discretization. The Green’s function is computed using the inverse Fourier operator of its spectral transform, applying an inverse FFT for the regular part, and removing the singularities analytically. Finally, some numerical results for the Green’s function and for a benchmark resonance problem are shown.

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. Abramowitz M. and Stegun I.A. (1965). Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York

    Google Scholar 

  2. Barton P. and Rawlins A. (1999). Acoustic diffraction by a semi-infinite plane with different face impedances. Q. J. Mech. Appl. Math. 52(3): 469–487

    Article  MATH  Google Scholar 

  3. Bateman H. (1954). Tables of Integral Transforms. McGraw-Hill Book Company Inc., New York

    Google Scholar 

  4. Chandler-Wilde S. (1997). The impedance boundary value problem for the Helmholtz equation in a half-plane. Math. Meth. Appl. Sci. 20: 813–840

    Article  MATH  Google Scholar 

  5. Chandler-Wilde S. and Peplow A. (2005). A boundary integral equation formulation for the Helmholtz equation in a locally perturbed half-plane. Z. Angew. Math. Mech. 85(2): 79–88

    Article  MATH  Google Scholar 

  6. Durán M. and Nédélec J.-C. (2000). Un Problème Spectral issu d’un Couplage Élasto-Acoustique. RAIRO Math. Model. Numer. Anal. 34: 835–857

    Article  Google Scholar 

  7. Durán M., Muga I. and Nédélec J.-C. (2005). The Helmholtz equation with impedance in a half-plane. C. R. Acad. Sci. Paris Ser. I 340: 483–488

    MATH  Google Scholar 

  8. Durán M., Muga I. and Nédélec J.-C. (2005). The Helmholtz equation with impedance in a half-space. C. R. Acad. Sci. Paris Ser. I 341: 561–566

    MATH  Google Scholar 

  9. Feng, K.: Finite element method and natural boundary reduction. In: Proceedings of the International Congress of Mathematicians, Warszawa, pp. 1439–1453 (1983)

  10. Feng K. (1984). Asymptotic radiation conditions for reduced wave equation. J. Comp. Math. 2(2): 130–138

    MATH  Google Scholar 

  11. Gasquet C. and Witomski P. (1999). Fourier Analysis and Applications. Springer, New York

    MATH  Google Scholar 

  12. Gradshteyn I.S. and Ryzhik I.M. (1980). Table of Integrals, Series and Products. Academic, New York

    MATH  Google Scholar 

  13. Hwang L. and Tuck E. (1970). On the oscillations of harbours of arbitrary shape. J. Fluid. Mech. 42: 447–464

    Article  Google Scholar 

  14. Joannopoulos J., Meade R. and Winn J. (1995). Photonic crystals: molding the flow of light. Princeton University Press, New Jersey

    MATH  Google Scholar 

  15. Lee J. (1971). Wave-induced oscillations in harbours of arbitrary geometry. J. Fluid Mech. 45: 372–394

    Article  Google Scholar 

  16. Mader C.L. (1979). Numerical Modeling of Detonation. University of California Press, Berkeley

    Google Scholar 

  17. Natroshvili, D., Arens, T., Chandler-Wilde, S.: (2003) Uniqueness, existence, and integral equation formulations for interface scattering problems. In: Memoirs on Differential Equations and Mathematical Physics, vol. 30. Georgian Academy of Sciences, pp. 105–146

  18. Nédélec J.-C. (2001). Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems. Springer, Berlin

    MATH  Google Scholar 

  19. Peplow A. and Chandler-Wilde S. (1999). Noise propagation from a cutting of arbitrary cross-section and Impedance. J. Sound Vib. 223: 355–378

    Article  Google Scholar 

  20. Sakoda K. (2001). Optical Properties of Photonic crystals. Springer Series in Optical Sciences 80. Springer, Berlin

    Google Scholar 

  21. Shaw, R.: Long Period Forced Harbor Oscillations, Topics in Ocean Engineering. Bretschneider C. (ed.) Gulf Publishing Company, Texas (1971)

  22. Tamrock (1988) Surface Drilling and Blasting. Naapuri, J. (ed.) Tamrock Oy, Finland

Download references

Author information

Authors and Affiliations

  1. Facultad de Ingeniería, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna, 4860, Macul, Santiago, Chile

    Mario Durán & Ricardo Hein

  2. CMAP, École Polytechnique, 91128, Palaiseau, France

    Jean-Claude Nédélec

Authors
  1. Mario Durán
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ricardo Hein
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jean-Claude Nédélec
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mario Durán.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Durán, M., Hein, R. & Nédélec, JC. Computing numerically the Green’s function of the half-plane Helmholtz operator with impedance boundary conditions. Numer. Math. 107, 295–314 (2007). https://doi.org/10.1007/s00211-007-0087-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-007-0087-9

Mathematics Subject Classification (2000)

Search

Navigation