# Analytical Expression for Sound Transmission Loss Calculation: An Improvement to the Existing Method after Singh and Katra

Authors: Iwan Yahya
Journal: Advances in Acoustics and Vibration
 (2009)

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## Abstract

An analytical expression for measuring of sound transmission loss (TL) has been developed by using two microphones, an impedance tube and an impulse sound source as a proposed improvement to the existing procedure after Singh and Katra (1978). The calculation procedure is based on the autospectrum of short-time signals captured by the two microphones placed on two opposite positions from test sample while the sound source is on its surface. No spectral decomposition is required and the  TL  is calculated directly from the autospectrums of captured signals.


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## Body

## 1. Introduction

Understanding  of acoustic parameter and/or properties of material is  very important in many applications. Nowadays it is ranging from tissue analysis for medical purposes to elastic properties analysis of material such as attenuation and TL in engineering applications. Researchers have developed methods and measurement techniques for these purposes. Such measurements simply categorized into two main groups: first, the standing waves method by using impedance tube with single microphone and single tone sound source; this method is inefficient due to its time consuming [1, 2]; second,  the spectral decomposition-based measurement, including a two-microphone  impedance tube [3–5] and its four microphones improvement for TL measurement purposes [6–8]; the second group uses random noise instead of single tone sound source which is more efficient compared to the first group.Another method for TL calculation is presented in this paper. The proposed method is an improvement to the previous one by Singh and Katra [9] and has been derived with two basic assumptions.  (i)  An impulse sound source is generated right on the sample surface while the two microphones are placed at two different opposite positions inside the tube. The microphone distance to the sample surface should be longer compared to the tube diameter to ensure that the propagating wave is plane wave. (ii)  Tube is sufficiently long, so there are no reflected waves from the both ends; for special purposes the utilization of anechoic termination is needed.


## 2. Existing TL Measurement Method by Singh and Katra

In 1978, Singh and Katra introduced a procedure for measuring muffler characteristics based on impulse techniques as depicted in Figure1.Figure 1
Setup for measuring muffler characteristics developed by Singh and Katra.The impulse sound is generated from loudspeaker at the end of the tube. For TL calculation purposes, the test sample is placed in certain position between both microphones. Upstream and downstream signals are captured by microphone number 1 and microphone number 2, respectively. A time window is set in an appropriate size to make sure that first detected signal captured properly and minimizing any part of reflected waves from both ends of the tube. The rule for this window sizing is expressed in detail by Singh and Katra.The TL is then giving by(1)TL=10log{(S12S22)(S12S22)*},
where S12 is cross power spectrum of signals captured by the microphones  and S22 is autopower spectrum of signal captured by microphone number two. The asterisk (*) indicates complex conjugates of  S12 and S22 ratios.


## 3. Proposed Direct Calculation Procedure

It   happens that reflected waves are also captured by microphone number one, so TL in (1) was not obtained directly from incident and transmitted waves power spectrum.To do this direct calculation, the modified configuration is depicted in Figure2.Figure 2
Proposed setup configuration for the new TL measurement.Short-time impulse sound is radiated from a source located directly on the left surface of the test sample. Time window is set in same manner as in Singh and Katra’s method.  Sound pressure values captured by the two microphones are given by(2)p1(x)=S11=p0eαx1,p2(x)=S22=pde-α(x2-d),
where α=jk is the complex propagation constant of the waves in air inside the tube, S11 and S22 are autospectrum of the short-time captured signal by microphones number one and number two, respectively, and  p0 and pd are the sound pressure levels of the signals at x=0 and x=d, and in this case the ejωt terms were eliminated from the equations for simplification purpose.An illustration of the short-time captured signal and its spectrum is depicted in Figure3. Both microphones capture the signal in a same size window. Autospectrum of these captured signals is then used for TL calculation purposes.Illustration of first arrival truncation process.
(a)(b)From   (2), one could write the expression for the sound pressure level on   both   surfaces of the test sample, for x=0  and x=d, as follows:(3)p0(x)=S11e-αx1,(4)pd(x)=S22eα(x2-d).
Let  x2-d=-x1,  then (4) becomes(5)pd(x)=S22e-αx1.
Expression for the sound pressure level in the test sample is given by(6)ps(x)=p0e-αsx,0≤x≤d,
where αs is the complex propagation constant for the test sample.TL is then calculated by comparing the intensity of the incident waves and transmission waves using the following relation:(7)TL=10logWiWt,
where(8)Wi=pi2ρcAi,(9)Wt=pt2ρcAo,
where pi and  pt are given by  (6) for x=0 and x=d, respectively, ρ is the  density of the air, and c is sound velocity. Ai and Ao are cross sections of   both sides of the tube.By using the relation in   (3), then   (8) changes to(10)Wi=pi2ρcAi=S11e-2αx1ρcAi.Taking the boundary condition forx=d and using   (5) and (6)  give pt=ps(d)=p0e-αsd=S22e-αx1. Equation (9) could be rewritten as(11)Wt=S22e-2αx1ρcAo.Since   both sides of the tube have an equal cross-section,Ai=Ao, substitution of   (10) and (11) into   (7) gives the expression for sound transmission loss as(12)TL=10logS11S22.Substituting (3) and (5) into  (6) for x=0 and x=d, we get the expression for αs as follows:(13)αs=-1dln(S22S11)=-1d[lnS22-lnS11].Since this method requires only calculation of autospectrum of both truncated signals from the microphones, (12) and (13) show  the effectiveness of the proposed method which is   more simple in analysis compared to the previous calculation method by Singh and Katra, or even with the four-microphone  technique.Properly fitted anechoic termination is highly recommended to ensure that no parts of reflected waves are captured by the microphones. But this is not such a critical problem by the advance on surface microphone technology: by using a pair of surface microphones, a similar result could be obtained by placing the microphones properly on   both surfaces of the test sample. An impulse sound source generated signal from the end of the tube or even the source placed on the sample surface, and TL could be derived directly from  (6) with(14)pi=S11=ps(0),(15)pt=S22=ps(d)e-αsd.It is obvious that (14) and (15) give similar results for TL and αs as obtained from  (12) and (13).


## 4. Conclusion

The proposed method is more efficient and faster compared to the existing procedure, even promising a better performance by the advance of new surface microphone technology. The ability of giving similar result to previous techniques is the advantage of the current proposed method.



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*Source: 574604-2010-01-12.xml*