Skip to main content
SpringerLink
Log in
Book cover

Ocean Acoustics pp 79–157Cite as

Numerical Models of Underwater Acoustic Propagation

  • Chapter

Part of the book series: Topics in Current Physics ((TCPHY,volume 8))

Abstract

In this chapter an attempt is made to summarize those models of propagation loss in the field of underwater acoustics which have been converted into an automated computer code capable of being executed by someone other than the originator for a wide variety of problems. No single model currently exists which is adequate for all applications. This is perhaps not surprising considering the diversity of the ocean environment and its boundaries, and the concomitant fact that the acoustic frequencies of interest span the regime from less than 10 Hz to greater than 100 kHz. As a result a large number of models, each with its own domain of validity which in many cases is difficult to precisely define, have been developed. Their sheer number makes an exhaustive summary impossible within these limited pages. Thus it was decided to limit consideration to those models which purport to be a solution of the wave equation found in Sect.2.2.1. Fundamentally these models consider the ocean to be a deterministic environment for which the speed of sound is only a function of the spatial coordinates. Non-deterministic effects, if accounted for at all, are included in an ad hoc fashion following the determination of the deterministic propagation loss result. Model development work for the more general problem is required and is in progress. However, this effort has not reached the point where “hands off” computer codes are available. This is due in part to a lack of available experimental/ environmental data and the need for larger and faster computers.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. F.R. DiNapoli, R.L. Deavenport: “Computer Models of Underwater Acoustic Propagation”; Tech. Rpt. 5867 ( Naval Underwater Systems Center, New London, Conn. 1978 )

    Google Scholar 

  2. H.F. Baker: Proc. London Math. Soc. (Ser. l) 35, 333 (1902)

    Google Scholar 

  3. R.A. Frazer, W.J. Duncan, A.R. Collar: Elementary Matrices ( Cambridge University Press, London 1960 )

    Google Scholar 

  4. F.R. Gantmacher: The Theory of Matrices, Vol. 2 ( Chelsea Publishing, New York 1959 )

    MATH  Google Scholar 

  5. F. Gilbert, G.E. Backus: Geophys. 31, 326 (1966)

    Article  Google Scholar 

  6. L.B. Felsen, N. Marcuvitz: Radiation and Scattering of Waves ( Prentice-Hall, Englewood Cliffs, NJ 1973 )

    Google Scholar 

  7. E.A. Coddington, N. Levinson: Theory of Ordinary differential Equations ( McGraw- Hill, New York 1955 )

    MATH  Google Scholar 

  8. H.W. Marsh, S.R. Elam: Internal Document, Raytheon Company, Marine Research Laboratory, New London, Conn. (1967)

    Google Scholar 

  9. F.R. DiNapoli: “A Fast Field Program for Multilayered Media”; Tech. Rpt. 4103 ( Naval Underwater Systems Center, New London, Conn. 1971 )

    Google Scholar 

  10. F.R. DiNapoli, M.R. Powers: “Recursive Calculation of Products of Cylindrical Functions”, Tech. Memo. PA-83-70 (Naval Underwater Systems Center, New London, Conn. 1970 )

    Google Scholar 

  11. F.R. DiNapoli: “The Collapsed Fast Field Program (FFP)”; Tech. Memo. TA11-317-72 (Naval Underwater Systems Center, New London, Conn. 1972 )

    Google Scholar 

  12. D.C. Stickler: J. Acoust. Soc. Am. 57, 856 (1975)

    Article  ADS  Google Scholar 

  13. W.M. Ewing, W.S. Jardetzky, F. Press: Elastic Waves in Layered Media ( McGraw-Hill, New York 1957 )

    MATH  Google Scholar 

  14. C.L. Bartberger:;AP2 Normal Mode Program, Report, Naval Air Development Center, Warminster, Pa. (1978)

    Google Scholar 

  15. C.L. Pekeris: Geol. Soc. Am. Mem. 27 (1948)

    Google Scholar 

  16. I. Tolstoy: J. Acoust. Soc. Am. 28, 1182 (1956)

    Article  ADS  Google Scholar 

  17. S.R. Santaniello, F.R. DiNapoli, R.K. Dullea, P. Herstein: “A Synopsis of Studies on the Interaction of Low Frequency Acoustic Signals with the Ocean Bottom”; Tech. Doc. 5337, Naval Underwater Systems Center, New London, Conn. (1976)

    Google Scholar 

  18. P. Debye: Phys. Z. 9, 775 (1908)

    Google Scholar 

  19. B. Van den Pol, H. Bremmer: Phil. Mag. 24, 141, 825 (1937)

    MATH  Google Scholar 

  20. D.V. Batorsky, L.B. Felsen: Radio Sci. 6, 911 (1971)

    Article  ADS  Google Scholar 

  21. H. Weinberg: J. Acoust. Soc. Am. 58, 97 (1975)

    Article  ADS  Google Scholar 

  22. C.W. Spofford: “The FACT Model”; Tech. Rpt. 109, Maury Center for Ocean Science, Washington D.C. (1974)

    Google Scholar 

  23. G.A. Leibiger: “The Acoustic Propagation Model RAYMODE: Theory and Numerical Treatment”; NUSC Tech. Rpt., Naval Underwater Systems Center, New London, Conn. (1978)

    Google Scholar 

  24. M.A. Pedersen, D.F. Gordon: J. Acoust. Soc. Am. 51, 323 (1972)

    Article  MATH  ADS  Google Scholar 

  25. C. Bartberger: “Normal Mode Solutions and Computer Programs for Underwater Sound Propagation”; Tech. Rpt. NADC-72001-AE, Naval Air Development Center, Warminster, Pa. (1973)

    Google Scholar 

  26. E.C. Titchmarsh: Introduction to the Theory of Fourier Integrals, 2nd ed. ( Oxford University Press, Oxford 1948 )

    Google Scholar 

  27. H. Bremmer: Terrestrial Radio Waves ( Elsevier Publishing, Amsterdam 1949 )

    Google Scholar 

  28. H.M. Nussenzveig: Ann. Phys. 34, 23 (1965)

    Article  ADS  MathSciNet  Google Scholar 

  29. F. Gilbert: Geophys. J. R. Astr. Soc. 44, 275 (1976)

    MATH  Google Scholar 

  30. A. Ben-Menahem: Bull. Seis. Soc. Am. 54, 1315 (1964)

    Google Scholar 

  31. C.L. Pekeris: Proa. Symp. Appl. Math., Vol.2 (American Mathematical Society, New York 1950) pp.71–75

    Google Scholar 

  32. D.S. Ahluwalia, J.B. Keller: “Exact and Asymptotic Representations of the Sound Field in a Stratified Ocean”, in Wave Propagation and Underwater Acoustics, ed. by J.B. Keller, J.S. Papadakis, Lecture Notes in Physics, Vol. 70 ( Springer, Berlin, Heidelberg, New York 1977 )

    Google Scholar 

  33. N.A. Haskell: J. Appl. Phys. 157 (1951)

    Google Scholar 

  34. L.M. Brekhovskikh: Sov. Phys. Acoust. 2, 124 (1956)

    Google Scholar 

  35. R.B. Lauer, B. Sussman: “A Methodology for the Comparison of Models for Sonar System Applications”; Vol.1, Naval Sea Systems Command Tech. Rpt. SEA 06H1/036- EVA/M0ST-10, Naval Underwater Systems Center, New London, Conn. (1976)

    Google Scholar 

  36. R.B. Lauer, B. Sussman: “A Methodology for the Comparison of Models for Sonar System Applications—Results for Low Frequency Propagation Loss in the Mediterranean Sea”; Vol.11, Naval Sea Systems Command Tech. Rpt. SEA 06H1/036- EVA/M0ST-11, Naval Underwater Systems Center, New London, Conn. (1978)

    Google Scholar 

  37. D.F. Yarger: “The User’s Guide for the Raymode Propagation Loss Program”; Tech. Memo. 222-10-76, Naval Underwater Systems Center, New London, Conn. (1976)

