The position of the sound source within the room is also an important parameter, since for many source positions certain types of modes may not be excited. For example, if the source is located in one of the corners of the room, then it is possible to excite every normal mode, while if the source is located at the center of a rectangular room then only the even modes (one eighth of the total number of possible modes) can be excited. Similarly if we keep the position of the source constant and measure the sound pressure throughout the room we see differences in level depending on where we are standing in the room relative to the normal modes. In this way the room superimposes its own acoustical response characteristics upon those of the source. Hence we cannot measure the true frequency response of a sound source (e.g. loudspeaker) in a reverberant room because of the effect of the modal response of the room. This interference can be removed by making all the wall surfaces highly sound‐absorbent. Then all the modes are sufficiently damped so we are able to measure the true output of the source [4]. Such rooms are called anechoic (see Figure 3.22).
3.18 Waveguides
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.
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
