Acoustic transfer admittance of cylindrical cavities
in infrasonic frequency range
Paul VINCENT(1),(2) , Dominique RODRIGUES(2) , Franck LARSONNIER(1) , Cécile GUIANVARC’H(2) , Stéphane
DURAND(4)(1) Commissariat à l’Énergie Atomique (CEA), France(2) Laboratoire Commun de Métrologie LNE-CNAM (LCM), France(3) Laboratoire d’Acoustique de l’Université du Maine (LAUM), France

Abstract
Demand for calibration at infrasonic frequencies has emerged in response to earth monitoring problems. The
primary standard for sound pressure is defined through the reciprocity calibration method specified in the International Electrotechnical Commission (IEC) Standard 61094-2:2009. This method is based on the use of closed
couplers and is routinely applied by the National Metrology Institutes for a large frequency range; however,infrasonic frequencies below 2 Hz have not been explored until recently. The acoustic transfer admittance of thecoupler, including the heat conduction effects of the fluid, must be modelled precisely to obtain accurate microphone sensitivity. IEC 61094-2:2009 provides two standardised solutions for the correction of heat conduction.
However, researchers have noted significant deviations between these corrections at low frequencies in plane
wave couplers, indicating that one or both techniques incorrectly calculate the influence of heat conduction. In
this paper, two alternative solutions are proposed. An experiment is also reported, which highlights the limitations of the standardised formulations for acoustic transfer admittance, while also demonstrating the validity ofthe proposed alternative formulations at frequencies down to 0.04 Hz.
Keywords: Infrasound, Calibration, Microphones, Reciprocity, Admittance1

INTRODUCTION

The pressure reciprocity calibration method as specified in the International Electrotechnical Commission (IEC)Standard 61094-2:2009 [3] is currently used worldwide for absolute pressure calibration of laboratory standard
microphones, and provides the basis for primary measurement standards for sound pressure. This method, which
is based on the use of closed couplers, is routinely applied by the National Metrology Institutes at frequenciesof up to 25 kHz and, recently, down to 2 Hz [4]. While the reciprocity method has been used for a long time todetermine microphone pressure sensitivity over an extended audible frequency range, the demand for calibration
at infrasonic frequencies below 2 Hz has not been identified until recently. This is revealed by the absenceof Calibration and Measurement Capabilities (CMCs) in the Bureau International des Poids et Mesures (BIPM)
database [7] for frequencies below 2 Hz, except for static pressures, for which CMCs have been obtained using
specific techniques such as the pressure balance [13] (see Figure 1).
To perform calibration at infrasonic frequencies, the validity and performance of the pressure reciprocity method
must be examined for this frequency range. In the most usual configuration, the pressure reciprocity method
requires three reciprocal microphones coupled by pairs using a cavity, generally with a cylindrical shape. The
coupler ends are closed by the microphone diaphragms, with one being used as a transmitter and the other one
as a receiver. The product of the microphone sensitivities is determined from electrical measurements and fromanalytical calculation of the acoustic transfer admittance of the system. This operation is repeated with three
microphone couples.
(1) paul.vincent@cea.fr

155

Calibration
capability

No primary standard

Static pressure

InfrasoundCalibration
capability
Acoustic pressure
frequency

0,1 Hz

1 Hz

2 Hz

20 Hz

100 Hzperiod10 s

1s

Figure 1. CMC status from static pressure range to acoustic pressure range.
Calculation of the acoustic transfer admittance is a key aspect of microphone pressure reciprocity calibration.
The acoustic transfer admittance, defined as the ratio of the short-circuit volume velocity produced by thetransmitter microphone to the sound pressure acting on the diaphragm of the receiver microphone has been
extensively explored and discussed considering both influence of heat conduction and viscous losses [5, 6, 8–
12, 14, 16–18]. In particular, the effects of heat conduction is an important issue of the calculation of theacoustic transfer admittance especially in small closed volumes and at low frequencies where the expansion andcompression processes of the gas are somewhere in between an isothermal process and adiabatic process, orsaid to be polytropic process. The IEC Standard 61094-2:2009 [3] provides two formulations for calculationof the acoustic transfer admittance. These formulations have been revised in the context of the above problem
since the first edition, IEC 327 [2], which was published in 1971. However, significant behavioural differences
between the standardised models at very low frequencies have recently been highlighted; these discrepancies
yield inconsistent calibration results [11, 14].
With the objective of achieving an acoustic primary standard in the infrasonic frequency range, the second
section of this paper proposes alternatives and reports an experimental study in the frequency range of 0.04–
100 Hz. A detailed presentation of the measurement setup and methodology is provided, and the experimentresults are reported and discussed.

2

ACOUSTIC TRANSFER ADMITTANCE: MODEL PRESENTATIONThe IEC Standard 61094-2:2009 [3] specifies the requirements for pressure reciprocity calibration of laboratory
standard (LS) microphones and includes models for calculating the acoustic transfer admittance of cylindricalcouplers. In Appendix A of the standard, two formulations for correcting heat conduction under polytropic
conditions are presented:
1. the ‘broadband solution’, which considers both thermal and viscous effects in plane wave couplers and isapplicable to higher frequencies;
2. the ‘low-frequency solution’, which considers only thermal effects for cylindrical coupler assuming uniform
pressure, and which is based on a solution presented by Gerber [9].
Recently, Jackett [11] highlighted significant deviations between these models at low frequencies for plane wave
couplers, indicating that one or both models incorrectly calculate the influence of heat conduction. To realise a
primary standard for the infrasonic frequency range, validation of an appropriate acoustic modelling techniqueappears to be an essential preliminary step. The limitations of the normalized models are reported in [19].
Two new theoretical formulation are also presented in [19]: the general alternative low-frequency solution andthe short-term alternative low-frequency solution.

156

The acoustic transfer admittance of cylindrical couplers Ya for these models is given byijωV hγ − (γ − 1)EP +Yr +Yt ,
Ya =
γP0

(1)

with V the coupler volume, ω the angular frequency, γ the ratio of the specific heat capacities, P0 the staticpressure under measurement conditions, Yr and Yt the acoustical admittances of receiver and transmitter microphones, respectively. For the general alternative low-frequency solution [19], EP is given by−1+∞ +∞ λn2 R2 + (m + 1/2)2 π 2 2
8/π 2
X
,
EP = ∑ ∑
F
,
with
F
=
1
+
m,n
P
2 2 m,n(1 + 2R)2m=0 n=1 (m + 1/2) λn(2)r
A 1 − j αt
√
XP =
,
V
ω
2
where λn corresponds to the zeros of integer-order Jn (x) Bessel function, A is the total internal area of thecylinder surface (length `, radius a), R = `/(2a) and αt the thermal diffusivity of the enclosed gas. A shortterm Laplace asymptotic development of the previous general solution can also be obtained, which gives theshort-term alternative low-frequency solution [19]:EP = 1 − XP +3

R3 − 6R2
πR2 + 8R 2 3 √
XP +
π √
X3 .
2
π(2R + 1)
4
3 π(2R + 1)3 P(3)VALIDITY TEST FOR ACOUSTIC TRANSFER ADMITTANCE FORMULATIONS

3.1 Methodology
The experimental protocol implemented to test the accuracy of the above formulations for the acoustic transfer
admittance was derived from [15] and was based on the pressure reciprocity method. Figure 2 presents an
overview of the measurement system. Two electrical transfer impedances (defined as the ratio of the open-circuit
voltage ur0 of the receiving microphone to the current it through the transmitter microphone) were measured
for a pair of microphones using two cavities of different lengths, hereafter referred to as the short and longcavities.
The products of the sensitivities Mr and Mt of the receiver and transmitter microphones, respectively, are givenby the well–known equations
Mt Mr |s = Ze,s Ya,s , and Mt Mr |` = Ze,` Ya,` ,(4)
for the short and long cavities (subscripts s and `), respectively. Here, Ya,(s,`) are the previously defined anddiscussed acoustic transfer admittances of the cavities. By considering the microphones as stable during theexperiment, the products of the sensitivities Mt Mr |s and Mt Mr |` should be invariant as functions of the cavity,
insofar as the models of the acoustic transfer admittances are perfectly valid. The objective of the experiment
was to test this validity. Therefore, the error estimator δm was defined as the ratioδm =Mt Mr sMt Mr `=

Ze,s Ya,s.
Ze,` Ya,`(5)

This ratio should tend towards unity (or 0 dB) for a perfect model of the acoustic transfer admittances Ya,(s,`) .
Otherwise, the estimated Mt Mr depends on the cavity dimensions, so the model is invalid.
During a reciprocity calibration, the electrical transfer impedance is measured using the insert voltage technique [3] to determine the ur0 of the receiver microphone. The current it through the transmitter microphone isdeduced from the voltage developed across a series-connected capacitor u = it /( jωC), knowing the value of C
(Figure 2). Thus, the electrical transfer impedance is measured based on two voltage ratios, as follows:Ze =−1 ur ut0,
jωC ut u0r

(6)

