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Practical Considerations when using the Two-Load Method to Determine the

Transmission Loss of Mufflers and Silencers

Article in SAE International Journal of Passenger Cars - Mechanical Systems · May 2013
DOI: 10.4271/2013-01-1881

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 2013-01-1881
Published 05/13/2013
Copyright © 2013 SAE International
doi:10.4271/2013-01-1881
saepcmech.saejournals.org

Practical Considerations when using the Two-Load Method to

Determine the Transmission Loss of Mufflers and Silencers
X. Hua and D. W. Herrin
University of Kentucky

ABSTRACT
The two-load method is commonly used to determine the transmission loss of a muffler or silencer. Several practical
measurement considerations are examined in this paper. First of all, conical adapters are sometimes used to transition
between impedance tubes and the muffler. It is demonstrated that the effect of adding the adapter can be quite significant at
low frequencies especially if the adapter is short in length. The effect of changing the length of the adapter was examined
via measurement and plane wave theory. Secondly, the effect of selecting the reference microphone was examined
experimentally. It was found that measurements are improved by selecting a downstream reference. Finally, the effect of
using different frequency response function estimation algorithms (H1, H2 and Hv) was compared sans flow. This had little
effect on the measurement.

CITATION: Hua, X. and Herrin, D., "Practical Considerations when using the Two-Load Method to Determine the
Transmission Loss of Mufflers and Silencers," SAE Int. J. Passeng. Cars - Mech. Syst. 6(2):2013, doi:10.4271/2013-01-1881.
____________________________________

INTRODUCTION The two-load method has been standardized in ASTM

E2611-09 [5]. Although this standard is geared towards
Insertion loss is the metric typically used to assess muffler determining the transmission loss through a sound absorbing
performance in the field. It is defined as the difference in material, the algorithm and methodology can be applied to
sound pressure at a point near the termination with and measuring muffler transmission loss. As noted in the
without a muffler installed. Though insertion loss is easy to standard, two different loads are selected.
measure, it is a property of the source and termination Both Munjal [4] and Åbom [6] pointed out that a potential
impedances as well as the lengths of the inlet and outlet ducts challenge is to find two different loads at all frequencies of
and the muffler. Hence, insertion loss is a measure of the interest. If the two loads are too close, the determined
installed muffler attenuation for a particular exhaust system. transmission loss is prone to error. The standard recommends
Transmission loss eliminates the effect of the inlet and using one absorptive load, which allows minimal reflection.
outlet ducts. Indeed, insertion loss is equal to the transmission The other preferred load is an open or closed termination
loss if the source and termination impedances are anechoic. where significant reflection of sound is anticipated.
Transmission loss is simpler to determine using analysis since Four microphones are mounted along the impedance tube
an anechoic termination is easily applied as a boundary with two upstream and two downstream of the muffler. A
condition below the plane wave cutoff. reference signal is selected prior to the test, which can be one
Measurement of transmission loss is trickier. Certainly, of the four microphones, the source signal for driving the
transmission loss is often roughly measured using either a loudspeaker or a fifth microphone. Based on the reference
very long downstream duct or a makeshift anechoic selected, three or four transfer functions for each load are
termination. However, it is difficult to create an anechoic measured. From these transfer functions; the so-called four-
termination that will be accurate over the full frequency pole parameters for the muffler can be determined along with
range. For precise determination of transmission loss, the the transmission loss.
muffler must be measured at two different conditions. The In this paper, the two-load method is investigated in a
two commonly used techniques are the two-load [1,2,3] and number of ways. The cross-sectional area of the impedance
two-source [4] methods. In this work, the two-load method is tube is fixed and cannot be easily adjusted to fit the
examined exclusively though many of the conclusions made dimensions for different mufflers. A pair of conical adapters
in this paper should be transferable to the two-source method.
 Hua et al / SAE Int. J. Passeng. Cars - Mech. Syst. / Volume 6, Issue 2(July 2013)

is sometimes used to transition between impedance tubes and

the muffler. The effect of using adapters is discussed.
Secondly, the selection of the reference is also
investigated by comparing results for selecting upstream and (1d)
downstream references. Reference signal selection is
unaddressed in the ASTM standard. For each load (a or b), pressure and particle velocity at
When measuring the transfer functions, three different two ends of the sample or muffler are expressed as
frequency response function algorithms can be adopted. The
effect of using each of the different algorithms is also
compared. All measurements discussed in the paper are sans (2a)
flow at room temperature.

