The Foundations of Acoustics: Basic Mathematics and Basic Acoustics

The Foundations of Acoustics: Basic Mathematics and Basic Acoustics
Eugen Skudrzyk
Springer-Verlag, NY, (1971)
790 pp., softbound, 99 USD
ISBN: 978-3-7091-8257-4

Overview and Purpose

This book is not a textbook in the sense that it is not designed to take the student from the basics of acoustics or acoustical mathematics to more advanced topics. It is more a collection of chapters on topics in acoustics and mathematics related to acoustics. There are no student problems included. The materials do proceed in a logical manner, building from basic principles to more advanced topics. However, it is organized in a manner that would be conducive to a student learning acoustics. I would classify this as a reference book on acoustics and on the mathematics related to acoustics. There is a heavy emphasis on the mathematics, but the author does a good job relating the applications in acoustics. If I were a graduate level student in acoustics or a researcher in the field, this book would be very valuable. I only wish I had had it when I was a student.

Chapters 1–8 (see the list of chapters below) provide a complete mathematical background for someone studying or working in acoustics. A thorough understanding of these chapters would provide an excellent foundation for anyone working in acoustics. The author builds from basic nomenclature to complex analyses through these chapters. With examples using electrical circuits, point mass systems, and different input functions, it is clear how the material is related to the field of acoustics. However, there are no problem exercises or practical examples provided. The use of the mathematics described is not shown in “real world” examples.

Chapters 10–12 provide an excellent foundation for signal analysis in acoustics. Including sampling theory and basic signal processing concepts, these chapters provide a good foundation of the mathematics and principles of signal analysis.

Chapters 13–28 are specific to defining the mathematics of sound, sound radiation, sources, diffraction, reflection, and other acoustic phenomena. The equations governing the sound radiation from shells, pistons, and geometries are treated in detail. The mathematics for reflections from various surface definitions are well described. Many complex radiation and diffraction problems are also defined.

This book provides a comprehensive treatment of a number of topics in acoustics and the mathematics of acoustics. It is well organized and clearly written. It clearly leans heavily to the mathematics and does not delve into practical applications and certainly not noise control. As noted above, chapters 1–8 provide an excellent foundation in the mathematics related to acoustics. Chapters 10–12 are an excellent start in signal processing. The remaining chapters, 13–28, provide detailed mathematical explanations of sound, sources, radiation, and diffraction. Where the topics align with the reader’s interests they can be very useful. However, these chapters are not practical instructions in the application of acoustics or noise control. In summary, I would say that this is an excellent foundation for the mathematics of acoustics and a great reference for the mathematics applicable to particular problems in acoustics.


Historical Introduction, pages 1–5

  1. Equations and Units, pages 6–16
  2. Complex Notation and Symbolic Methods, pages 17–32
  3. Analytic Functions: Their Integration and the Delta Function, pages 33–77
  4. Fourier Analysis, pages 78–94
  5. Advanced Fourier Analysis, pages 95–122
  6. The Laplace Transform, pages 123–130
  7. Integral Transforms and the Fourier Bessel Series, pages 131–136
  8. Correlation Analysis, pages 137–148
  9. Wiener’s Generalized Harmonic Analysis, pages 149–151
  10. Transmission Factor, Filters, and Transients (Küpfmüller’s Theory), pages 152–200
  11. Probability Theory, Statistics, and Noise, pages 201–235
  12. Signals and Signal Processing, pages 236–269
  13. Sound, pages 270–283
  14. The One-Dimensional Wave Equation and Its Solutions, pages 284–294
  15. Reflection and Transmission of Plane Waves at Normal Incidence, pages 295–312
  16. Plane Waves in Three Dimensions, pages 313–325
  17. Sound Propagation in Ideal Channels and Tubes, pages 326–343
  18. Spherical Waves, Sources, and Multipoles, pages 344–377
  19. Solution of the Wave Equation in General Spherical Coordinates, pages 378–391
  20. Problems of Practical Interest in General Spherical Coordinates, pages 392–422
  21. The Wave Equation in Cylindrical Coordinates and Its Applications, pages 423–454
  22. The Wave Equation in Spheroidal Coordinates and Its Solutions, pages 455–488
  23. The Helmholtz Huygens Integral, pages 489–511
  24. Huygens Principle and the Rubinowicz–Kirchhoff Theory of Diffraction, pages 512–556
  25. The Sommerfeld Theory of Diffraction, pages 557–592
  26. Sound Radiation of Arrays and Membranes, pages 593–640
  27. The Green’s Functions of the Helmholtz Equation and Their Applications, pages 641–662
  28. Self and Mutual Radiation Impedance, pages 663–676

James K. Thompson
Williamsburg, VA, USA