THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

14.14 Mufflers and Silencers 399Figure 14.19. Expansion chamber of a reactive muffler.interference. Both reactive and dissipative principles may be combined in a singlemuffler or silencers in order to ensure effectiveness over a broad frequency range.Reactive Mufflers and SilencersConsider the basic reactive muffler in Figure 14.19. Sound waves transmit from leftto right in the inlet pipe and reflections occur in the expansion chamber, causingdestructive interference under the appropriate conditions. In this analysis (Daviset al., 1984), the following assumptions are made:1. Sound pressure is small in comparison with absolute pressure in the expansionchamber.2. No reflected waves occur in the tailpipe (i.e., the outlet pipe).3. The expansion chamber walls do not transmit nor conduct sound.4. Only plane waves exist.5. Viscosity effects are negligible.We denote the following subscripts in the description of incident and reflectedwaves: I for incident waves and R for reflected waves. The particle displacementsξ of the incident and reflected waves are described in complex format as follows:andξ I = A I e i(ωt−kx) (14.48)ξ R = A R e i(ωt+kx) (14.49)Here A I and A R are complex amplitudes. Particle velocities are obtained by differentiatingequations (14.48) and (14.49) with respect to time:u I = iω A I e i(ωt−kx) (14.50)