    Google Scholar 

  38. C.L. Bartberger: “PLRAY, A Ray Propagation Loss Program”; Tech. Rpt. Naval Air Development Center, Warminster, Pa. (1978)

    Google Scholar 

  39. D.F. Gordon: “Normal Mode Computation of Propagation Loss for an Arbitrary Number of Layers”; Tech. Pub. 236, Naval Undersea Center, San Diego, Calif. (1971)

    Google Scholar 

  40. D.W. Hoffman: “LORA, A Model for Predicting the Performance of Long-Range Active Sonar Systems”; Tech. Pub. 541, Naval Undersea Center, San Diego, Calif. (1976)

    Google Scholar 

  41. W.H. Watson, R. McGirr: “Raywave-II, A Propagation Loss Model for the Analysis of Complex Ocean Environments”; Tech. Note 1516, Naval Undersea Center, San Diego, Calif. (1975)

    Google Scholar 

  42. E.B. Wright: “Acoustic Transmission Loss by Single-Profile Ray Tracing, Program RTRACE”; Tech. Rpt. 7815, Naval Research Laboratory, Washington D.C. (1974)

    Google Scholar 

  43. I.M. Blatstein: “Comparisons of Normal Mode Theory, Ray Theory, and Modified Ray Theory for Arbitrary Sound Velocity Profiles Resulting in Convergence Zones”; Tech. Rpt. 74–95, Naval Ordnance Laboratory, White Oak, Silver Spring, Md. (1974)

    Google Scholar 

  44. H. Weinberg: J. Acoust. Soc. Am. 50, 975 (1971)

    Article  ADS  Google Scholar 

  45. R.P. Porter, H.D. Leslie: J. Acoust. Soc. Am. 58, 812 (1975)

    Article  ADS  Google Scholar 

  46. F.R. DiNapoli: “The Inverse Fast Field Program (IFFP): An Application to the Determination of the Acoustic Parameters of the Ocean Bottom”; Tech. Memo. 771160, Naval Underwater Systems Center, New London, Conn. (1977)

    Google Scholar 

  47. H.W. Kutschale, F.D. Tappert: J. Acoust. Soc. Am. 62, S518 (1977)

    Article  Google Scholar 

  48. H.W. Kutschale, F.R. DiNapoli: J. Acoust. Soc. Am. S518 (1977)

    Google Scholar 

  49. F.D. Tappert: “The Parabolic Approximation Method”, in Wave Propagation and Underwater Acoustics, ed. by J.B. Keller, J.S. Papadakis, Lecture Notes in Physics, Vol. 70 ( Springer, Berlin, Heidelberg, New York 1977 )

    Google Scholar 

  50. R.H. Hardin, F.D. Tappert: SIAM Rev. 15, 423 (1973)

    Google Scholar 

  51. F.D. Tappert, R.H. Hardin: Proc. 8 Intern. Cong, on Acoustics, Goldcrest, London, Vol. 2, 452 (1974)

    Google Scholar 

  52. R.M. Wilcox: J. Math. Phys. 8, 962 (1967)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  53. F. Jensen, H. Krol: “The Use of the Parabolic Equation Method in Sound Propagation Modeling”; Tech. Rpt. Sm 72, NATO SACLANTCEN, La Spezia, Italy (1975)

    Google Scholar 

  54. J.S. Papadakis, D. Wood: “A Parabolic Decomposition of Helmholtz Equation”; Tech. Rpt., Naval Underwater Systems Center, New London, Conn. (1978)

    Google Scholar 

  55. M.C. Smith: J. Acoust, Soc. Am. 46, 233 (1969)

    Article  ADS  Google Scholar 

  56. S.T. McDaniel: J. Acoust. Soc. Am. 58, 1178 (1975)

    Article  ADS  Google Scholar 

  57. D. Lee, J.S. Papadakis: “Numerical Solutions of Underwater Acoustic Wave Propagation Problems”; NUSC Tech. Rpt. 5929, Naval Underwater Systems Center, New London, Conn. (1978)

    Google Scholar 

  58. D. Lee: “Nonlinear Multistep Methods for Solving Initial Value Problems in Ordinary Differential Equations”; Ph.D. dissertation, Polytechnic Institute of New York, N.Y. (1977)

    Google Scholar 

  59. P. Henrici: Discrete Variable Methods in Ordinary Differential Equations ( Wiley and Sons, New York 1962 )

    MATH  Google Scholar 

  60. W.G. Kanabis: “A Shallow Water Acoustic Model for an Ocean Stratified in Range and Depth”; NUSC Tech. Rpt. 4887–1, Naval Underwater Systems Center, New London, Conn. (1975)

    Google Scholar 

  61. W.G. Kanabis: “Computer Programs to Calculate Normal Mode Propagation and Applications to Analysis of Explosive Sound Data in the BIFI Range”; NUSC Tech. Rpt. 4319, Naval Underwater Systems Center, New London, Conn. (1972)

    Google Scholar 

  62. C.W. Spofford, H.M. Garon: “Deterministic Methods of Sound-Field Computation”; in Proc. NATO Conf. on Oceanic Acoustic Modeling„ ed. by W. Bachmann, R.B. Williams (SACLANTCEN, La Spezia, Italy, 1975) pp.40–1 to 43

    Google Scholar 

  63. J.J. Cornyn: “Grass, A Digital-Computer Ray-Tracing and Transmission-Loss-Prediction System”; NRL Rpt. 7621, Vol. 1, Naval Research Laboratory, Washington D.C. (1973)

    Google Scholar 

  64. H.P. Bucker: “The RAVE (Ray Wave) Method”; in Proc. NATO Conf. on Geometrical Acoustics, ed. by B.W. Conolly, R.H. Clark ( SACLANTCEN, La Spezia, Italy 1971 ) pp. 32–36

    Google Scholar 

  65. A.J. Kalinowski: The Shock and Vibration Digest, 11, 12 (March 1979)

    ADS  Google Scholar 

  66. M.J. Turner, R.W. Clough, H.C. Martin, L.J. Topp: J. Aeronaut. Sci. 23, 805 (1956)

    MATH  Google Scholar 

  67. J.T. Oden: Finite Elements of Nonlznear Continuum ( McGraw-Hill, New York 1972 )

    Google Scholar 

  68. O.C. Zienkiewicz, Y.K. Cheung: “Finite Elements in the Solution of Field Problems”, The Engineer 220, 507 (1965)

    Google Scholar 

  69. R.H. Gallagher, J.T. Oden, C. Taylor, O.C. Zienkiewicz: Finite Elements in Fluids—Vol.1, Viscous Flow and Hydrodynamics ( Wiley and Sons, New York 1975 )

    MATH  Google Scholar 

  70. R.H. Gallagher, J.T. Oden, C. Taylor, O.C. Zienkiewicz: Finite Elements in Fluids, Vol.2, Mathematical Foundations3 Aerodynamics and Lubrication ( Wiley and Sons, New York 1975 )

    Google Scholar 

  71. J.T. Oden, O.C. Zienkiewicz, R.H. Gallagher, C. Taylor: Finite Element Methods in Flow Problems (University of Alabama Huntsville Press, Huntsville, Ala. 1974 )

    Google Scholar 

  72. O.C. Zienkiewicz, Y.K. Cheung: The Finite Element Method in Structural and Continuum Mechanics ( McGraw-Hill, New York 1967 )

    MATH  Google Scholar 

  73. L.J. Segerland: Applied Finite Element Analysis ( Wiley and Sons, New York 1976 )

    Google Scholar 

  74. K.H. Huebner: The Finite Element Method for Engineers ( Wiley and Sons, New York 1975 )

    Google Scholar 

  75. R.D. Cook: Concepts and Applications of Finite Element Analysis ( Wiley and Sons, New York 1974 )

    Google Scholar 

  76. J.T. Oden, J.N. Reddy: An Introduction to the Mathematical Theory of Finite Elements ( Wiley and Sons, New York 1976 )

