Читать книгу Engineering Acoustics - Malcolm J. Crocker - Страница 138

Waveguides can occur naturally where sound waves are channeled by reflections at boundaries and by refraction. Even the ocean can sometimes be considered to be an acoustic waveguide that is bounded above by the air–sea interface and below by the ocean bottom (see chapter 31 in the Handbook of Acoustics [1]). Similar channeling effects are also sometimes observed in the atmosphere [34]. Waveguides are also encountered in musical instruments and engineering applications. Wind instruments may be regarded as waveguides. In addition, waveguides comprised of pipes, tubes, and ducts are frequently used in engineering systems, for example, exhaust pipes, air‐conditioning ducts and the ductwork in turbines and turbofan engines. The sound propagation in such waveguides is similar to the three‐dimensional situation discussed in Section 3.17 but with some differences. Although rectangular ducts are used in air‐conditioning systems, circular ducts are also frequently used, and theory for these must be considered as well. In real waveguides, airflow is often present and complications due to a mean fluid flow must be included in the theory.

For low‐frequency excitation, only plane waves can propagate along the waveguide (in which the sound pressure is uniform across the duct cross‐section). However, 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.