Skip to main content
SpringerLink
Log in

Solution of the Helmholtz-Poincaré Wave Equation Using the Coupled Boundary Integral Equations and Optimal Surface Eigenfunctions

  • Chapter
Boundary Element Technology VII

  • 467 Accesses

Abstract

The Helmholtz-Poincaré Wave Equation (H-PWE) arises in many areas of classical wave scattering theory. In particular it can be found for the cases of acoustical scattering from submerged bounded objects and electromagnetic scattering from objects. The extended boundary integral equations (EBIE) method1–6 is derived from considering both the exterior and interior solutions of the H-PWE’s. This coupled set of expressions has the advantage of not only offering a prescription for obtaining a solution for the exterior scattering problem, but it also obviates the problem of irregular values corresponding to fictitious interior eigenvalues. Once the coupled equations are derived, they can be obtained in matrix form by expanding all relevant terms in partial wave expansions, including a bi-orthogonal expansion of the Green’s function. However some freedom in the choice of the surface expansion is available since the unknown surface quantities may be expanded in a variety of ways so long as closure is obtained. Out of many possible choices, we develop an optimal method to obtain such expansions which is based on the optimum eigenfunctions related to the surface of the object. In effect, we convert part of the problem (that associated with the Fredholms integral equation of the first kind) an eigenvalue problem of a related Hermitian operator. The methodology will be explained in detail and examples will be presented.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Waterman, P. C. ‘New formulation of acoustic scattering’ J. Acoust. Soc. Am., Vol. 45 (6), p. 1417, 1969.

    Article  MATH  Google Scholar 

  2. Waterman, P. C. ‘Matrix Theory of Elastic Wave Scattering’ J. Acoust. Soc. Am., Vol. 60 (1), p. 567, 1976.

    Article  MathSciNet  MATH  Google Scholar 

  3. Waterman, P. C. ‘Survey of T-Matrix Methods and Surface Field Representation.’ Acoustic, Electromagnetic, and Elastic Wave Scattering — Focus on the T-Matrix Approach eds. Varadan, V. K. and V. V. pp. 61–78, Pergammon Press, New York, 1980.

    Google Scholar 

  4. Werby, M. F. and Green, L. R. ‘An Extended Unitary Approach for Acoustical Scattering from Elastic Shells Immersed in Fluids’ J. Acoust. Soc. Am., Vol. 74 (2), p. 625, 1983.

    Article  MATH  Google Scholar 

  5. Werby, M. F. and Chin-Bing, S. ‘Numerical Techniques and Their Use in the Extension of the T-Matrix and Null-Field Approaches to Scattering’ Int. J. Comput. Math. Appls., Vol. 11 (7/8), p. 717, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  6. Werby, M. F. ‘A Coupled High-Order T-Matrix’ J. Acoust. Soc. Am., (to appear) 1987.

    Google Scholar 

  7. Tricomi, F. Integral Equations Wiley Interscience, New York, 1967.

    Google Scholar 

  8. Kleinman, R. E. and Roach, G. F. Boundary integral equations for the three-dimensional Helmholtz equation, SIAM Review, 16, 214 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  9. Martin, P. A. ‘On the Null-Field Equations for the Exterior Problems of Acoustics’ Quart. J. Mech. Appl. Math., Vol. 33, pp. 385, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  10. Ramm, A. G. ‘Convergence of the T-Matrix Approach to Scattering Theory’ J. Math. Phys., Vol. 23, pp. 1123, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  11. Wall, D. N. ‘Methods of Overcoming Numerical Instabilities Associated with the T-Matrix Method.’ Acoustic, Electromagnetic, and Elastic Wave Scattering — Focus on the T-Matrix Approach eds. Varadan, V. K. and V. V. pp. 269–286, Pergammon Press, New York, 1980.

    Google Scholar 

  12. Goldberger, M. L. and Watson, K. M. Collision Theory John Wiley, New York, 1964.

    Google Scholar 

  13. Werby, M. F., Tango, G. J., and Green, L. H. ‘Eigenvalue and Similarity Transformation Methods in the Solution of Acoustical Scattering Problems.’ Computational Acoustics: Algorithms and Applications eds. Lee, D., Sternberg, R. L., and Schultz, M. H. Vol. 2, pp. 257–278, Elsevier Science Publishers B. V., North Holland, 1988.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Naval Research Laboratory Detachment, Theoretical Acoustics Branch, Stennis Space Center, MS, 39529–5004, USA

    M. F. Werby & M. K. Broadhead

  2. Oak Ridge National Lab, Oak Ridge, TN, 37831, USA

    M. R. Strayer & C. Bottcher

Authors
  1. M. F. Werby
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. K. Broadhead
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. R. Strayer
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. C. Bottcher
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Wessex Institute of Technology, University of Portsmouth, Ashurst Lodge, Ashurst, Southampton, SO4 2AA, UK

    C. A. Brebbia

  2. Dept. of mechanical Engineering, University of New Mexico, Albuquerque, NM, 87131, USA

    M. S. Ingber

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Computational Mechanics Publications

About this chapter

Cite this chapter

Werby, M.F., Broadhead, M.K., Strayer, M.R., Bottcher, C. (1992). Solution of the Helmholtz-Poincaré Wave Equation Using the Coupled Boundary Integral Equations and Optimal Surface Eigenfunctions. In: Brebbia, C.A., Ingber, M.S. (eds) Boundary Element Technology VII. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2872-8_17

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-2872-8_17

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-85166-782-6

  • Online ISBN: 978-94-011-2872-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation