Frequency response and design parameters for ...

Frequency response and design parameters for differential microbarometers Johan H. Mentinka) and La¨slo G. Evers Seismology Division, Royal Netherlands Meteorological Institute, De Bilt, The Netherlands

(Received 24 June 2010; revised 12 May 2011; accepted 12 May 2011) The study of infrasound is experiencing a renaissance since it was chosen as a verification technique for the Comprehensive Nuclear-Test-Ban Treaty. Source identification is one of the main topics of research which involves detailed knowledge on the source time function, the atmosphere as medium of propagation, and the measurement system. Applications are also foreseen in using infrasound as passive probe for the upper atmosphere, taking the field beyond its monitoring application. Infrasound can be conveniently measured with differential microbarometers. An accurate description of the instrument response is an essential need to be able to attribute the recorded infrasound to a certain source or atmospheric properties. In this article, a detailed treatment is given of the response of a differential microbarometer to acoustic signals. After an historical introduction, a basic model for the frequency response is derived with its corresponding poles and zeros. The results are explained using electric analogs. In addition, thermal conduction is added to the model in order to capture the transition between adiabatic and isothermal behavior. Also discussed are high-frequency effects and the effect of external temperature variations. Eventually, the design parameters for differential microbarC 2011 Acoustical Society of America. [DOI: 10.1121/1.3596718] ometers are derived. V PACS number(s): 43.28.Dm, 43.38.Kb, 43.28.Tc [AJZ]

I. INTRODUCTION

Infrasound was first discovered after the violent eruption of the Krakatoa in Indonesia (1883). Small pressure disturbances, with respect to the ambient pressure, were noted on traditional barographs. Further analysis of the worldwide recordings revealed that these low-frequency acoustic waves, i.e., inaudible sound, traveled around the globe up to four times.1 The first microbarograph recordings of an infrasound event date from 1908 when a large meteoroid, or asteroid, exploded over Tunguska in Siberia (Russia).2 These microbarographs were actually constructed to measure small air pressure fluctuations associated with severe weather. In their 1904 article, Shaw and Dines3 end the introduction with: it is proposed to call the apparatus a Micro-Barograph. A hollow cylindrical bell was used which floated in a vessel containing mercury. The interior communicated through a thin pipe with a closed reservoir containing air. A very small leak was allowed, i.e., the low-frequency cut-off. The reference volume was enclosed in a larger cylinder where the intervening space was packed with feathers or some other insulating material to avoid pressure fluctuations due to temperature changes. A decrease in atmospheric pressure would raise the cylindrical bell in the mercury. This change is recorded on paper by pen.3 This design was based on the earlier work by Wildman Whitehouse who modified the sympiesometer invented by Alexander Adie (Edinburgh, 1818).4 Societal and scientific interest in infrasound increased during World War I and onward. A remarkable development

a)

Author to whom correspondence should be addressed. Present address: Institute for Molecules and Materials, Radboud University, Nijmegen, The Netherlands. Electronic mail: [email protected]

J. Acoust. Soc. Am. 130 (1), July 2011

Pages: 33–41

took place during 1939 when two seismologists, Benioff and Gutenberg, combined their knowledge from seismology with their interest in atmospheric processes. Benioff had designed an electromagnetic seismograph and Gutenberg was interested in the structure of the earth and of the layering of the atmosphere. Their instrument, a loudspeaker mounted in a wooden box, connected very easily to the equipment that was in use in the seismological community.5 Benioff and Gutenberg discovered oscillations on their microbarograph, which they called microbaroms,6 a name derived from microseisms that were used in seismology. Microbaroms are almost continuously present in the atmosphere and follow from the non-linear interaction of interfering oceanic waves. In the nuclear testing era, infrasound was used to monitor atmospheric nuclear explosions7 and to obtain yield estimates.8 This interest diminished as nuclear tests were confined to the subsurface under the Limited Test Ban Treaty in 1963. Recently, the study of infrasound gained renewed attention as it was chosen as a verification technique for the Comprehensive Nuclear-Test-Ban Treaty (CTBT) that opened for signing in 1996. A worldwide network of 60 infrasound arrays is being established to verify the treaty with respect to atmospheric nuclear tests.9 Arrays are deployed to reduce the effect of noise due to wind and to determine the direction of the arrival of the wave and its apparent propagation velocity. Additional noise reduction at each array element is achieved by a wind barrier, porous hose, or pipe array with discrete inlets10 (see Fig. 1). An infrasound sensor, as an array element, must be broadband to be able to measure energy from a wide variety of sources. It should also sense the associated air pressure fluctuations, which are in the range of millipascals to tens of pascals. Among anthropogenic sources are explosions, supersonic flying, nuclear tests, mining, and military activities.

0001-4966/2011/130(1)/33/9/$30.00

C 2011 Acoustical Society of America V

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FIG. 1. Schematic drawing of the field setup with the KNMI-mb. T1, T2, and T3 indicate customarily placed temperature sensors to monitor temperature variations. Six porous hoses are applied in a star-like configuration, each hose has a length of 6 m.

Natural sources comprise: avalanches, oceanic waves, severe weather, sprites, earthquakes, meteors, lightning, volcanoes, and aurora.11 In addition, infrasound sensors are capable of detecting non-acoustic phenomena, like gravity waves. Various instruments have been developed that have the ability to detect these anthropogenic and natural sources, such as differential12 and absolute microbarometers,11,13 vector sensors,14 loudspeakers,5 and piezo-electrical15 and fiber-optical devices.16 Some other earlier developments for measuring infrasound include the microbarovariograph,17 the solion (electrochemical) infrasonic microphone,18 an electric variometer used in sailplanes,19 and a mercury tilt meter.20 Infrasound can not only be used for verification purposes but also has the potential to image the atmosphere. Recent efforts are directed toward acquiring actual observations of upper atmospheric winds and temperatures from infrasound recordings.21 The aim is to validate upper atmospheric models with these observations and obtain information on a fine temporal and spatial scale.22 Here we show the constructional efforts for a differential microbarometer that is capable of measuring infrasound in the range of 1000 s up to 20 Hz. Not only acoustic waves and events of CTBT interest occur in this passband, but also gravity waves, i.e., internal waves with gravity as restoring force.23 The instrument was developed at the authors’ institute and will be referred to as the KNMI microbarometer (KNMI-mb). A differential microbarometer was chosen because it has several advantages when compared to other instruments. These are its simple configuration, low price reproducibility, tunable response, durability, and minimal mechanical response.24 The latter will not be considered in detail in this article since the lack of masses in the KNMImb prohibits a seismic response.25 In this article, attention 34

J. Acoust. Soc. Am., Vol. 130, No. 1, July 2011

will be paid to the response of the KNMI-mb and considerations regarding the adiabatic and isothermal behavior. Temperature stability of a differential microbarometer is a concern and will be addressed based on field measurements. This work will be concluded with a summary of the design parameters as follows from the various topics that have been studied. II. FREQUENCY RESPONSE

The basic theory for the linear response of a differential barometer was derived by Burridge.26 In this section we review this theory by giving a convenient interpretation using electric analogs and derive the frequency response. The significance of non-linear effects is discussed in the Appendix. Empirical treatments of the response of differential microbarometers are discussed by Cook and Bedard,12 Christie et al.,27 and Richiardone.28 The starting point for the derivation of the response is the ideal gas law for a perfect gas: pvc ¼ constant. Here, p is the pressure, v is the specific volume of the gas, and c ¼ cp =cv is the ratio of specific heats at constant pressure and volume, respectively. By setting c ¼ 1 isothermal behavior can be derived similarly. Since acoustic pressure disturbances p0 are typically small compared to the static ambient pressure PA, we can work in the linear regime. Differentiation of the ideal gas law with respect to time then gives 1 dp0 1 dv þc ¼ 0: PA dt v dt

(1)

In the frequency range of interest, acoustic wave lengths are much larger than the dimensions of the enclosure. J. H. Mentink and L. G. Evers: Design parameters for microbarometers

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Therefore, we can neglect the spatial variation of the specific gas volume over the instrument. Hence, we replace v by the total enclosure volume V and the volume change dv=dt by the net volume flux f through the opening of the enclosure. This is the so-called lumped-parameter model. Together with the linearization, this allows for a convenient representation with electrical analogs, where acoustical pressure p0 and volume flux f are interpreted as electric voltage u and current i. The relations between the acoustic components and their electrical equivalents are defined as follows. The equivalent electrical resistance R ¼ u=i is defined as the ratio q ¼ p0 =f between the acoustical pressure and the volume flux. The equivalent electric capacitance of a physical volume V is defined as C ¼ V=ðcPA Þ. Also the mechanical sensitivity of the diaphragm is represented as capacitance Cd ¼ DV=DP, where DV is the volume change and DP is the pressure change. Figure 2 shows both the physical and electric analog schema of a differential microbarometer. Note that in principle the inlet impedance also has an inductive component, see also Sec. III, which can be neglected in first approximation. The quantity of interest is the response function H  ðu1  u2 Þ=u, which gives the relation between the signal u1  u2 sensed by the diaphragm and the input signal u. The response can be derived in the harmonic approximation d=dt ¼ ix by application of the familiar Kirchoff laws. For the electric analog circuitry this results in five linear equations for the five unknowns i; i1 ; i2 ; u1 ; u2 , which can be expressed in terms of u. Using that R2 and Cd are in parallel, we obtain

ixs2

HðixÞ ¼

1 þ ixs2 A þ ðixÞ2 s1 s2 B A ¼ 1 þ s1 =s2 þ R1 =R2 þ Cd =C2 ;   1 B ¼ 1 þ Cd C1 1 þ C2 :

i2 ¼ ixC2 u2 ;

(2)

u1  u2 ¼ i2 R2 =ð1 þ ixR2 Cd Þ; u  u1 ¼ iR1 : Solving the set of equations for the frequency response HðixÞ gives

(3)

The essential functioning of the microbarometer can be understood by considering the time constants s1 ¼ R1 C1 and s2 ¼ R2 C2 , where C1 and C2 are the capacities of the fore and backing volume, respectively. R1 and R2 are the resistances of the inlet and capillary. For low frequencies we obtain HðixÞ ! ixs2 ðx ! 0Þ;

(4)

which illustrates that frequencies much smaller than 1=s2 are averaged out. In the high-frequency limit we find HðixÞ ! ðixs1 BÞ1 ðx ! 1Þ:

(5)

Hence, 1=2ps1 determines the high-frequency cutoff, provided that the diaphragm has a small volumetric displacement per unit pressure compared to the fore and reference volume, such that Cd  C1;2 . This analysis shows that a differential microbarometer acts as a bandpass filter in the frequency domain. Next it is of interest to determine the behavior of the response function in the passband: HðixÞ !

1 ðs1 < x < s1 1 Þ: (6) 1 þ s1 =s2 þ R1 =R2 þ Cd =C2 2 |ffl{zffl} |fflffl{zfflffl} |fflfflffl{zfflfflffl} 1

i ¼ i1 þ i2 ; i1 ¼ ixC1 u1 ;

;

2

3

In this regime the response is constant, and the question is how to design the microbarometer such that this constant is unity. There are three contributions in the denominator: (1) s1  s2 is necessary for a proper broadband instrument, such that the pressure in the reference volume is approximately constant in the frequency range of interest. (2) R1  R2 is required to have the measured signal closest to the actual signal at the inlet. For this, the voltage drop over R1 should be negligible compared to the voltage drop over R2. (3) Cd =C2  1 ensures that the volumetric displacement of the diaphragm is negligible compared to the total reference volume. From these small terms the third contribution is largest in the KNMI-mb: Cd =C2  0:07, for the isothermal case, while the other terms satisfy 5s1 =s2  R1 =R2  104 . The value of Cd follows from the sensor specifications. Note that R1  R2 is required even when C1  C2 is satisfied. Conversely, when R1  R2 , C1 and C2 are allowed to be in the same order. The inlet resistance R1 is determined by the porous hoses from the wind-noise reducer and is determined experimentally.29 The resistance R2 is made from a capillary tube, and the value can be estimated with Poiseuille’s law:

FIG. 2. Schematic drawing of a differential microbarometer (a), and its electric analog (b). J. Acoust. Soc. Am., Vol. 130, No. 1, July 2011

R2 ¼

8ll : pr 4

J. H. Mentink and L. G. Evers: Design parameters for microbarometers

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(7) 35

TABLE I. Parameter values and standard conditions used in the computations. Fore volume Reference volume Capillary length Capillary radius Porous hose resistance Capillary resistance Diaphragm sensitivity Temperature Ambient pressure Density Cp =Cv Dynamic viscosity Thermal diffusivity

V1 ¼ 2:20  105 m3 V2 ¼ 1:03  104 m3 l ¼ 1:00  101 m r ¼ 1:00  104 m q1 ¼ 6:33  106 kg m4 s1 q2 ¼ 4:63  1010 kg m4 s1 Cd ¼ 7:5  1011 m4 s2 kg1 T ¼ 20:0  C PA ¼ 1:01  105 Pa . ¼ 1:21 kg m3 c ¼ 1:40 l ¼ 1:82  105 kg m1 s1 a ¼ 2:12  105 m2 s1

Here l is the dynamic viscosity, l the length of the capillary, and r the inner radius of the capillary. The relevant numbers for the KNMI-mb are summarized in Table I. The full response for these numbers is shown in Fig. 3 for both isothermal (dashed line) and adiabatic (dash-dotted line) cases.

FIG. 4. Schematic illustration of the poles () and zero () of the transfer function. The dashed lines show how s changes due to high-frequency inlet effects. The pole sD is due to the wind-noise reducer of diameter D.

III. POLES, ZEROS, AND HIGH-FREQUENCY EFFECTS

Poles and zeros provide a representation of the response that is convenient when discussing adaptions to models. In this section we derive the poles and zeros of the differential microbarometer and use it to discuss the importance of highfrequency effects. Poles and zeros are properties of the so-called transfer function HðsÞ  HðixÞ, where s ¼ ix and HðixÞ is the frequency response of Eq. (3). The transfer function is a second order polynomial in s, hence we can write it in the general form: HðsÞ ¼

Ds : ðs  s Þðs  sþ Þ

(8)

By equating Eq. (8) with Eq. (3) we obtain D1 ¼ s1 B

(9)

and A 1 6 s6 ¼  2s1 B 2s1 B

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s  s1 R1 Cd 2 R1 1  þ þ 4 : (10) s2 R2 C2 R2

FIG. 3. Frequency response HðixÞ of the KNMI-mb for amplitude [top (a)] and phase [bottom (b)]. s1 1;2 indicate the approximate values of the corner frequencies. The deviation from a unit response in the passband is then determined by small quantities as discussed in the main text. Thermal conduction induces a transition from adiabatic behavior at high frequencies to isothermal behavior at low frequencies in the vicinity of the transition frequency x2 .

The constants s6 are called poles since the transfer function diverges for these values of s. The only zero of the transfer function is s ¼ 0. Figure 4 illustrates the poles and zero of the transfer function in the complex s-plane. Both poles are situated on the negative real s-axis, as expected for a passive instrument. Since the poles have no imaginary part, oscillatory behavior is absent. A single pole on the negative real axis corresponds to a low pass filter. A combination of a zero at s ¼ 0 and a pole with a (small) negative real s-value is equivalent to a high pass filter. Hence, these poles and zero also show that the microbarometer acts as a bandpass filter in the frequency domain. The poles and therefore the corner frequencies are closely related to the time constants s1 and s2 . This becomes apparent when we expand s6 to first order in the small quantities R1 =R2 and Cd =C1;2 :

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J. H. Mentink and L. G. Evers: Design parameters for microbarometers

J. Acoust. Soc. Am., Vol. 130, No. 1, July 2011

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sþ  ð1  R1 =R2  Cd =C2 Þ=s2 ;

(11)

s  ð1 þ R1 =R2  Cd =C1 Þ=s1 :

(12)

Note that the non-zero Cd shifts both poles to larger (less negative) values. Next we use the poles and zeros to illustrate high-frequency effects. The inlet resistance shows inductive behavior when frequency is increased. This can be modeled by replacing R1 ! R1 þ ixL1 , with L1 as the inductance associated with the inlet. For a long cylindrical pipe of length l and cross section A, the value can be estimated as L1 ¼ .l=A. This inductance induces Helmholtz-resonator like behavior with a characteristic frequency of about xc1 ¼ ðL1 C1 Þ1=2 ’ 103 s1 . The corresponding change of the transfer function can be represented in terms of zeros and poles. Introduction of the inductance splits the high-frequency pole s in two complex conjugate poles shifted to the right, as illustrated in Fig. 4. The effect on sþ can be neglected in first approximation as it involves R1 =R2 , which is a small term. It is important that such high-frequency signals are not picked up by the microbarometer. In practice this is ensured by the porous hoses of the wind-noise reducer. Porous hoses are used in a star-like geometry to reduce wind noise, see Haak and de Wilde29 and Evers and Haak.30 For acoustic signals the wind-noise reducer acts as a low pass filter since most of the acoustic energy propagates in the horizontal plane. Wavelengths shorter than roughly twice the diameter D of the noise reducer are attenuated, resulting in a corner frequency xD ’ pc=D, where c is the speed of sound. The total response changes due to the additional pole sD of the wind-noise reducer, as indicated in Fig. 4. In practice, D ’ 10 m, resulting in xD ’ 102 s1 . Consequently, acoustic resonances are not excited. A similar analysis can be carried out to also show that mechanical resonances of the sensor are not excited. IV. THERMAL CONDUCTION

Due to the presence of heat conduction inside the microbarometer, the response of the microbarometer can be considered neither isothermal nor adiabatic. Generally a transition between these two types of behavior occurs inside the frequency range of interest.31 In this section we explain the effect of heat conduction, show how it can be incorporated in the model of Sec. II, and discuss its implications on the microbarometer response. Effects of thermal conduction in relation with calibration experiments are discussed by Richiardone.28 The difference between adiabatic and isothermal behavior is determined by difference in the compression (and expansion) of the gas. Isothermal behavior (c ¼ 1) corresponds to compression at constant temperature and occurs in small volumes. Conversely, when the entropy is constant during compression and the compression is reversible, adiabatic behavior (c ¼ cp =cv ) is obtained. The latter resembles the case of a sound wave in free space. The effect of thermal conduction on the sound wave in an enclosure is governed by the ratio of the characteristic J. Acoust. Soc. Am., Vol. 130, No. 1, July 2011

size L ¼ V=S of the enclosure and the thermal penetration pffiffiffiffiffiffiffiffiffiffiffi depth dT ¼ 2a=x. Here V and S are volume and inner surface of the enclosure, respectively, and a ¼ j=.cp is the thermal diffusivity of air, defined as the ratio of thermal conductivity j to heat capacity per unit volume .cp . Adiabatic behavior is obtained in the high-frequency limit when dT  L, whereas isothermal behavior corresponds to low frequencies such that L  dT . The effect of thermal conduction on acoustic compliances (capacitance in terms of electric analogs) was derived by Daniels.32 For temperature fluctuations that are small compared to the ambient temperature again the harmonic approximation can be employed, and analytical results can be obtained for specific enclosure shapes. The result is the replacement of c with a complex-valued c ¼ cK, where the frequency-dependent prefactor K is a function of dT =L, and therefore depends on x and V/S. The prefactor has the following structure:  1=2 ; jKj ¼ X2 þ Y 2

argðKÞ ¼

(13)

p X þ arctan ; 2 Y

(14)

where X ¼ xðc  1Þ  c;

(15)

Y ¼ yðc  1Þ:

(16)

Specific forms for xðdT =LÞ and yðdT =LÞ are given in Table II. These idealized geometries model the microbarometer components rather well, i.e., the typical length scales are representative. The correction factor K becomes real-valued in the adiabatic and isothermal limit, with limiting values K ! 1 and K ! c1 , respectively. In between these limits, the imaginary part of K damps and phase shifts the measured signals. The maximum effect is seen at about V=ðSdT Þ ¼ 1, and we use this to define the transition frequency x ¼ 2a=L2 . The imaginary part of K can be seen as a frequency-dependent resistance parallel to the enclosure capacitance.33 As a consequence, the position of the poles becomes frequency dependent and will deviate from the negative real axis.

TABLE II. Definition of x þ iy for three different enclosure types. The length scale L is defined as the ratio of volume V and surface S. b ¼ ð1 þ iÞ=dT and f ¼ ð2iÞ1=2 L=dT , with dT the thermal penetration depth. J0 and J1 are order zero and one Bessel functions of the first kind. Enclosure type characteristic size

L ¼ V=S

x þ iy

Sphere radius r Rectangular box thickness 2r Long cylinder radius r

r=3

3 1  Lb coth Lb þ L23b2

r

Lb 1  tanh Lb

r=2

ðfÞ 1  2f JJ01 ðfÞ

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Differential microbarometers are sensitive to temperature gradients across the microbarometer. This can be easily understood from the ideal gas law: dp=dT ¼ P=T ¼ 345 Pa/ C under standard conditions of 1010 hPa and 20  C. Sudden temperature variations appear as noise on infrasound measurements. By proper thermal insulation the amplitude and frequency of temperature-induced pressure variations can be