    MATH  Google Scholar 

  77. G. Strang, G. Fix: An Analysis of the Finite Element Method (Prentice-Hall, Englewood Cliffs, N.J. 1973 )

    MATH  Google Scholar 

  78. D. Norrie, Gerard de Vries: Finite Element Bibliography ( IFI/Plenum Data Company, New York 1976 )

    MATH  Google Scholar 

  79. A.J. Kalinowski: “Fluid-Structure Interaction”, in Shock and Vibration Computer Programs, Review and Summaries, ed. by Pi 1 key and Pi 1 key, SVM-10, The Shock and Vibration Information Center, Washington D.C. (1975)

    Google Scholar 

  80. L.H. Chen, V.M. Pierucci: “Underwater Fluid-Structure Interaction”, Parts I and II, The Shock and Vibration Digest (April issue, pp.23–24; May, pp. 17–21, 1977 )

    Google Scholar 

  81. R.L. Kuhlemeyer, J. Lysmer: J. Soil Mech. Found. Div., ASCE, 99, 421 (1973)

    Google Scholar 

  82. J. Lysmer, R.L. Kuhlemeyer: J. Engr. Mech. Div., ASCE, 95, 859 (1969)

    Google Scholar 

  83. J. Lysmer, L. Drake: “A Finite Element Method for Seismology”, Methods in Computational Physics, ed. by B. Bolt ( Academic Press, New York 1972 )

    Google Scholar 

  84. J.A. Gutierrez: “A Substructure Method for Earthquake Analysis of Structure-Soil Interaction”, Earthquake Enqineering Research Center, Rpt. No. 76–9 (April 1976)

    Google Scholar 

  85. J. Lysmer: Bull. Scism. Soc. Am. 60, 89 (1970)

    Google Scholar 

  86. J. Lysmer, G. Waas: J. Eng. Mech. Div., ASCE, 98, 85 (1972)

    Google Scholar 

  87. E. Kausel, J. Roesset, G. Waas: J. Eng. Mech. Div., ASCE, 101, 679 (1975)

    Google Scholar 

  88. E. Kausel, J. Roesset: J. Eng. Mech. Div., ASCE, 103, 569 (1977)

    Google Scholar 

  89. P. Chakrabarti, A.K. Chopru: Earthquake Engineering and Structural Dynamics 2, 107 (1973)

    Google Scholar 

  90. G.J. Fix, S.P. Marin: “Variational Methods for Underwater Acoustics Problems”, ICASE Rpt. 77–16 (August 1977)

    Google Scholar 

  91. A. Carlson, A.J. Kalinowski, J. Patel: “Solutions of a General Class of Fluid Structure Problems by the Finite Element Method”, Naval Underwater Systems Center TR-5361 (Conf.) (June 1976)

    Google Scholar 

  92. “The NASTRAN User’s Manual”, NASA SP-222(03) (March 1976)

    Google Scholar 

  93. O.C. Zienkiewicz, D.W. Kelly, P. Bettess: International Journal of Numerical Methods in Engineering, 11, 355 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  94. C.D. Mote, Jr.: Int. J. Numerical Methods in Engineering, 3, 565 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  95. E.A. Rukos: International Journal for Numerical Methods in Engineering, 12, 11 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  96. M. Petyt, J. Lea, Koopman: Journal of Sound and Vibration, 45, 495 (1976)

    Article  ADS  Google Scholar 

  97. G.J. Fix, M.H. Gunzburger: “On the Use of Modern Numerical Methods in Acoustics”, ICASE Rpt., no date

    Google Scholar 

  98. A.J. Kalinowski: Shock and Vibration Bulletin, 48, 62 (1977)

    Google Scholar 

  99. A.J. Kalinowski: “Application of the Finite Element Method to Acoustic Propagation in the Ocean”; NUSC Tech. Rpt. S891, Naval Underwater Systems Center, New London, Conn. (1979)