157

where ur /ut and ut0 /u0r are the ratios of voltages measured at the outputs of the microphone power supply of thereceiver (ur and u0r ) and transmitter (ut and ut0 ) microphones, respectively, during the main measurement phaseand voltage insertion phase.
A defined measurement process with two cavities is implemented by fixing the following variables: (a.) the
microphone, preamplifier, and conditioner combinations for the receiver and transmitter; (b.) the respective
settings of the conditioners, assuming that both measurement channels are stable.
Here, the error estimator δm is given byur,s /ut,s Ya,s
δm =
,
(7)
ur,` /ut,` Ya,`
where the subscripts s and ` represent the short and long cavities, respectively, in the voltage ratios. Note that
this simplification of the measurement process is of interest in the context of infrasound measurement, as thesemeasurements are time-consuming.
❶ Reciprocity setup❷ Static pressurechamber𝒖𝒓𝟎

𝒁∞

Receiverpreamplifier

Receivermicorphone

SealedvolumeSapphiresealed cavity
Transmitter
microphone

~

𝑪
𝒖

𝒁∞

Transmitter
preamplifier❸ Thermal control area

Figure 2. Measurement setup: Ê reciprocity setup with microphones, preamplifiers, and sealed sapphire cavity;
Ë static pressure chamber; Ì thermal control area.

3.2 Measurement Setup
As it was essential to perform the experiment in the infrasonic frequency range, i.e., from 0.04 Hz to 100 Hz,
some changes were required to the measurement setup employed in this study. The measurements were performed inside a regulated static pressure chamber (Figure 2) installed in a laboratory with a dedicated thermallycontrolled area. This controlled environment was required to avoid microphone instability due to static pressure
and temperature changes. Note that this is particularly important for measurements at infrasonic frequencies forwhich very long integration times are required.
The reciprocity system was composed of two microphones and their preamplifiers, which were sequentiallycoupled by two sapphire cavities. The transmitter and receiver microphones were two B&K Type 4160 1 inch
microphones, which are usually used for pressure reciprocity calibration (LS1p microphones). Two cavities were
especially designed and manufactured for the purposes of this study: a short (6 mm long) and long (10 mmlong) cavity. Their diameters fit the microphone membranes (18.6 mm) (see Figure 2).

158

The cavity lengths were chosen to be sufficiently different to allow measurement of the deviation between thethermal corrections γ − (γ − 1)E p incorporated in (1) for both cavities. For these cavity lengths, the deviationof the thermal corrections reached 0.3 dB in the isothermal-adiabatic transition frequency range; this could bemeasured with the given reciprocity system accuracy.
The receiver microphone was connected to a B&K 2669-L-004 preamplifier. The transmitter microphone wasconnected to a specific preamplifier designed and manufactured for the purposes of this study (Figure 2). The
latter had a cut-off frequency of approximately 0.005 Hz given by a 500 GΩ polarisation resistor, with additionof a 100 pF capacitance in parallel with the microphone. The output preamplifiers were connected to a 4 channel B&K Type 2829 microphone power supply. This conditioner was modified by bypassing the high-pass filter.
The signals were digitised by a VTI Instruments CMX09 chassis and an EMX4350 digitiser card. The digitisingsystem had a negligible noise level compared to that of the signal to be measured. The amplitude and phase of
the signals were computed using a standardised method given in [1].
To avoid acoustic short-circuiting and to obtain a sufficiently high signal-to-noise ratio at lower frequencies,
special attention was paid to the sealing of the reciprocity system. That is, the back cavity vents of both
microphones were sealed and the cavities were designed with gaskets to ensure optimal sealing conditions.
Another reason for sealing the microphones is to simplify the modelling of the microphone acoustic admittances,which is required for all acoustic transfer admittance formulations. Consequently, complex modelling [14] of
the microphone vent effects at low frequencies is not required, which places the experiment focus on validationof the thermal effects on the acoustic transfer admittance only. Given the frequency range of interest (lower
than 100 Hz), the microphone admittance (Yt and Yr ) is given in its simplest form [3] by
Yr,t =jωVeq,(r,t),γre f Pre f(8)

where Veq,(r,t) is the equivalent volume of the microphones, and γre f and Pre f are the specific heat ratio andstatic pressure at reference environmental conditions, respectively. For a fully rigorous discussion, is should
be noted that the back cavity of the microphone is also subject to thermal effects and its equivalent volume
should be dependent on the frequency when the acoustic behaviour is no longer adiabatic. However, in theexperiment conducted in this study, the equivalent volumes of the LS1p microphones were much lower than thevolume of the smallest cavity (V /Veq,(r,t) ≈ 20). Therefore, these complex effects were assumed to be negligible.

3.3 Measurement processing
As the cavities and microphones are necessarily sealed, the local environmental variations inside the reciprocity
system (i.e., those of the back cavities of the microphones and coupler) have an important effect on its stability.
This is true even if the environment inside the static pressure chamber is controlled.
To overcome this problem, a specific measurement process was implemented, explained in [19].

