Acoustics of Ducts and Mufflers
Dr. M. L. Munjal is a Mechanical Engineer specializing in Engineering Acoustics. Over a four-decade career at the Indian Institute of Science, he has published over 190 papers in refereed journals, carried out over 100 industrial consultancy projects, and has run graduate courses on 'Industrial Noise Control' and 'Acoustics of Ducts and Mufflers'. He is Chairman of the National Committee for Noise Pollution Control that advises on noise control measures and legislations in India. He is also a Distinguished International Member of the Institute of Noise Control Engineering, USA, and Honorary Fellow of the International Institute of Acoustics and Vibration. He has been a member of the Editorial Board of several journals including the International Journal of Acoustics and Vibration .
Acoustics of Ducts and Mufflers
Propagation of Waves in Ducts
Exhaust noise of internal combustion engines is known to be the biggest pollutant of the present-day urban environment. Fortunately, however, this noise can be reduced sufficiently (to the level of the noise from other automotive sources, or even lower) by means of a well-designed muffler (also called a silencer). Mufflers are conventionally classified as dissipative or reflective, depending on whether the acoustic energy is dissipated into heat or is reflected back by area discontinuities.
However, no practical muffler or silencer is completely reactive or completely dissipative. Every muffler contains some elements with impedance mismatch and some with acoustic dissipation. In fact, combination mufflers are getting increasingly popular with designers.
Dissipative mufflers consist of ducts lined on the inside with an acoustically absorptive material. When used on an engine, such mufflers lose their performance with time because the acoustic lining gets clogged with unburnt carbon particles or undergoes thermal cracking. Recently, however, better fibrous materials such as sintered metal composites have been developed that resist clogging and thermal cracking and are not so costly. Besides, long strand unglued glass fibers can stand high temperatures. Nevertheless, no such problems are encountered in ventilation ducts, which conduct clean and cool air. The fan noise that would propagate through these ducts can well be reduced during propagation if the walls of the conducting duct are acoustically treated. For these reasons the use of dissipative mufflers is much more common in air-conditioning systems.
Reflective mufflers, being nondissipative, are also called reactive mufflers. A reflective muffler consists of a number of tubular elements of different transverse dimensions joined together so as to cause, at every junction, impedance mismatch and hence reflection of a substantial part of the incident acoustic energy back to the source. Most of the mufflers currently used on internal combustion engines, where the exhaust mass flux varies strongly, though periodically, with time, are of the reflective or reactive type. In fact, even the muffler of an air-conditioning system is generally provided with a couple of reflective elements at one or both ends of the acoustically dissipative duct.
Clearly, a tube or pipe or duct is the most basic and essential element of either type of muffler. A study of the propagation of waves in ducts is therefore central to the analysis of a muffler for its acoustic performance (transmission characteristics). This chapter is devoted to the derivation and solution of equations for plane waves and three-dimensional waves along rectangular ducts, circular tubes and elliptical shells without and with mean flow, without and with viscous friction, with rigid unlined walls and compliant or acoustically lined walls. We start with the simplest case and move gradually to the more general and involved cases.
1.1 Plane Waves in an Inviscid Stationary Medium
In the ideal case of a rigid-walled tube with sufficiently small cross dimensions filled with a stationary ideal (nonviscous) fluid, small-amplitude waves travel as plane waves. The acoustic pressure perturbation (on the ambient static pressure) p and particle velocity u at all points of a cross-section are the same. The wave front or phase surface, defined as a surface at all points of which p and u have the same amplitude and phase, is a plane normal to the direction of wave propagation, which in the case of a tube is the longitudinal axis.
The basic linearized equations for the case are:
Energy equation (isentropicity)