Comfort for the Computationally Crippled

The difference between engineering and science, and all other human activity, is the fact that engineers and scientists make quantitative predictions about measurable outcomes and can specify their uncertainty in such predictions. Because those predictions are quantitative, they must employ mathematics. This chapter is intended as review of some of the more useful mathematical concepts, strategies, and techniques that are employed in the description of vibrational and acoustical systems and in the calculation of their behavior. Topics in this review include techniques such as Taylor series expansions, integration by parts, and logarithmic differentiation. Equilibrium and stability considerations lead to relations between potential energies and forces. The concept of linearity leads to superposition and Fourier analysis. Complex numbers and phasors are introduced along with the techniques for their algebraic manipulation. The discussion of physical units is extended to include their use for predicting functional dependencies of resonance frequencies, quality factors, propagation speeds, flow noise, and other system behaviors using similitude and the Buckingham Π-theorem to form dimensionless variables. Linearized least-squares fitting is introduced as a method for extraction of experimental parameters and their uncertainties and error propagation is presented to allow those uncertainties to be combined.

Radiation and Scattering

At this point, we have made a rather extensive investigation into the sounds that excite Helmholtz resonators as well as the departures from equilibrium that propagate as plane waves through uniform or inhomogeneous media. We have not, as yet, dealt with how those sounds are actually produced in fluids. Our experience tells us that sound can be generated by vibrating objects (e.g., loudspeaker cones, stringed musical instruments, drums, bells), by modulated or unstable flows (e.g., jet engine exhaust, whistles, fog horns, speech), by electrical discharges in the atmosphere (i.e., thunder), or by optical absorption (e.g., modulated laser beams). In this chapter, we will develop the perspective and tools that will be used for the calculation of the radiation efficiency of various sources and combinations of sources, like the sound reinforcement system shown in Fig. 12.1.

Attenuation of Sound

We will capitalize on our understanding of thermoviscous loss to develop an understanding of the attenuation of sound waves in fluids that are not influenced by proximity to solid surfaces. Such dissipation mechanisms are particularly important at very high frequencies and short distances (for ultrasound) or very low frequencies over geological distances (for infrasound). The Standard Linear Model of viscoelasticity introduced the nondimensional frequency, ωτR, that controlled the medium’s elastic (in-phase) and dissipative (quadrature) responses. Those response curves were “universal” in the sense that causality linked the elastic and dissipative responses through the Kramers-Kronig relations. That relaxation-time perspective is essential for attenuation of sound in media that can be characterized by one or more relaxation times related to those internal degrees of freedom that make their equation of state a function of frequency. Examples of these relaxation-time effects include the rate of collisions between different molecular species in a gas (e.g., nitrogen and water vapor in air), the pressure dependence of ionic association-dissociation of dissolved salts in sea water (e.g., MgSO4 and H3BO3), and evaporation-condensation effects when a fluid is oscillating about equilibrium with its vapor (e.g., fog droplets in air or gas bubbles in liquids).

Nonlinear Acoustics

A fundamental assumption of linear acoustics is that the presence of a wave does not have an influence on the properties of the medium through which it propagates. By extension, the assumption of linearity also means that a waveform is stable since any individual wave does not interact with itself. Small modifications in the sound speed due to wave-induced fluid convection (particle velocity) and to the wave’s effect on sound speed through the equation of state can lead to effects that could not be predicted within the limitations imposed by the assumption of linearity. Although a wave’s influence on the propagation speed may be small, those effects are cumulative and create distortion that can produce shocks. These are nonlinear effects because the magnitude of the nonlinearity’s influence is related to the square of an individual wave’s amplitude (self-interaction) or the product of the amplitudes of two interacting waves (intermodulation distortion). In addition, the time-average of an acoustically induced disturbance may not be zero. Sound waves can exert forces that are sufficient to levitate solid objects against gravity. The stability of such levitation forces will also be examined along with their relation to resonance frequency shifts created by the position of the levitated object.

Ideal Gas Laws

This is the first chapter to explicitly address fluid media. For springs and solids, Hooke’s law, or its generalization using stress, strain, and elastic moduli provided an equation of state. In fluids, we have an equation of state that relates changes in pressure (stresses) to changes in density (strain). The simplest fluidic equations of state are the Ideal Gas Laws. Our presentation of these laws will combine microscopic models that treat gas atoms as hard spheres with phenomenological (thermodynamic) models that combine the variables that describe the gas with conservation laws that restrict those variables. The combination of microscopic and phenomenological models will give us the important characteristics of gas behavior under isothermal or adiabatic conditions and will provide relationships between gas heat capacities and their constituent particles when augmented with elementary concepts from quantum mechanics. The chapter ends by adding a velocity field to the pressure, temperature, and density, thus providing the equations of hydrodynamics that will guide all of the subsequent development of acoustics in fluids.