4Results and discussion

Figure 3 shows the error estimator δm as defined in (7) as a function of frequency, for the acoustic transfer
admittances derived from the formulations discussed. The uncertainties are mainly due to the repeatability
process. Analysis of the error estimator δm results yields the following findings:
1. If δm does not tend towards zero (unit: dB), the estimated Mt Mr depends on the cavity dimensions.
Thus, the formulation of the acoustic transfer admittance is invalid.
2. If δm tends towards zero (unit: dB), the estimated Mt Mr does not depend on the cavity dimensions. Thus,
the formulation of the acoustic transfer admittance is valid.
3. Other unknown and unaccounted for effects somehow compensate for each other by coincidence, for the
chosen coupler sizes. However, such possibility appears to be unlikely.

159

1.0

Amplitude de−iation / dB

0.5
0.0
10.5
11.0

General alternati−e lo.-frequency solution
Short-term alternative low-frequency solution
IEC 61094-2:2009; "Low-frequency solution"
IEC 61094-2:2009; "Broadband solution"

−1.5

−2.0

0.04

0.1

8

10

100

General alternative low-frequency solution

Short-term alternative low-frequency solution
IEC 61094-2:2009; "Low-frequency solution"
IEC 61094-2:2009; "Broadband solution"

6
Phase deviation / degree1Frequency / Hz

4
2
0
−2
−4

0.04

0.1

1
Frequency / Hz

10

100

Figure 3. Amplitude (unit: dB, upper graph) and phase (unit: degrees, lower graph) of error estimator δm asfunction of frequency for four different acoustic transfer admittance formulations.
It is worth noting that, for case (2.), the formulation can be considered as valid provided the cavity lengths
are sufficiently different that the effects under investigation (here, the heat conduction effects) can be measured.
This hypothesis was verified in the present study (see Section 3.2).
It is clearly apparent from Figure 3 that, among the studied formulations, the general alternative low-frequency
solution (1) is the unique valid model in the targeted frequency range (for amplitude and phase). It is also
reminded here that the results for the phase were obtained without applying the environmental correction processdescribed in the previous section. These results are comparable to those obtained for the amplitude where
the correction process has been applied. The polytropic condition that occurs in the frequency range of 0.1–
10 Hz (for these cavity dimensions) was well corrected by this formulation. The result provided by the generalalternative low-frequency solution was calculated using (m, n) = (100, 100) for EP in (2); convergence study
shows results within 0.01 dB for (m, n) = (17, 17) at 100 Hz and (m, n) = (2, 2) at 0.04 Hz.

160

As expected, the short-term alternative low-frequency solution (3) provided a better result than the standardised‘low-frequency solution’ and ‘broadband solution’. These results highlight the limitations of the current standardised formulations of acoustic transfer admittance for the purpose of microphone infrasound calibration.
To obtain information on the possible error of microphone calibration using the pressure reciprocity technique,a method is to present the acoustic transfer admittances relative to the general alternative low-frequency solution,
taken as a reference. As an example, for the standardised ‘low-frequency solution’, the possible errors in the
acoustic transfer admittance, and thus, in Mt Mr reached 0.1 dB and 0.5 degrees at 1 Hz, and up to 3 dB and
30 degrees at 0.04 Hz. The error in the sensitivity estimation was potentially half these values. Therefore,
traceability to the International System of Units (SI) for current calibrations is possibly incorrect.

5

CONCLUSION

The main motivation of this study was to perform groundwork for future primary calibration of microphones
in the infrasonic frequency range. Therefore, it was essential to verify the validity of the acoustic transfer
admittance formulations for cylindrical cavities at infrasonic frequencies, which are currently standardised andused for primary reciprocity calibration of microphones.
An experiment performed to test the validity of the formulations discussed in this paper clearly indicated thatthe general alternative low-frequency solution (1) is the only valid model among the studied formulations in thetargeted frequency range (for amplitude and phase). From the experiment results, it was also concluded that the
short-term alternative low-frequency solution (3) yields lower errors than the standardised solutions. Finally, theexperiment highlighted the limitations of the current standardised formulations of acoustic transfer admittancefor infrasound calibration of microphones.
In conclusion, the models quoted in IEC Standard 61094-2:2009 are not suitable at low frequencies. The following recommendations can be made for future revision of the IEC standard:
1. The current standardised ‘low-frequency solution’ should be modified by the short-term alternative lowfrequency solution as defined in (1) and (3), as the validity of the former solution is limited at low
frequencies by the asymptotic development of the general formulation of EP , presented in (2).
2. At lower frequencies, where the previous solution is no longer valid, the general alternative low-frequency
solution should be implemented, as defined in (1) and (2).
The findings of this work have implications for calibration of infrasound sensors, which is particularly importantfor earth monitoring applications. It would be advisable that in the near future, calibrations of sensors at
infrasonic frequencies through reciprocity method as well as others methods based on closed couplers, such as
the laser pistonphone, take into consideration these recommendations.

161

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