REVIEW OF TRANSMISSION LOSS (2b)

MEASUREMENT
Figure 1 shows the transmission loss measurement setup
(2c)
schematically. A speaker is placed at the end of the
impedance tube. Two microphones are mounted upstream
and the other two microphones are mounted downstream.
Two different termination loads are applied and four transfer (2d)
functions are measured for each load, noted as H1,ref, H2,ref, The four-pole matrix based on pressure and particle
H3,ref, and H4,ref. velocity can be written as

(3)

Figure 1. Two-load transmission loss test setup. where the subscripts a and b denote the two different loads.
Then, the transmission loss is expressed as

Wave Decomposition Data Processing

There are two methods to process the measured transfer
functions to obtain transmission loss. The first method, which (4)
has been standardized in Reference 5, is based on wave
decomposition. By applying wave decomposition at both the The algorithm used is identical to that for the two-source
upstream and downstream tubes, the wave strength A, B, C, method.
and D can be determined as
Four-Pole Matrix Data Processing
The other well-known method for processing data follows
References 4 and 7 and was originally used for the two-
source method. Instead of the four-pole matrix for the
(1a) sample, the four-pole matrix between microphones 2 and 3 is
calculated. The four-pole parameters can be written as

(1b)

(5a)
(1c)
 Hua et al / SAE Int. J. Passeng. Cars - Mech. Syst. / Volume 6, Issue 2(July 2013)

(9b)
(5b)

(5c)
(9c)

(5d)
where Hij = pj / pi. Aij, Bij, Cij, and Dij are the corresponding (9d)
four-pole parameters between microphone i and j, and Δij =
AijDij - BijCij. Since additional tube extensions at the inlet and
outlet do not modify the transmission loss, the transmission
loss can be expressed as

(6) Figure 2. The dimensions of a divergent conical adapter.

In the authors' opinion, the wave decomposition method is
simpler to use and program. Moreover, it can be used to
directly determine the four-pole parameters for the muffler. The transfer matrix of the convergent conical tube is
similar and can be written as
CONICAL ADAPTER EFFECT
Plane Wave Theory with Conical Adapters
Assuming a pair of conical adapters is utilized to
transition between the impedance tube and the muffler in the (10)
measurement, the measured transfer matrix [Ttotal] below the By solving Equation 7, the transfer matrix of the muffler
cutoff frequency includes the pair of adapters and the muffler is
itself. It is expressed as

(7) (11)
where [Tcone1] and [Tcone2] are the transfer matrices of the The transmission loss of the muffler is then calculated
upstream and downstream adapters respectively. [Tmuffler] is using
the transfer matrix of the muffler. The transfer matrix of the
divergent conical tube with length l, and radii ru and rd
(shown in Figure 2) can be written as [8]

(8) (12)
where where Si and So are the cross-sectional area of the inlet and
outlet of the muffler, respectively. ρ is the air density and c is
the speed of sound.
The measured transmission loss is only valid below the
(9a) cutoff frequency. If the muffler is smaller than the impedance
 Hua et al / SAE Int. J. Passeng. Cars - Mech. Syst. / Volume 6, Issue 2(July 2013)

tube, the cutoff frequency of the muffler is controlled by the transmission loss below 200 Hz is noisy, which is
cross-sectional size of the inlet and outlet of the impedance understandable because the transmission loss is high.
tube. If the muffler is larger than the impedance tube, the Moreover, the source power is not sufficient because a
cutoff of the muffler is determined by the size of the inlet and compression driver loudspeaker is used. The results agree
outlet of the muffler itself. very well between 200 Hz and 600 Hz. The shift at high
frequency is understandable due to the geometry mismatch
Effect on Measurements between the barrel and the simple expansion chamber. In
Although the theory above is straightforward Figure 5, the results agree well with a slight shift at high
mathematically, it can be problematic in practice. One frequencies.
common concern is that the inlet and outlet diameter to the By applying transfer matrix theory (Equation 11), the
muffler are not the same diameter as those for the impedance effect of the cones can be removed, and the transmission loss
tubes with microphone mountings. of the expansion chamber itself is compared in Figure 6. It is
In order to investigate the conical adapter effect on shown that the results are very noisy and do not agree well
measurement, a barrel was used as an expansion chamber with the analytical transmission loss at low frequencies
with 152 mm diameter inlet and outlet holes on each side. (below 200 Hz) for both cases. However, the transmission
Since the impedance tube diameter is only 34.8 mm, two loss using long adapters compares well with the analytical
pairs of conical adapters with different lengths were built as solution above 200 Hz, whereas the transmission loss using
shown in Figure 3. The dimensions of each part are listed in short adapters compares well only above 700 Hz.
table 1.

