Skip to main content

Validity of the HELS Method

  • Chapter
  • First Online:
The Helmholtz Equation Least Squares Method

Part of the book series: Modern Acoustics and Signal Processing ((MASP))

  • 1828 Accesses

Abstract

The validity challenges came at the joint meetings of the 136th Meeting of the Acoustical Society of America (ASA), the 2nd Convention of the European Acoustics Association (EAA), and the 25th German Annual Conference on Acoustics (DAGA) held in Berlin, Germany, 1999 [58]. The major questions were as follows: “How can the acoustic field on the surface of any non-spherical structure be described by the spherical wave functions?” “Is this a Rayleigh hypothesis in NAH that pushes a solution formulation beyond its region of validity?”

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. S. F. Wu, in An overview of reconstruction of radiated acoustic pressures from complex vibrating structures by using the HELS method, The joint meeting of the meeting of the 136th Acoustical Society of America, 2nd Convention of the European Acoustics Association: Forum Acusticum 99, and 25th German Acoustics DAGA Conference, Berlin, Germany, March 1999

    Google Scholar 

  2. L. Rayleigh (J. W. Strutt), On the dynamical theory of gratings. Proc. R. Soc. Lond. A, 79, 399–416 (1907)

    Google Scholar 

  3. L. Rayleigh (J. W. Strutt), The Theory of Sound, vol. 2 (Dover, New York, 1945), pp. 89–96

    Google Scholar 

  4. R.F. Millar, Rayleigh hypothesis in scattering problems. (With author’s reply). Electron. Lett. 5, 416–417 (1969)

    Article  MathSciNet  Google Scholar 

  5. R.F. Millar, On the Rayleigh assumption in scattering by a periodic surface. Proc. Camb. Philos. Soc. 65, 773–791 (1969)

    Article  MathSciNet  Google Scholar 

  6. R.F. Millar, The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatters. Radio Sci. 8, 785–796 (1973)

    Article  MathSciNet  Google Scholar 

  7. R.F. Millar, On the Rayleigh assumption in scattering by a periodic surface. II. Math. Proc. Camb. Philos. Soc. 69, 217–225 (1971)

    Article  MathSciNet  Google Scholar 

  8. N.R. Hill, V. Celli, Limits of convergence of the Rayleigh method for surface scattering. Phys. Rev. B 17, 2478–2481 (1987)

    Article  Google Scholar 

  9. P.M. van den Berg, J.T. Fokkema, The Rayleigh hypothesis in the theory of diffraction by a perturbation in a plane surface. Radio Sci. 15, 723–732 (1980)

    Article  Google Scholar 

  10. P.M. van den Berg, J.T. Fokkema, The Rayleigh hypothesis in the theory of diffraction by a cylindrical obstacle. IEEE Trans. Antennas Propag. AP-27, 577–583 (1979)

    Article  Google Scholar 

  11. B.V. Apel’tsin, On the method of nonorthogonal series in exterior problems in the theory of steady-state oscillations. Sov. Phys. Dokl. 26, 831–833 (1982)

    Google Scholar 

  12. R.F. Millar, Singularities and the Rayleigh hypothesis for solutions to the Helmholtz equation. IMA J. Appl. Math. 37, 155–171 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  13. D. Maystre, M. Cadilhac, Singularities of the continuation of fields and validity of Rayleigh’s hypothesis. J. Math. Phys. 26, 2201–2204 (1985)

    Article  MathSciNet  Google Scholar 

  14. J.B. Keller, Singularities and Rayleigh’s hypothesis for diffraction gratings. J. Opt. Soc. Am. 17, 456–457 (2000)

    Article  Google Scholar 

  15. P. Davis, The Schwarz function and its applications, in The Carus Mathematical Monographs, vol. 17 (Mathematical Association of America, Buffalo, 1974)

    Google Scholar 

  16. T. Semenova, On the behavior of HELS solutions for acoustic radiation and reconstruction, Ph.D. dissertation, Department of Mechanical Engineering, Wayne State University, May 2004

    Google Scholar 

  17. T. Semenova, S.F. Wu, The Helmholtz equation least squares method and the Rayleigh’s hypothesis in nearfield acoustical holography. J. Acoust. Soc. Am. 115(4), 1632–1640 (2004)

    Article  Google Scholar 

  18. N. E. Rayess, Development of acoustic holography using the Helmholtz Equation-Least Squares (HELS) Method, Ph.D. dissertation, Department of Mechanical Engineering, Wayne State University, May 2001

    Google Scholar 

  19. V. Isakov, S.F. Wu, On theory and applications of the HELS method in inverse acoustics. Inverse Probl. 18, 1147–1159 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. V. Isakov, Inverse Problems for Partial Differential Equations (Springer, New York, 1990)

    Google Scholar 

  21. R. Pike, P. Sabatier (eds.), Scattering (Academic, New York, 2001), pp. 794–769

    Google Scholar 

  22. D. Colton, R. Kress, Inverse acoustic and electromagnetic scattering theory, in Applied Mathematical Sciences, vol. 93 (Springer, Berlin, 1992)

    Google Scholar 

  23. V. Isakov, Inverse Source Problems (AMS, Providence, 1990)

    Book  MATH  Google Scholar 

  24. A.N. Tikhonov, On stability of inverse problems. Dokl. Akad. Nauk SSSR 39, 176–179 (1943)

    MathSciNet  MATH  Google Scholar 

  25. J. Prager, Approximate reconstruction of sound fields close to the source surface using spherical nearfield acoustic holography. J. Acoust. Soc. Am. 122, 2067–2073 (2007)

    Article  Google Scholar 

  26. J. Prager, Approximate reconstruction of sound fields close to the source surface using spherical nearfield acoustic holography. J. Acoust. Soc. Am. 122, 2067–2073 (2007)

    Article  Google Scholar 

  27. J. Prager, Approximate reconstruction of sound fields close to the source surface using spherical nearfield acoustic holography. J. Acoust. Soc. Am. 122, 2067–2073 (2007)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Science+Business Media New York

About this chapter

Cite this chapter

Wu, S.F. (2015). Validity of the HELS Method. In: The Helmholtz Equation Least Squares Method. Modern Acoustics and Signal Processing. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1640-5_4

Download citation

  • DOI: https://doi.org/10.1007/978-1-4939-1640-5_4

  • Published:

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4939-1639-9

  • Online ISBN: 978-1-4939-1640-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics