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(1948, 1981) used Rayleigh’s “energy method” to determine the modes of a stiff string, he might have realized that his analysis gave the wrong frequencies (Garrett, 2020).
Unlike the damped simple harmonic oscillator in the Morse and the Kinsler and Frey treatments, the addi- tion of the resistive element to the mass-spring oscillator opens a two-way street for the exchange of energy with the environment (Uhlenbeck and Goudsmit, 1929). Heat is generated in the mechanical resistance (i.e., dashpot; Rm,) that escapes to the surroundings, but that path also connects the oscillator to “the environment,” which must share energy with the oscillator by virtue of the fact that the absolute (Kelvin) temperature of the environment and Rm are both nonzero.
For example, the mean potential energy of a spring with stiffness K that is in thermal equilibrium with its sur- roundings at absolute temperature T has an average mean-squared displacement of , where kB ≡ 1.380649 × 10−23 J/K (Boltzmann’s constant). The mass never comes to rest! In vibroacoustic systems, our “uncer- tainty principle” is controlled by Boltzmann’s constant, not Planck’s constant. The growing availability of microphones and accelerometers based on microelectromechanical sys- tems (MEMS) has renewed interest in the fundamental limitations imposed by thermal noise (Gabrielson, 1993).
The general lack of awareness of the coupling between fluctuations and dissipation has led some investigators to spurious conclusions in their evaluation of acousti- cal sensor performance: “...it appears that fiber sensors operating at room temperature offer detection sensi- tivities comparable to or exceeding cryogenic SQUID technology, which normally operate between 4 and 10 K” (Giallorenzi et al., 1982).
Similar errors arise resulting from the Morse/Kinsler and Frey failure to demonstrate the interrelationships of elas- tic moduli, particularly for isotropic solids, which have an elastic response that is completely specified by only two independent elastic moduli. Good evidence of the need for a new acoustics textbook is the large number of professionals, including acoustics faculty, who do not realize that a plane wave involves both hydrostatic com- pression and shear deformations: “If the propagation is truly planar, then shear stress is zero.” That statement is not correct and is followed in a recent acoustics textbook
by other justifications for sound attenuation due to “vis- cous effects [that] also arise as frictional resistance to expansion and contraction” (Ginsberg, 2019).
A separate chapter on elasticity in Understanding Acoustics also provides the opportunity to introduce vis- coelasticity and a single-relaxation-time model using a simple combination of a spring and dashpot in series placed in parallel with another spring. Such an analysis leads to the “discovery” of the Kramers-Kronig relation- ships (Kronig and Kramers, 1928) that is important for understanding attenuation due to “bulk viscosity,” which is the “resistance to expansion and contraction” through a relaxing variable in the equation of state (Landau and Lifshitz, 1959), not “viscous effects.” It also allows discus- sion of rubber springs that simultaneously provide both stiffness and damping. Rubber springs play an important
role in commercial vibration-isolation products.
Another change in the traditional sequence of topics places the theory of Helmholtz resonators before intro- duction of the wave equation, starting with a chapter that is dedicated to the ideal gas laws as a prototypical equa- tion of state. Derivation of the isothermal and adiabatic gas laws also provides the opportunity to demonstrate the complimentary functions of the microscopic theory (i.e., kinetic theory and quantum mechanics) and the phenomenological theory (i.e., thermodynamics).
The linearized continuity equation is associated with the concept of acoustical compliance (i.e., the gas spring), and the linearized Euler equation introduces the con- ceptofacousticalinertance.Thecombinationprovidesan example of the fluidic equivalent of the simple harmonic oscillator known as a Helmholtz resonator (Helmholtz, 1885). More importantly, it provides a firm understand- ing of the equation of state, the continuity equation, and the momentum conservation equation before they are linearized and combined to produce the wave equation. If masses and springs are always analyzed before the vibra- tion of strings, wouldn’t it make sense to study Helmholtz resonators before introducing one-dimensional wave propagation in a fluid?
Discussion of the dissipative processes in fluids due to irreversibility, quantified by thermal conductivity and viscosity, is another area that is overlooked in the Morse/ Kinsler and Frey treatment. The diffusion equation is just
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