Figure 3. Photos of barrel and adapters.

Table 1. Dimensions of barrel and conical adapters Figure 4. Transmission loss of the expansion chamber
with short conical adapters.

The transmission losses of the expansion chamber with

the different adapter pairs were measured using the wave
decomposition algorithm. The transmission losses with long
and short adapters are shown in Figures 4 and 5 respectively.
The cutoff frequency of the measurement is around 5650 Hz,
which is determined by the size of the ACUPRO impedance
tube provided by Spectronics Inc. [9]. Sidlab [10] was used to
model the duct system using 1-D plane wave theory. The
analytical solutions by Sidlab for both cases are also provided
for comparison. The higher order behavior in the chamber is
considered [11] but plane wave behavior is assumed in the
adapters. Accordingly, the cutoff frequency for the analytical Figure 5. Transmission loss of the expansion chamber
solutions is governed by the size of the adapters and should with long conical adapters.
be around 1300 Hz.
The measured transmission loss generally agrees well
with the analytical solution for both cases. In Figure 4, the
 Hua et al / SAE Int. J. Passeng. Cars - Mech. Syst. / Volume 6, Issue 2(July 2013)

The dominant lobe is still the first but with a lower

transmission loss.

Figure 6. Transmission loss of expansion chamber

without conical adapters.

The most likely explanation is that conical adapters Figure 8. Transmission loss of reversed conical pairs
introduce additional transmission loss especially at low with different lengths and area ratios. l is the length of a
frequencies. After the effect of the conical pairs is removed, single adapter.
the error from the measurement remains and is significantly
amplified due to the inversion process.
Figure 7 shows the transmission loss of conical adapters By selecting a suitable length for the conical adapters, the
with different sizes and lengths connected with each other. measured transmission loss will not be compromised by the
The first lobe in transmission loss is much higher than the cones. Figure 9 shows dimensions of such a real muffler. The
following lobes. The length of the cone determines the width area ratios on the inlet and outlet sides are approximately 4
of the first lobe, whereas the height of the lobe is determined and 1 respectively. The length of the conical adapters is .178
by the area ratio of the cone. Since the area ratio for the m. Figure 10 shows the comparison of the measured and the
conical adapters is 19, the peak of the first lobe is simulated transmission loss sans conical adapters using
approximately 15 dB, which is much higher than the Equation (11). The results compare well even at low
transmission loss of the expansion chamber itself at low frequencies.
frequencies. Hence, the conical adapter behavior dominates
the transmission loss at low frequencies.

Figure 9. Dimensions of a real muffler and its adapters.

For mufflers very close in diameter to the impedance

tube, the conical area ratio is very low and the adapter effect
is minimal. In those cases, using different lengths of adapters
will lead to a similar accuracy in the transmission loss. This
Figure 7. Transmission loss of conical pairs with will be the case even for a sudden expansion and/or
different lengths and area ratios. l is the length of a contraction, which can be thought of as a very short conical
single adapter. adapter.
For mufflers with a large area ratio, a pair of long
adapters is recommended for a smooth and accurate
When small mufflers are measured, a pair of reversed transmission loss at low frequencies. With long conical
adapters may be used. The corresponding transmission loss adapters, the high transmission loss of the adapters is pushed
plots with different sizes and lengths are shown in Figure 8. to lower frequencies. This suggests that the effect of the
 Hua et al / SAE Int. J. Passeng. Cars - Mech. Syst. / Volume 6, Issue 2(July 2013)

cones connected to each other should first be examined muffler is shown in Figure 12. When microphone 1 is
before measuring the transmission loss of the muffler itself. selected as reference, the measured transmission loss is noisy
at 225 Hz, 330 Hz, 620 Hz and so forth. When reference 3 is
selected, the transmission loss is very smooth for the entire
frequency band with the only exception at around 700 Hz.

Figure 10. Comparison of measured and simulated

muffler transmission loss without conical adapters.

Figure 11. Measured transmission loss of the simple

EFFECT OF REFERENCE SIGNAL expansion chamber with different reference signals.
Using the wave decomposition algorithm, four transfer
functions are measured for each load, which are H1,ref, H2,ref,
H3,ref and H4,ref. For convenience, the reference signal is
selected as one of the four microphones. And hence the
number of transfer functions measured is reduced to three for
each load because one of the four transfer functions is unity.
The reference signal selection will affect the measured
transmission loss, although they are equivalent theoretically.
In order to investigate the effect of reference signal, a
simple expansion chamber was constructed with 11.5 mm
thick plastic. The length was 200 mm and the inner diameter
was 150 mm. The inlet and outlet diameter was 34.8 mm,
which exactly matched the impedance tube. Although this
muffler was geometrically symmetric so that only one load
was required, two loads were used.
The test setup and microphones positions are shown in
Figure 1. Both microphone spacings (s1 and s2) are 29.2 mm. Figure 12. Measured transmission loss of a reactive
muffler with different reference signals
The lengths of the upstream and downstream tubes are 787
mm and 406 mm, respectively. Load a is a 100 mm thick
acoustic foam with a blocked end which minimizes This can be explained in the following way. Note that pd
reflections. Load b is an open tube.
and ud occur in the denominator for each term in the transfer
The transmission loss was measured using different
matrix (Equation 3) where errors may be amplified
references and results are compared in Figure 11. The four
significantly. pd and ud are determined from C and D which
curves generally agree with each other. However, there is
significant noise at 100 and 400 Hz if microphone 1 or 2 is depend solely on H3,ref or H4,ref.
selected as a reference whereas the measurement is smooth If an upstream microphone (1 or 2) is selected as
using reference 3 or 4 for the entire frequency band. reference, either H11 or H22 is reduced to 1 without any
For the impedance tube with an open load, a phase jump measured error. Similarly, the errors for either H33 or H44 are
occurs for the transfer functions at 100 and 400 Hz eliminated if a downstream microphone is selected as a
(indicating a likely resonance). Accordingly, the coherence reference. Additionally, the coherence will be higher for
drops to below 0.5. Although the coherence is also low for either H34 or H43 if a downstream microphone is selected for
transfer functions using microphone 3 or 4 as a reference, the a reference because microphones 3 and 4 are not separated by
transmission loss is less noisy. This phenomenon is apparent the muffler. This in turn reduces the errors in the
not only for the simple expansion chamber, but is also present denominators of the transfer matrix (Equation 3) because the
for other reactive mufflers. The transmission loss of one such
 Hua et al / SAE Int. J. Passeng. Cars - Mech. Syst. / Volume 6, Issue 2(July 2013)

terms in the denominator depend solely on H33 or H43 (or

H34 or H44).
Another potential and convenient reference signal is the
source, which is the output signal of the testing software used
to drive the speaker. A comparison test was done with the
same simple expansion chamber mentioned above. However,
the transmission loss is also noisy as shown in Figure 13.

Figure 15. Transmission loss comparison using H1, H2

and Hv.