Reflection, Transmission, and Refraction

The behavior of one-dimensional waves propagating through media that are not homogeneous will be the focus of this chapter. We start with an examination of the behavior of planewaves impinging on a planar interface between two fluid media with different properties and then extend that analysis to multiple interfaces and to waves that impinge on such an interface from an angle that is not perpendicular to that surface. The extent of those boundaries separating regions with different acoustical properties will be much larger than the wavelength of the sound. Many cases to be examined here will assume that the extent of the boundary is infinite and the wave incident on such an interface will be both reflected back into the medium from which it originated and be transmitted into the second medium on the other side of the interface. This exploration concludes with consideration of wave propagation through a medium whose properties change slowly and continuously through space resulting in curved ray paths. If the variation of sound speed is linear with height or depth, then the ray paths are arcs of circles. Complicated sound speed profiles will be approximated by piecewise-linear segments that have constant sound speed gradients.

Membranes, Plates, and Microphones

The restoring forces on membranes are due to the applied tension, while the restoring forces for plates are due to the flexural rigidity of the plate’s material. The transition to two dimensions introduces some features that did not show up in our analysis of one-dimensional vibrating systems. Instead of applying boundary conditions at one or two points, those constraints will have to be applied along a line or a curve. In this way, incorporation of the boundary condition is linked inexorably to the choice of coordinate systems used to describe the resultant normal mode shape functions. For two-dimensional vibrators, two indices are required to specify the frequency of a normal mode, fm,n, with the number of modes in a given frequency interval increasing in proportion to the center frequency of the interval, even though that interval remains a fixed frequency span. It is also possible that modes with different mode numbers might correspond to the same frequency of vibration, a situation that is designated as “modal degeneracy.” A membrane’s response to sound pressures provides the basis for broadband condenser microphone technology that produces signals related to the electrical properties of that capacitor and the charge stored on its plates.

Three-Dimensional Enclosures

In this chapter, solutions to the wave equation that satisfies the boundary conditions within three-dimensional enclosures of different shapes are derived. This treatment is very similar to the two-dimensional solutions for waves on a membrane of Chap. 10.1007/978-3-030-44787-8_6. Many of the concepts introduced in Sect. 10.1007/978-3-030-44787-8_6#Sec1 for rectangular membranes and Sect. 10.1007/978-3-030-44787-8_6#Sec5 for circular membranes are repeated here with only slight modifications. These concepts include separation of variables, normal modes, modal degeneracy, and density of modes, as well as adiabatic invariance and the splitting of degenerate modes by perturbations. Throughout this chapter, familiarity with the results of Chap. 10.1007/978-3-030-44787-8_6 will be assumed. The similarities between the standing-wave solutions within enclosures of different shapes are stressed. At high enough frequencies, where the individual modes overlap, statistical energy analysis will be introduced to describe the diffuse (reverberant) sound field.

One-Dimensional Propagation

Having already invested in understanding both the equation of state and the hydrodynamic equations, only straightforward algebraic manipulations will be required to derive the wave equation, justify its solutions, calculate the speed of sound in fluids, and derive the expressions for acoustic intensity and the acoustic kinetic and potential energy densities of sound waves. The “machinery” developed to describe waves on strings will be sufficient to describe one-dimensional sound propagation in fluids, even though the waves on the string were transverse and the one-dimensional waves in fluids are longitudinal. These results are combined with the thermal and viscous penetration depths to calculate the frequencies and quality factors in standing wave resonators. The coupling of those resonators to loudspeakers will be examined. The introduction of reciprocal transducers that are linear, passive, and reversible will allow absolute calibration of transducers using only electrical measurements (i.e., currents and voltages) by the reciprocity method, if the acoustic impedance that couples the source and receiver is calculable. Reflection and transmission at junctions between multiple ducts and other networks will be calculated and applied to the design of filters. The behavior of waves propagating through horns will provide useful impedance matching but introduce a low-frequency cut-off.

String Theory

The vibrating string has been employed by nearly every human culture to create musical instruments. Although the musical application has attracted the attention of mathematical and scientific analysts since the time of Pythagoras (570 BC–495 BC), we will study the string primarily because its vibrations are easy to visualize and string vibrations introduce concepts and techniques that will recur throughout our study of the vibration and the acoustics of continua. In this chapter, we will develop continuous mathematical functions of position and time that describe the shape of the entire string. The amplitude of such functions will describe the transverse displacement from equilibrium, y(x, t), at all positions along the string. The importance of boundary conditions at the ends of strings will be emphasized, and techniques to accommodate both ideal and “imperfect” boundary conditions will be introduced. Solutions that result in all parts of the string oscillating at the same frequency which satisfy the boundary conditions are called normal modes, and the calculation of those normal mode frequencies will be a focus of this chapter.