Page 28 - Fall2021

A HISTORY OF ACOUSTICS TEXTBOOKS
as easy to solve using a harmonic substitution as is the wave equation. Just as the wave equation introduces the wavelength as a “scale length” for propagation, the Fou- rier diffusion equation produces the thermal penetration depth and the Navier-Stokes equation introduces the viscous penetration depth as their relevant scale lengths (Landau and Lifshitz, 1959).
Failure to take these effects into account led Kinsler and Frey to calculate the quality factor of a Helmholtz reso- nator based only on radiation losses (Kinsler and Frey, 1962). In a typical Helmholtz resonator, viscous dissipa- tion in the neck and thermal relaxation at the surface of the volume overwhelm the losses due to radiation, which comes in a distant third in its contribution to the reduc- tion of the quality factor. Having been on the physics faculty at the NPS from 1982 to 1995, I was able to con- vince Coppens and Sanders to correct that error in the fourth edition.
Fundamental Defenses Against Erroneous Results
“Thermodynamics is the true testing ground of physi- cal theory because its results are model independent. It is the only physical theory of universal content which I am convinced will never be overthrown, within the framework of applicability of its basic con- cepts” (Einstein, 1979).
What do all of the discussions in this article have to do with the statement that was placed at its start: “A computer can provide the wrong answer with 7-digit precision a thousand times each second” (Garrett, 2020)? In this era where “Virtually every engineering problem is [now] solved with an ‘Enter’ key,” it is more important than ever to have fundamental principles that are “model independent,” thus not depending on any specific algo- rithm, to provide a check on computer-generated results. To paraphrase the comedian P. J. O’Rourke, “without those principles, giving a student access to a computer is like giving a teenager a bottle of whisky and the keys to a Ferrari.”
For example, the Kramer-Kronig relationships restrict the real (i.e., in-phase) and imaginary (i.e., quadrature) components of any “susceptibility” that links stimulus to response in a linear-response theory. That result is dependent only on causality; an effect cannot precede
its cause. Although Kramers and Kronig applied their discovery to the absorption and dispersion of X-ray spec- tra (Kronig and Kramers, 1928), it applies equally to the elastic moduli and loss tangents of an elastomer, the radi- ation resistance and hydrodynamic mass of a vibrating piston, and the sound speed and attenuation in a relax- ing medium, among many other systems of interest to acousticians. In the Rudnick group, I was first introduced to its consequences when trying to understand the maxi- mum attenuation per wavelength in a porous waveguide filled with superfluid helium. That maximum was related only to the speed of propagation in the limit where the flow resistance of the porous medium was zero (i.e., first sound) or infinite (i.e., fourth sound). The energy differ- ence between X-rays (~10 keV) and a quantum liquid close to absolute zero temperature (~100 μeV) is gargan- tuan (Tarantino, 2004).
Similitude depends only on the units that are used to express parameters and variables in a model. Adiabatic invariance is a consequence of any change to a vibrating system’s constraints that are made slowly enough that the normal mode shape remains unchanged (Strutt, 1902). It guarantees that the ratio of the energy to the frequency remains constant (Landau and Lifshitz, 1960). Adiabatic invariance applies to the transformation of a mode that is a solution in an enclosure with boundaries that can be expressed in one of the 11 coordinates in which the wave equation is separable (Eisenhart, 1934), to a shape like that of the Space Shuttle’s cargo bay, having a cross section described by a hemiellipse on top of a truncated irregular octagon. It can also be used to relate the frequency shift in a resonator, due to the position of an object, to the radia- tion force on that object (Putterman et al., 1989).
Conclusions
The Morse and the Kinsler and Frey textbooks have launched the careers of many of us who now use acous- tics in our careers. I used it as the primary textbook in the introductory acoustics courses I taught at the NPS (1982 to 1995) and in the Graduate Program in Acoustics at Penn State (1995 to 2010). I would tell my students that
“Kinsler and Frey is the Listerine of acoustics; nobody likes the taste, but they use it twice each day.” It contained most of the necessary results, but few of the reasons. On occasion, I would refer to it as the “satanic verses” (Rush- die, 1988), for example, when it incorrectly calculated the quality factor of a Helmholtz resonator, a result that
28 Acoustics Today • Fall 2021