Engineering Acoustics. Malcolm J. Crocker

as the frequency is increased, the so‐called first cut‐on frequency is reached above which there is a standing wave across the duct cross‐section caused by the first higher mode of propagation.

For excitation just above this cut‐on frequency, besides the plane‐wave propagation, propagation in higher order modes can also exist. The higher mode propagation in a rectangular duct can be considered to be composed of four traveling waves in each direction. Initially, these vectors (rays) are almost perpendicular to the duct walls and with a phase speed along the duct that is almost infinite. As the frequency is increased, these vectors point increasingly toward the duct axis, and the phase speed along the duct decreases until at very high frequency it is only just above the speed of sound c. However, for this mode, the sound pressure distribution across duct cross‐section remains unchanged. As the frequency increases above the first cut‐on frequency, the cut‐on frequency for the second higher order mode is reached and so on. For rectangular ducts, the solution for the sound pressure distribution for the higher duct modes consists of cosine terms with a pressure maximum at the duct walls, while for circular ducts, the solution involves Bessel functions. Chapter 7 in the Handbook of Acoustics [1] explains how sound propagates in both rectangular and circular guides and includes discussion on the complications created by a mean flow, dissipation, discontinuities, and terminations. Chapter 161 in the Encyclopedia of Acoustics [19] discusses the propagation of sound in another class of waveguides, that is, acoustical horns.

3.19 Other Approaches

      3.19.1 Acoustical Lumped Elements

      When the wavelength of sound is large compared to physical dimensions of the acoustical system under consideration, then the lumped‐element approach is useful. In this approach it is assumed that the fluid mass, stiffness, and dissipation distributions can be “lumped” together to act at a point, significantly simplifying the analysis of the problem. The most common example of this approach is its use with the well‐known Helmholtz resonator (see Chapter 9 of this book) in which the mass of air in the neck of the resonator vibrates at its natural frequency against the stiffness of its volume.

      A similar approach can be used in the design of loudspeaker enclosures and the concentric resonators in automobile mufflers in which the mass of the gas in the resonator louvers (orifices) vibrates against the stiffness of the resonator (which may not necessarily be regarded completely as a lumped element). Dissipation in the resonator louvers may also be taken into account. Chapter 21 in the Handbook of Acoustics [1] reviews the lumped‐element approach in some detail.

      3.19.2 Numerical Approaches: Finite Elements and Boundary Elements

      In cases where the geometry of the acoustical space is complicated and where the lumped‐element approach cannot be used, then it is necessary to use numerical approaches. In the late 1960s, with the advent of powerful computers, the acoustical finite element method (FEM) became feasible. In this approach, the fluid volume is divided into a number of small fluid elements (usually rectangular or triangular), and the equations of motion are solved for the elements, ensuring that the sound pressure and volume velocity are continuous at the node points where the elements are joined. The FEM has been widely used to study the acoustical performance of elements in automobile mufflers and cabins (See Chapter 10 of this book.).

      The boundary element method (BEM) was developed a little later than the FEM. In the BEM approach the elements are described on the boundary surface only, which reduces the computational dimension of the problem by one. This correspondingly produces a smaller system of equations than the FEM. BEM involves the use of a surface mesh rather than a volume mesh. BEM, in general, produces a smaller set of equations that grows more slowly with frequency, and the resulting matrix is full; whereas the FEM matrix is sparse (with elements near and on the diagonal). Thus computations with FEM are generally less time‐consuming than with BEM. For sound propagation problems involving the radiation of sound to infinity, the BEM is more suitable because the radiation condition at infinity can be easily satisfied with the BEM, unlike with the FEM. However, the FEM is better suited than the BEM for the determination of the natural frequencies and mode shapes of cavities (See Chapter 10 of this book.).

      Recently, FEM and BEM commercial software has become widely available. The FEM and BEM are described in Refs. [35, 36] and in chapters 12 and 13 in the Handbook of Acoustics [1].

      3.19.3 Acoustic Modeling Using Equivalent Circuits

Electrical analogies have often been found useful in the modeling of acoustical systems. There are two alternatives. The sound pressure can be represented by voltage and the volume velocity by current, or alternatively the sound pressure is replaced by current and the volume velocity by voltage. Use of electrical analogies is discussed in chapter 11 of the Handbook of Acoustics [1]. They have been widely used in loudspeaker design and are in fact perhaps most useful in the understanding and design of transducers such as microphones where acoustic, mechanical, and electrical systems are present together and where an overall equivalent electrical circuit can be formulated (see Handbook of Acoustics [1], chapters 111–113).

      Beranek makes considerable use of electrical analogies in his books [10, 11]. In chapter 14 in the Handbook of Acoustics [1] their use in the design of automobile mufflers is described. Chapter 10 in this book also reviews the use of electrical analogies in muffler and silencer acoustical design.

References

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      2 2 Malecki, I. (1969). Physical Foundations of Technical Acoustics. Oxford: Pergamon Press.

      3 3 Skudrzyk, E. (1971). The Foundations of Acoustics. New York: Springer (reprinted by the Acoustical Society of America in 2008).

      4 4 Crocker, M.J. and Price, A.J. (1975). Noise and Noise Control, vol. I. Cleveland, OH: CRC Press.

      5 5 Lighthill, M.J. (2001). Waves in Fluids, 2e. Cambridge: Cambridge University Press.

      6 6 Pierce, A.D. (1981). Acoustics: An Introduction to Its Physical Principles and Applications. New York: McGraw‐Hill (reprinted by the Acoustical Society of America, 1989).

      7 7 Crocker, M.J. and Kessler, F.M. (1982). Noise and Noise Control, vol. II. Boca Raton, FL: CRC Press.

      8 8 Morse, P.M. and Ingard, K.U. (1986). Theoretical Acoustics. Princeton, NJ: Princeton University Press.

      9 9 Junger, M.J. and Feit, D. (1986). Sound, Structures, and Their Interaction. Cambridge, MA: MIT Press.

      10 10 Beranek, L.L. (1986). Acoustics. New York: Acoustical Society of America (reprinted with changes).

      11 11 Beranek, L.L. (1988). Acoustical Measurements, rev. ed. New York: Acoustical Society of America.

      12 12 Crighton, D.G., Dowling, A.P., Ffowcs Williams, J.E. et al. (1992). Modern Methods in Analytical Acoustics. Berlin: Springer‐Verlag.

      13 13 Fahy, F.J. (1995). Sound Intensity, 2e. London: E&FN Spon, Chapman & Hall.

      14 14 Fahy, F.J. and Walker, J.G. (eds.) (1998). Fundamentals of Noise and Vibration. London and New York: E/FN Spon.

      15 15 Filippi, P., Habault, D., Lefebvre, J., and Bergassoli, A. (1999). Acoustics: Basic Physics Theory & Methods. San Diego, CA: Academic Press.

      16 16 Kinsler, L.E., Frey, A.R., Coppens, A.B., and Sanders, J.V. (1999). Fundamentals of Acoustics, 4e. New York: Wiley.

      17 17 Blackstock, D.T. (2000). Fundamental of Physical Acoustics. New York: Wiley.

      18 18 Bruneau, M. and Scelo, T. (2006).