NISTIR 7526 

Determination of Wave Motion 

Correction Values Required for 

Comparison Calibrations of a New 

Working Standard Microphone 

Randall P. Wagner 

Victor Nedzelnitsky
NISTIR 7526 

Determination of Wave Motion 

Correction Values Required for 

Comparison Calibrations of a New 

Working Standard Microphone 

Randall P. Wagner 

Victor Nedzelnitsky 

Sensor Development & Application Group 

Manufacturing Metrology Division 

Manufacturing Engineering Laboratory

National Institute of Standards and Technology 

November 2008 

U.S. Department of Commerce 

Carlos M. Gutierrez, Secretary 

National Institute of Standards and Technology 

Patrick D. Gallagher, Deputy Director 
Determination of Wave Motion Correction 

Values Required for Comparison Calibrations of 

a New Working Standard Microphone 

Randall P. Wagner and Victor Nedzelnitsky 

National Institute of Standards and Technology, 

Gaithersburg, MD 20899-8220 

Comparison calibrations of microphones based on the reciprocity method are performed with 

both the microphone under test and a reference standard microphone inserted into an acoustic 

coupler. For these calibrations, the wave motion correction W is needed to account for the 

departure from spatially uniform sound pressure amplitude in the coupler at frequencies for 

which the coupler interior dimensions are not small compared to a wavelength of sound. This 

internal report describes the procedure used to determine the values of W4192 applicable to 

comparison calibrations of a relatively new working standard microphone, the Bruel and Kjaer 

model 41921, and provides these values. 

Key words: acoustic calibration; calibration of microphones; microphone calibration; NIST 

microphone calibration; reciprocity calibration; standard microphone; wave motion correction; 

working standard microphone. 

 

1

 Certain commercial equipment, instruments, or materials are identified in this paper in order to specify the 

experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by 

the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment 

identified are necessarily the best available for the purpose. 
TABLE OF CONTENTS

Page

1. INTRODUCTION/BACKGROUND ........................................................................................................... 1 

Fig. 1. Sectional view of reference standard microphone and test microphone inserted into an 

acoustic coupler for a reciprocity-based comparison calibration of the test microphone. .................... 2 

2. EXPERIMENTAL SETUP ........................................................................................................................ 3 

Fig. 2. Block diagram of the system used to measure the voltage ratio U2/U1 of receiver voltage to 

transmitter voltage. ................................................................................................................................ 5 

3. PROCEDURES FOR ACQUIRING WAVE MOTION CORRECTION DATA .............................................. 6 

Table 1a. List of test frequencies at which measurements were made in the 4 cm3

 coupler to 

determine values of W4192, which are values of the wave motion correction applicable to reciprocitybased comparison calibrations of a Bruel and Kjaer model 4192 microphone. .................................... 7 

Table 1b. List of test frequencies at which measurements were made in the 20 cm3

 coupler to 

determine values of W4192, which are values of the wave motion correction applicable to reciprocitybased comparison calibrations of a Bruel and Kjaer model 4192 microphone. .................................... 7 

4. DATA REDUCTION TO OBTAIN WAVE MOTION CORRECTION VALUES AT CALIBRATION 

FREQUENCIES IN HYDROGEN-FILLED COUPLERS ............................................................................. 8 

5. RESULTS AND SUMMARY ..................................................................................................................... 9 

Table 2. Values of the wave motion correction W4192, and 20 log W4192, applicable to the NIST 

fixed-cost comparison calibrations of a Bruel and Kjaer model 4192 microphone performed with the 

hydrogen-filled 20 cm3

 coupler.. ......................................................................................................... 10 

Table 3. Values of the wave motion correction W4192, and 20 log W4192, applicable to the NIST 

fixed-cost comparison calibrations of a Bruel and Kjaer model 4192 microphone performed with the 

hydrogen-filled 4 cm3

 coupler.. ........................................................................................................... 10 