strongly reduced. To this end, the KNMI-mb is placed in a pit below the earth surface, as illustrated in Fig. 1. In this section we present temperature measurements of the insulated KNMImb to show that there is no correlation between temperature changes and pressure changes in the passband. We subsequently discuss experimental setup, show the measurements, and discuss the results. We end the discussion with some remarks on influence of temperature on the response function itself. The experimental setup consists of the standard field setup, shown schematically in Fig. 1. Temperature sensors are customarily placed beneath the cover, the upper slice of Styrofoam insulation material, and at the outer-side of the reference volume, indicated by T1, T2, and T3, respectively. The sensors are type K (chrome-aluminum) thermocouples. The output of the sensors is recorded using a Testo 177-T4 datalogger at a sampling rate of 30 s. The experiment was carried out in the period from June 18, 2007 until June 20, 2007 in pit 17 of the Deelen Infrasound Array.34 The results of the temperature measurements are shown in Fig. 5.36 The lower two panels show the temperature and mean pressure of the microbarometer. The mean is calculated using a sliding window with a fixed length of 1000 s. The window length equals the longest period of interest. Using the same window length, we also calculated the standard deviation of the mean temperature as function of time. This gives us a quantitative measure of the temperature fluctuations. The same is done for the pressure and the results are shown in the upper two panels of Fig. 5. Next we discuss the results. The bottom panel of Fig. 5 shows that temperature fluctuations in the bottom of the pit are significantly reduced compared to the temperature variations in the top of the pit. This is also true for the temperature fluctuations. For the bottom of the pit, we have 0.2  C as maximum standard deviation, while this is 1.9  C for the top of the pit. As a consequence, temperature-induced pressure fluctuations are due to temperature variations in the fore volume and hoses of the noise reducer. We observe a clear correlation between mean microbarometer pressure and the gradual changes of temperature on June 19. However, there is no correlation between variations in temperature and microbarometer pressure on that day. Hence, this gradual temperature variation is not reflected in the frequency range of interest. Note also that the amplitude of the fluctuations is not affected by the gradual external temperature variation. Moreover, no causal correlation exists between the largest pressure fluctuations and the largest temperature fluctuations, which occur on day 18 of the observation period. Checks with longer window lengths, up to 10 000 s, show similar results. Finally, note that the temperature fluctuations in the bottom of the pit are equal to the resolution of the thermometer and therefore cannot be related to pressure fluctuations. Based on the results shown, we conclude that temperature gradients may induce pressure variations in the microbarometer, but these occur far outside the frequency range of interest. We therefore conclude that the arrangements made to reduce temperature fluctuations are sufficient. Note that temperature changes outside the frequency range of interest are reflected as changes of equilibrium temperature. We end the discussion on temperature effects with some

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J. H. Mentink and L. G. Evers: Design parameters for microbarometers

A differential microbarometer consists of a reference volume and a fore volume, which are generally of different size. Hence the transition between adiabatic and isothermal behavior also occurs at different frequencies. In the KNMI-mb the fore volume has a circular shape with a radius much larger than its thickness. Hence a reasonable model is the rectangular box with 2r1 ¼ L1 ¼ 6 mm, giving x1 ’ 1 s1 . The backing volume is also circular, but height and radius are in the same order. Here the sphere with r2 ¼ 3L2 ¼ 31 mm is a better approximation, resulting in x2 ’ 0:4 s1 . Using the appropriate formulas for K, the total response is computed by substituting Cj ¼ Cj =K for the capacity Cj. The resulting response is shown in Fig. 3 by the solid line, and illustrates that including thermal conduction smoothly connects the adiabatic and isothermal limiting cases. Next, we discuss how we can explain this behavior in terms of the transition frequencies x1 and x2 of fore and reference volume. The response at frequencies above s1 1 is adiabatic since x1 s1  1. Similarly, because x2 s2  1 the response at frequencies below s1 2 is isothermal. Hence, the main effect of thermal conduction appears in the passband, where the response is determined by HðixÞ !

1 : 1 þ R1 =R2 ð1 þ C1 =C2 Þ þ Cd =C2

(17)

Because the microbarometer must be broadband with unit response in the passband, we have R1 =R2  1. Therefore, the effect of thermal conduction in the second term of the denominator is negligible. Consequently, effects of thermal conduction are due to the term Cd =C2 . Interestingly, it is advantageous to work with a large reference volume to reduce the effect of thermal conduction. This seems to contradict previous literature,12 where it was suggested to fill the reference volume with steel wool to ensure an isothermal response over the whole frequency range of interest. However, for the KNMI-mb, the effect of thermal conduction on the amplitude is DjHj ¼ ðc  1ÞCd =C2  3% and on the phase less than 1 degree. Hence the effect of thermal conduction is small and filling of the reference volume with steel wool is not needed. Such a filling can be beneficial to suppress convectional noise in the backing volume. It should be noted that Cd =C2  1 implies that the diaphragm is sufficiently stiff such that the stiffness of air does not significantly contribute to the sensitivity of the diaphragm. We conclude that although the value of c changes with 40%, the net effect is small since the adiabatic to isothermal transition occurs in the passband where the difference between adiabatic and isothermal behavior is small. V. TEMPERATURE STABILITY

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FIG. 5. Measured temperatures and differential pressure (lower two panels), and their corresponding fluctuations binned in 1000 s. windows (upper two panels) for June 18 up to June 20, 2007.

remarks on the influence of the equilibrium temperature on the response function. The dominant effect is due to the temperature-dependent dynamic viscosity, which appears in the capillary resistance R2. The viscosity increases with temperature, thereby inducing a decrease of the corner frequency of about Ds2 =s2 < 0:3%/ C. Since R1  R2 , this has no significant effect in the passband. In addition, the sensor’s sensitivity depends on the temperature. We use a DP 45 sensor from Validyne engineering. Both from the specifications and from the above-shown measurements we find that the influence of the equilibrium temperature on the sensitivity is negligible.35 VI. SUMMARY AND DESIGN PARAMETERS

Differential microbarometers are conveniently used for measuring infrasound. An accurate description of the instruJ. Acoust. Soc. Am., Vol. 130, No. 1, July 2011

ment response is an essential need to be able to attribute the recorded infrasound to a certain source or atmospheric properties. In this article a detailed treatment is given of the response of a differential microbarometer. For acoustic signals, a differential microbarometer acts as a bandpass filter. The high cutoff frequency is determined by the response time s1 of the inlet resistance R1 and capacitance of fore volume C1. The low cutoff frequency is determined by the relaxation time s2 of the capillary resistance R2 and capacitance of reference volume C2. Deviations from the designed low cut-off frequency are due to thermal conduction, non-zero volumetric displacement Cd of the diaphragm, viscosity of air changes with temperature, and high-frequency effects of the inlet. Thermal conduction is needed to model the transition between adiabatic behavior (c ¼ cp =cv ) at high frequencies and isothermal behavior (c ¼ 1) at low frequencies. The

J. H. Mentink and L. G. Evers: Design parameters for microbarometers

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parameters x1 and x2 of the fore and reference volume characterize the frequency at which the influence of thermal conduction is most apparent. Although c changes by 40%, the net effect is small, since the transition occurs in the passband, where the difference between isothermal and adiabatic behavior is small. Remarkably, the effect of thermal conduction can be reduced by increasing the reference volume, since the dominant effect is caused by the term that involves the non-zero volumetric displacement of the diaphragm. Inlet effects occur at characteristic frequencies xc1 due to acoustic resonances in the small pipes of the microbarometer. These are suppressed by the wind-noise reducer, which acts as a low pass filter for acoustic signals with corner frequency xD . Since it is effectively xD that determines the high cut-off frequency, effects due to variations in the resistance of the porous hoses themselves are negligible as well. An ideal differential microbarometer and its noise reducer is broadband with a unit response in the passband. With this criterion we can derive the following design parameters:

linear effects may be of importance even when pressure amplitudes are small compared to barometric pressure, since at low frequencies flow effects are of importance. This is particularly the case for large flow velocities and when the flow in the microbarometer becomes turbulent. Here we argue that such effects are not significant. Undesired pressure fluctuations can occur inside pipes and as end effects. First we discuss the flow inside pipes, where the undesired pressure fluctuations are due to turbulent effects. Then we discuss end effects, which can induce pressure fluctuations for both laminar and turbulent flow. The stability of a fluid flow is determined by the Reynolds number Re. For small Re, the flow is laminar, while for large Re turbulence appears. For a long circular pipe, it has been established empirically that the flow remains laminar when Re < 2300. Here Re is defined as

(1) s1  s2 : Pressure variations in reference volume for signals in passband are negligible. (2) R1  R2 : Damping of signals in passband due to inlet resistance is negligible. (3) x1 s1  1, x2 s2  1: Effect of thermal conduction on corner frequencies is negligible. (4) Cd  C2 : Damping of signals due to displacement of diaphragm is negligible and the effect of thermal conduction in passband is negligible. (5) xD  xc : No excitation of acoustic resonances in the microbarometer. (6) s1 xD  1: Effect of variations of inlet resistance is negligible.

with mean flow velocity u ¼ f =ðpr2 Þ, pipe radius r, and kinematic viscosity  ¼ l=.. Here l is the dynamic viscosity and . the density of the gas. The right-hand side of Eq. (A1) is added to illustrate that Re becomes large for small pipe radii or when the volume velocity f becomes large. Given a pressure drop Dp over a pipe, the volume velocity is related to the pipe radius by the resistance R of the pipe:

Temperature effects are found to be reduced sufficiently by placing the differential microbarometer below the surface in an isolated pit. Results of temperature measurements showed that there is no correlation between temperature and pressure changes inside the passband. We expect that our findings are relevant for future design and construction of differential microbarometers, and for development of calibration procedures. In addition, our findings can be used to extract the instrument response from measured signals, in order to retrieve more accurate information of the amplitude and phase of low-frequency acoustic and gravitational pressure disturbances. This will enhance the capabilities of using infrasound for source identification and for passive probing of the atmosphere. ACKNOWLEDGMENTS

This work was carried out as part of the LOFAR project. The authors would like to thank Mico Hirschberg for fruitful discussions regarding the influence of thermal conduction. APPENDIX: NON-LINEAR EFFECTS

In this article, all acoustical modeling is restricted to the linear regime, where amplitudes of the excess pressure are small compared to their background values. However, non40

J. Acoust. Soc. Am., Vol. 130, No. 1, July 2011

Re ¼

f ¼

ur . f ¼ ;  pl r

Dp : R

(A1)

(A2)

Assuming Poiseuille’s law for the resistance, R ¼ 8ll=pr 4 , we obtain Re ¼

Dpr3 8. 2 l

(A3)

with pipe length l. The latter equation is only valid when the flow in the tube is laminar. Consequently, if Eq. (A3) renders Re < 2300, we can conclude that turbulent effects are absent. The small pipes in the microbarometer are the capillary tube and the inlet. For the flow inside the capillary, we compute Rec with the values r; l; .;  from Table I, using Dpc ’ 2  102 Pa we obtain Rec ’ 1:

(A4)

We find that for the pressure range of interest, Poiseuille’s law is indeed consistent with laminar flow in the capillary. In fact, the small capillary radius and long capillary length impedes a large volume velocity and thereby suppresses instabilities. For the inlet no non-linear effects are expected either, since the pressure drop over the inlet (porous hose and pipes) is much smaller than the pressure drop over the capillary. In the passband we can approximate Dpinlet ¼ ðR1 =R2 ÞDpc  104  102 Pa ¼ 102 Pa. Considering a single pore of the porous hose of length lphc ¼ 2 mm and radius rphc ¼ 0:1 mm, we find Rephc ’ 103 :

(A5)

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The same argument can be used for the resistance of the pipes of the microbarometer, since the porous hose resistance is much larger than the resistance of the pipes. It is therefore unlikely that vortices occur in the inlet. Next we discuss end effects. Here, undesired pressure disturbances are due to the appearance of a jet at the low pressure side of the pipe for large flow velocities. For both laminar and turbulent flow, we can estimate the excess pressure due to the jet as pjet ’ .u2 =2, with u the mean velocity in the pipe. To quantify the influence of the jet, we compare this excess pressure with the maximum pressure amplitude Dp ’ 2  102 Pa of interest. Using the equations introduced before we can write  2 pjet 1 f ¼ . Dp 2Dp pr 2  2 1 Dpr 2 Rec r . ’ 6  105 : ¼ ¼ (A6) 8ll 16 l 2Dp The pressure disturbance of the jet is more than four orders of magnitude smaller than the maximum pressure amplitude, hence in practice this is a negligible effect. Since the pressure drop over the inlet is much smaller, end effects are expected to be even smaller. Note, however, that the occurrence of a jet at the end of the capillary limits the dynamic range of the differential microbarometer. For example, consider a simultaneous measurement of a low-frequency signal with large amplitude and a high-frequency signal with small amplitude. In this case, unambiguous resolution of the smallamplitude signal will only be possible when its amplitude is larger than .u2 =2. In fact, the dynamic range can be estimated as the reciprocal of Eq. (A6): pmax Dp l2  / : pmin pjet Dpr 4

(A7)

Hence, the dynamic range decreases when the maximum pressure of interest is increased. For the KNMI-mb the dynamic range is approximately 102  102 Pa. In practice also pressure amplitudes p ’ 103 Pa, such as microbaroms, will be measurable in absence of signals with large amplitude, see, e.g., Ref. 30. 1

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J. H. Mentink and L. G. Evers: Design parameters for microbarometers

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