SUMMARY
Several aspects of using the two-load method to
determine the transmission loss have been examined in this
paper. First of all, the effect of using conical adaptors was
Figure 13. Measured transmission loss of the simple examined. It was demonstrated that conical adapters
expansion chamber with the microphone reference and significantly affect the measurement of transmission loss at
the source reference. low frequencies. In order to minimize the effect, the area ratio
should be minimized and the length of the cone maximized.
The effect of using conical adapters can be easily evaluated
EFFECT OF FREQUENCY RESPONSE prior to an experiment by determining the transmission loss,
either analytically or experimentally, of the two cones placed
FUNCTION ESTIMATIONS together as shown in Figures 7 and 8.
When measuring transfer functions, three frequency The choice of reference was also investigated. It was
response function estimation algorithms can be selected: H1, shown that choosing a downstream microphone as a reference
H2 or Hv, which assume the noise is on the input, output or improves the measurement quality. In prior work, Tao and
Seybert [7] had noted that the two-source method might be
both respectively [12-13]. Figure 14 illustrates how noise or
superior to the two-load. However, this conclusion should be
errors can be introduced. u is the noise on the input, and n is
reevaluated since an upstream microphone was chosen as a
the noise on the output. H2 minimizes the noise on the input
reference. The two-load method may well be comparable to
(u = 0), and H1 on the output (y = 0). Hv compensates for
the two-source method if a downstream reference is selected.
error on both the input and output. See References 12-13 for It was also shown that the choice of frequency response
more information. function estimation algorithm has little effect on the quality
of the measurement.

REFERENCES
1. To, C. W. S. and Doige, A. G., “A Transient Testing Technique for the
Determination of Matrix Parameters of Acoustic Systems, 2:
Experimental Procedures and Results,” Journal of Sound and Vibration,
62, 223-233 (1979).
2. To, C. W. S. and Doige, A. G., “A Transient Testing Technique for the
Determination of Matrix Parameters of Acoustic Systems, 1: Theory and
Figure 14. Frequency response function estimation Hv. Principles,” Journal of Sound and Vibration, 62, 207-222 (1979).
3. Lung, T. Y. and Doige, A. G., “A Time-averaging Transient Testing
Method for Acoustic Properties of Piping System and Mufflers,” J.
Acoust. Soc. Am. 73, 867-876 (1983).
The measured transmission loss using the three different 4. Munjal, M. L. and Doige, A. G., “Theory of a Two Source-location
Method for Direct Experimental Evaluation of the Four-pole Parameters
frequency response function algorithms is compared using of an Aeroacoustic Element,” Journal of Sound and Vibration, 141(2),
the aforementioned simple expansion chamber. The results 323-333 (1990).
5. ASTM E2611-09, “Standard Test Method for Measurement of Normal
(Figure 15) show that using different frequency response Incidence Sound Transmission of Acoustical Materials Based on the
function estimation algorithms does not improve the Transfer Matrix Method” (2009).
6. Åbom, M., “A Note on the Experimental Determination of Acoustical
measured accuracy of the transmission loss. All Two-port Matrices,” Journal of Sound and Vibration 155(1), 185-188
measurements were made without flow. (1992).
 Hua et al / SAE Int. J. Passeng. Cars - Mech. Syst. / Volume 6, Issue 2(July 2013)

7. Tao, Z. and Seybert, A., “A Review of Current Techniques for

Measuring Muffler Transmission Loss,” SAE Technical Paper
2003-01-1653, 2003, doi:10.4271/2003-01-1653.
8. Munjal, M. L. “Acoustics of Ducts and Mufflers,” Chapter 2 and 3,
Wiley-Interscience, New York, (1987).
9. ACUPRO Measurement System, Spectronics Inc. USA.
www.spectronics.net.
10. SIDLAB Acoustics User's Manual, Version 2.6.
11. Åbom, M., “Derivation of the four-pole parameters including higher
order mode effects for expansion chamber mufflers with extended inlet
and outlet,” Journal of Sound and Vibration, 137(3), 403-418 (1990).
12. White, P. R., Tan, M. H. and Hammond, J. K., “Analysis of the
Maximum Likelihood, Total Least Squares and Principal Component
Approaches for Frequency Response Function Estimation,” Journal of
Sound and Vibration, 290, 676-689 (2006).
13. Boden, H., Ahlin, K. and Carlsson, U., Signal Analysis Chapter 8,
Marcus Wallenberg Laboratoriet (2011).

CONTACT INFORMATION
David W. Herrin
University of Kentucky
Department of Mechanical Engineering
151 Ralph G. Anderson Building
Lexington, KY 40506-0503
dherrin@engr.uky.edu

ACKNOWLEDGMENTS
The authors would like to acknowledge the support of the
Vibro-Acoustics Consortium at the University of Kentucky.

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