6. UNCERTAINTIES OF WAVE MOTION CORRECTION MEASUREMENT RESULTS ............................. 11 

Table 4. Summary of uncertainties of the measured wave motion correction values .................... 12 

7. REFERENCES ...................................................................................................................................... 13

i 
1. INTRODUCTION/BACKGROUND

The National Institute of Standards and Technology (NIST) fixed-cost acoustic 

measurement services SP25010C through SP25040C [1] determine the pressure sensitivity 

level of a customer’s microphone (the test microphone) by using a reciprocity-based method 

for comparison calibration with reference standard microphones. Each such reference 

standard microphone is a NIST-owned International Electrotechnical Commission (IEC) and 

American National Standards Institute (ANSI) Type LS1Pn laboratory standard microphone 

[2,3] that has been calibrated by the reciprocity method [4-6] to determine both its sensitivity 

and its driving-point electrical impedance, which are used in the comparison calibration. 

During this comparison calibration, the reference standard microphone and the test 

microphone are inserted into ports on opposite sides of an acoustic coupler, which has an 

internal cavity of known volume between the microphone diaphragms, as shown in Fig. 1. 

The reference standard microphone is electrically driven and used as a sound source, called 

the transmitter (microphone 1), and the test microphone is used as a receiver (microphone 2), 

which produces an output voltage in response to the sound pressure at its diaphragm in the 

coupler. The ratio of open-circuit voltage U2 at the output of the test microphone, to the 

voltage U1 driving the reference standard microphone is measured. At each frequency of 

calibration, measurement results are combined with the acoustic transfer impedance 

calculated for the gas filling the coupler cavity. This transfer impedance calculation is based 

on the model of an acoustic compliance, using the equivalent volumes of the microphones, 

the geometric volume of the coupler cavity including the front cavity volumes of the 

microphones, and the properties of the gas filling the coupler. The other parameters that are 

included are the calculated frequency-dependent corrections to this impedance for the effects 

of heat conduction at the interior boundaries of the coupler with microphones, and for the 

effects of acoustic wave motion. The wave motion correction W is needed to account for the 

departure from a spatially uniform sound pressure amplitude in the coupler at frequencies for 

which the coupler interior dimensions are not small compared to a wavelength of sound. To 

cover the entire frequency range of the NIST measurement services, couplers of different 

sizes are used and are filled with air or hydrogen, as appropriate, for the measurement 

frequency range and coupler size. Hydrogen is used as a fill gas at high frequencies because 

the larger speed of sound in hydrogen than in air results in a longer wavelength at a given 

frequency. The upper frequency limit of a given coupler is extended because of the increase 

in wavelength when the coupler is filled with hydrogen rather than air. 

At each frequency of calibration, the calculation procedure uses a parameter B (in 

decibels) that contains the effect on the test microphone sensitivity level of all other 

parameters except the voltage ratio and static ambient (barometric) pressure data. The test 

microphone sensitivity level is determined from B and the measured data for the voltage ratio 

and barometric pressure. For a frequently used test configuration in which both the reference 

standard and test microphones are Bruel and Kjaer model 41601

 microphones, B is denoted 

 

1

 Certain commercial equipment, instruments, or materials are identified in this paper in order to specify the 

experimental procedure adequately. Such identification is not intended to imply recommendation or 

endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the 

materials or equipment identified are necessarily the best available for the purpose. 

1 
Fig. 1. Sectional view of reference standard microphone and test microphone inserted into 

an acoustic coupler for a reciprocity-based comparison calibration of the test microphone. 

Reference standard 

microphone drive voltage 

Reference standard 

microphone used as 

a sound source 

(transmitter) 

Test microphone 

used as a receiver 

of sound 

Internal cavity 

of acoustic 

coupler 

Front cavities of 

microphones 

Reference standard 

microphone diaphragm 

_ + U1 

Acoustic 

coupler 

Test microphone 

diaphragm 

_ 

U2

+

Test microphone output voltage 

2 
as B4160, and values of B4160 are used in this calculation procedure. The Bruel and Kjaer 

model 4160 microphones are of IEC and ANSI Type LS1Pn. For other test microphones, 

such as IEC and ANSI Type LS2aP microphones [2,3], and a number of manufacturers’ 

models of IEC Type WS2 microphones [7], modifications are applied to the values of B4160 to 

account for the differences in the test microphone characteristics. 

