# 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.

**Organization**

Historical Introduction, pages 1–5

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

James K. Thompson

Williamsburg, VA, USA

jktprof@outlook.com