    Google Scholar 

  100. R. Dungar, P.J.L. Eledred: Earthquake Eng. Struct. Dynamics 6, 123–138 (1978)

    Article  Google Scholar 

  101. J.M. Rosset, M.M. Ettouney: International Journal for Numerical and Analytical Methods in Geomechanics, 1, 151 (1977)

    Article  ADS  Google Scholar 

  102. R. Kuhlemeyer: “Vertical Vibrations of Footings Embedded in Layered Media”, Ph.D. Thesis, University of Calif., Berkeley (1969)

    Google Scholar 

  103. O.C. Zienkiewicz, R.E. Newton: “Coupled Vibrations of a Structure Submerged in a Compressible Fluid”, Proc. Int. Symp. on Finite Element Techniques (Stuttgart, F.R. Germany 1969 )

    Google Scholar 

  104. P. Bettess, O.C. Zienkiewicz: International Journal for Numerical Methods in Engineering, 11, 1271 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  105. P. Bettess: International Journal for Numerical Methods in Engineering, 11, 53 (1977)

    Article  MATH  ADS  Google Scholar 

  106. O.C. Zienkiewicz, P. Bettess: 2nd Intern. Symp. Computing Meth. Appl. Science and Engng. ( Versailles, France 1975 )

    Google Scholar 

  107. B. Engquist, A. Majda: Mathematical of Computation, 31, 629 (1977)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  108. J.T. Hunt, M. Knittel, C.S. Nichols, D. Barach: J. Acoust. Soc. Am. 57, 287 (1975)

    Article  MATH  ADS  Google Scholar 

  109. J.T. Hunt, M.P. Knittel, D. Barach: J. Acoust. Soc. Am. 55, 269 (1974)

    Article  ADS  Google Scholar 

  110. H.A. Schenck: J. Acoust. Soc. of Am. 44, 41 (1967)

    Article  ADS  Google Scholar 

  111. O.C. Zienkiewicz: The Finite Element Method in Engineering Science ( McGraw-Hill, London 1971 )

    MATH  Google Scholar 

  112. G.C. Everstine, E.M. Schroeder, M.S. Marcus: “The Dynamic Analysis of Submerged Structures”, 4th NASTRAN User’s Colloquium, Langley Research Center, Hampton, Virginia (1975)

    Google Scholar 

  113. D. Shantaram, D.R.J. Owen, O.C. Zienkiewicz: Earthquake Engineering and Structural Dynamics, 4, 561 (1976)

    Google Scholar 

  114. E.P. Sorensen, D.V. Marcal: “A Solid Mechanics Approach to the Solution of Fluid-Solid Vibration Problems by Finite Elements”, Brown University Tech. Rpt. N00014-0007/13 (May 1976)

    Google Scholar 

  115. L. Kiefling, G.C. Feng: AIAA J. 14, 199 (1976)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  116. G.J. Fix, M.D. Gunzburger, R.A. Nicolaides: “On Mixed Finite Element Methods The Kelvin Principle”, ICASE Rpt. ( Dec 1977 )

    Google Scholar 

  117. G.J. Fix, M.D. Gunzburger, R.A. Nicolaides: “On Mixed Finite Element Methods The Least Squares Method”, ICASE Rpt. 77–18 (Dec 1977)

    Google Scholar 

  118. A.J. Kalinowski: “Transmission of Shock Waves into Submerged Fluid Filled Vessels”, in Fluid Structure Interaction Phenomena in Pressure Vessel and Piping Systems, ed. by M. Wang, S.J. Brown (ASME, New York 1977 ) pp. 83–105

    Google Scholar 

Download references

Authors
  1. F. R. DiNapoli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. L. Deavenport
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Naval Resarch Laboratory, 20375, Washington, DC, USA

    John A. DeSanto

Rights and permissions

Reprints and permissions

Copyright information

© 1979 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

DiNapoli, F.R., Deavenport, R.L. (1979). Numerical Models of Underwater Acoustic Propagation. In: DeSanto, J.A. (eds) Ocean Acoustics. Topics in Current Physics, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-81294-1_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-81294-1_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-81296-5

  • Online ISBN: 978-3-642-81294-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation