Theory

The run stores the scattering matrix $S$ of the muffler and of its reference pipe: for every frequency, the waves leaving every port for a unit wave arriving at every port, for all modes. A source is described by the wave it sends, $A_s$ , and by the reflection coefficient $\Gamma_s$ of its plane mode for the wave coming back. The wave arriving at the inlet is then $A_s + \Gamma_s B$ , where $B$ is the wave leaving the inlet, and the closed loop of source, model and outlet termination is solved exactly from the stored $S$ . The muffler and the reference pipe get the same source, so their radiated powers form the insertion loss $IL = 10\log_{10}(W_{pipe}/W_{muffler})$ .

An anechoic source absorbs everything that comes back. A constant-velocity source (a rigid piston) reflects it fully in phase, a constant-pressure source in opposite phase. A table gives the measured normalised source impedance $z_s(f)$ , interpolated linearly between its rows and held at its end values outside them. With a reflecting source the reference pipe's power depends on its length (Reference pipe length).

Equations

\Gamma_s = \frac{z_s - 1}{z_s + 1}, \qquad \Gamma_s = 0\ \text{(anechoic)},\quad +1\ \text{(constant velocity)},\quad -1\ \text{(constant pressure)}

(1)

where	meaning	unit
$\Gamma_s$	Plane-mode reflection coefficient of the source	-
$z_s$	Source impedance normalised by $\rho c$ of the inlet gas	-

B_{out} = \left(I - S\,(G_s + G_o)\right)^{-1} S\,A_s

(2)

where	meaning	unit
$B_{out}$	Waves leaving the ports	Pa
$S$	Stored scattering matrix (all ports, all modes)	-
$G_s, G_o$	Reflection matrices of the source (inlets) and of the outlet terminations	-
$A_s$	Waves sent by the source(s)	Pa

Default and rule

Default: Anechoic (matched)
Rule: Anechoic.
Source: The algebra of the program on its finite-element scattering matrices. It was checked against the plane-wave four-pole insertion loss of Munjal (1987) (section 2.10.1, eqs. 2.58-2.65, pp. 55-56, with the uniform-tube matrix eq. 2.134, p. 77) on an expansion chamber for every source and radiation choice; that definition refers to the source connected straight to the radiation impedance, so the comparison subtracts the same expression for the reference pipe.

Valid range anechoic | constant velocity | constant pressure | table of z_s(f)

No solve

Changing the source re-evaluates the stored matrices in well under a second.

Table format

Three numbers per line: frequency in Hz, real and imaginary part of the normalised source impedance.

References 1

Munjal, M. L. (1987). Acoustics of ducts and mufflers with application to exhaust and ventilation system design. 1st ed., Wiley-Interscience, New York, xv + 328 pp., ISBN 0-471-84738-0. Web page

Theory

Equations

Default and rule

Valid range anechoic | constant velocity | constant pressure | table of z_s(f)

Related 4

References 1