This Demonstration represents analytical, closed-form solutions for the transmission loss of two common reactive mufflers, an expansion chamber and a quarter-wave resonator. These closed-form solutions are assumed to be accurate only at low frequency where only plane waves are involved. The analytical models were derived using continuity conditions at each change of cross-sectional area . The analytical models assume rigidity.
Muffler performance determined by transmission loss is governed by the cross-sectional area ratio of expansion chamber to pipe , the cross-sectional area ratio of quarter-wave resonator to pipe , wavenumber , and muffler length . As ratios and increase, the performance of each respective muffler improves. Resonances and anti-resonances are also clearly recognizable. A comparison plot is also presented in order to give the reader a generalized idea about muffler performance of a combined expansion chamber/quarter-wave resonator.
Transmission loss represents the reduction of sound power level. For our models, the pipe cross-sectional area is assumed to be constant and equal at both ends.
After applying various boundary conditions based on the continuity of acoustic pressure and volume velocity for each change in cross-sectional area (i.e., where the pipe interfaces with the muffling device), the closed-form solution for an expansion chamber is given (in dB) by
and the analytical closed-form solution for a quarter-wave resonator is
: transmission loss ()
: the ratio of expansion chamber cross-sectional area to pipe cross-sectional area
: the ratio of quarter-wave resonator cross-sectional area to pipe cross-sectional area
: wavenumber dependent on the speed of sound of the medium and frequency
: length of expansion chamber or quarter-wave resonator
 D. A. Bies and C. H. Hansen, Engineering Noise Control: Theory and Practice, 4th ed., New York: CRC Press, 2009 pp. 456–464.