J. Walker), 61–114. London: E & FN Spon.13 13 Kelly, S.G. (2000). Fundamentals of Mechanical Vibrations, 2e. McGraw‐Hill.
14 14 Leissa, A.W. (1993). Vibration of Plates. New York: Acoustical Society of America.
15 15 Berry, A., Guyader, J.L., and Nicolas, J. (1990). A general formulation for the sound radiation from rectangular, baffled plates with arbitrary boundary conditions. J. Acoust. Soc. Am. 88: 2792–2802.
16 16 Arenas, J.P. (2001) Analysis of the acoustic radiation resistance matrix and its applications to vibro‐acoustic problems. PhD thesis. Auburn University.
17 17 Blevins, R.D. (2015). Formulas for Dynamics, Acoustics and Vibration. New York: Wiley.
18 18 Malecki, I. (1969). Physical Foundations of Physical Acoustics. Oxford: Pergamon Press.
19 19 Arenas, J.P. (2003). On the vibration analysis of rectangular clamped plates using the virtual work principle. J. Sound Vib. 266: 912–918.
20 20 Tao, J.S., Liu, G.R., and Lam, K.Y. (2001). Sound radiation of a thin infinite plate in light and heavy fluids subject to multi‐point excitation. Appl. Acoust. 62: 573–587.
3 Sound Generation and Propagation
3.1 Introduction
The fluid mechanics equations, from which the acoustics equations and results may be derived, are quite complicated. However, because most acoustical phenomena involve very small perturbations from steady‐state conditions, it is possible to make significant simplifications to these fluid equations and to linearize them. The results are the equations of linear acoustics. The most important equation, the wave equation, is presented in this chapter together with some of its solutions. Such solutions give the sound pressure explicitly as functions of time and space, and the general approach may be termed the wave acoustics approach. This chapter presents some of the useful results of this approach but also briefly discusses some of the other alternative approaches, sometimes termed ray acoustics and energy acoustics, that are used when the wave acoustics approach becomes too complicated. The main purpose of this chapter is to present some of the most important acoustics formulas and definitions, without derivation, which are used in the other chapters of this book.
3.2 Wave Motion
Some of the basic concepts of acoustics and sound wave propagation used throughout the rest of this book are discussed here. For further discussion of some of these basic concepts and/or a more advanced mathematical treatment of some of them, the reader is referred to the Handbook of Acoustics [1] and other texts [2–18] which are also useful for further discussion on fundamentals and applications of the theory of noise and vibration problems.
Wave motion is easily observed in the waves on stretched strings and as ripples on the surface of water. Waves on strings and surface water waves are very similar to sound waves in air (which we cannot see), but there are some differences that are useful to discuss. If we throw a stone into a calm lake, we observe that the water waves (ripples) travel out from the point where the stone enters the water. The ripples spread out circularly from the source at the wave speed, which is independent of the wave height.
Somewhat like the water ripples, sound waves in air travel at a constant speed, which is proportional to the square root of the absolute temperature and is almost independent of the sound wave strength. The wave speed is known as the speed of sound. Sound waves in air propagate by transferring momentum and energy between air particles. Sound wave motion in air is a disturbance that is imposed onto the random motion of the air molecules (known as Brownian motion). The mean speed of the molecular random motion and rate of molecular interaction increases with the absolute temperature of the gas. Since the momentum and sound energy transfer occurs through the molecular interaction, the sound wave speed is dependent solely upon the absolute temperature of the gas and not upon the strength of the sound wave disturbance. There is no net flow of air away from a source of sound, just as there is no net flow of water away from the source of water waves. Of course, unlike the waves on the surface of a lake, which are circular or two‐dimensional, sound waves in air in general are spherical or three‐dimensional.
As water waves move away from a source, their curvature decreases, and the wavefronts may be regarded almost as straight lines. Such waves are observed in practice as breakers on the seashore. A similar situation occurs with sound waves in the atmosphere. At large distances from a source of sound, the spherical wavefront curvature decreases, and the wavefronts may be regarded almost as plane surfaces.
Plane sound waves may be defined as waves that have the same acoustical properties at any position on a plane surface drawn perpendicular to the direction of propagation of the wave. Such plane sound waves can exist and propagate along a long straight tube or duct (such as an air‐conditioning duct). In such a case, the waves propagate in a direction along the duct axis and the plane wave surfaces are perpendicular to this direction (and are represented by duct cross‐sections). Such waves in a duct are one‐dimensional, like the waves traveling along a long string or rope under tension (or like the ocean breakers described above).
Although there are many similarities between one‐dimensional sound waves in air, waves on strings, and surface water waves, there are some differences. In a fluid such as air, the fluid particles vibrate back and forth in the same direction as the direction of wave propagation; such waves are known as longitudinal, compressional, or sound waves. On a stretched string, the particles vibrate at right angles to the direction of wave propagation; such waves are usually known as transverse waves. The surface water waves described are partly transverse and partly longitudinal, with the complication that the water particles move up and down and back and forth horizontally. (This movement describes elliptical paths in shallow water and circular paths in deep water. The vertical particle motion is much greater than the horizontal motion for shallow water, but the two motions are equal for deep water.) The water wave direction is, of course, horizontal.
Surface water waves are not compressional (like sound waves) and are normally termed surface gravity waves. Unlike sound waves, where the wave speed is independent of frequency, long‐wavelength surface water waves travel faster than short‐wavelength waves, and thus water wave motion is said to be dispersive. Bending waves on beams, plates, cylinders, and other engineering structures are also dispersive. There are several other types of waves that can be of interest in acoustics: shear waves, torsional waves, and boundary waves (see chapter 12 in the Encyclopedia of Acoustics [19]), but the discussion here will concentrate on sound wave propagation in fluids.
3.3 Plane Sound Waves
The propagation of sound may be illustrated by considering gas in a tube with rigid walls and having a rigid piston at one end. The tube is assumed to be infinitely long in the direction away from the piston. We shall assume that the piston is vibrating with simple harmonic motion at the left‐hand side of the tube (see Figure 3.1) and that it has been oscillating back and forth for some time. We shall only consider the piston motion and the oscillatory motion it induces in the fluid from when we start our clock. Let us choose to start our clock when the piston is moving with its maximum velocity to the right through its normal equilibrium position at x = 0. See the top of Figure 3.1, at t = 0. As time increases from t = 0, the piston straight away starts slowing down with simple harmonic motion, so that it stops moving at t = T/4 at its maximum excursion to the right. The piston then starts moving to the left in its cycle of oscillation, and at t = T/2 it has reached its equilibrium position