In order to calibrate a relatively new model of IEC Type WS2 microphone, the Bruel 

and Kjaer model 4192, it was necessary to determine the values of the frequency-dependent 

correction Bcor,4192. This correction is applicable to the comparison calibration of a Bruel and 

Kjaer model 4192 test microphone with a Bruel and Kjaer model 4160 reference standard 

microphone and is used to modify the values of B4160. The geometrical configuration of the 

model 4192 microphone differs slightly from that of the other IEC Type WS2 microphones 

calibrated at NIST. In order to determine Bcor,4192, it was particularly necessary to determine 

experimentally the wave motion correction W4192 for the couplers configured with a Bruel 

and Kjaer model 4192 test microphone and a Bruel and Kjaer model 4160 reference standard 

microphone. This internal report describes the procedure used to determine the values of 

W4192, and provides these values. 

 

2. EXPERIMENTAL SETUP

The experimental setup used to perform the measurements required to determine the 

values of W4192 is the same setup currently used to perform reciprocity-based comparison 

calibrations of microphones for NIST customers. This setup consists of one of two 

cylindrical couplers designed to accommodate the transmitter-receiver pair of microphones, a 

system for measuring the voltage ratio U2/U1 at each frequency, and a hydrogen supply and 

retention system for filling the couplers with hydrogen. The gas grade purity of the hydrogen 

used is Matheson Purity (99.9995 %) supplied by Matheson Tri-Gas. Equipment for 

measuring the ambient environmental conditions, particularly the barometric pressure, is also 

included. 

 One acoustic coupler has a nominal cavity volume of 20 cm3

. For customer 

calibrations, this coupler is air-filled for frequencies from 50 Hz to 700 Hz, and hydrogenfilled for frequencies from 1 kHz to 10 kHz. A smaller coupler, which has a nominal cavity 

volume of 4 cm3

, is hydrogen-filled for calibrations at frequencies from 11 kHz to 20 kHz. 

Each coupler includes a pair of small-diameter capillary tubes connecting the interior volume 

of the coupler to the ambient air. Such capillary tubes are necessary to prevent the unwanted 

effects of changes in ambient barometric pressure. These effects would produce differences 

between the static pressures within the coupler and within the back-cavity volumes of the 

microphones, which are vented to ambient pressure through equalization vents on the sides of 

each microphone. Differences in static pressure between the exterior and interior surfaces of 

the microphone diaphragms would apply a static force to the diaphragms of the microphones 

and mechanically "bias" them to abnormal sensitivities in response to sound pressure. These 

couplers are filled with hydrogen by using the apparatus described in Ref. [8] connected to 

both capillary tubes in the case of the 4 cm3

 coupler, or only one such tube in the case of the 

20 cm3

 coupler. 

3 
 Both couplers were constructed with ports that readily accept IEC and ANSI Type 

LS1Pn microphones without adapters. IEC Type WS2 microphones fit into the couplers by 

means of an adapter having annulus and front-cavity geometries similar to that of an IEC and 

ANSI Type LS1Pn microphone. This adapter uses an O-ring of synthetic rubber with 

petroleum jelly to provide a seal at the outer diameter of the microphone, so that only the 

diaphragm is exposed to the sound field within the coupler cavity. With this adapter, it is 

possible to seal the cavity sufficiently well so that hydrogen can be used in the same 

frequency ranges as for the IEC and ANSI Type LS1Pn microphone calibrations. 

A block diagram of the system used to measure U2/U1 is shown in Fig. 2. To 

determine this ratio at a given frequency, the sound/attenuator switch is first set to the sound 

position. This setting causes the signal generator output to drive the transmitter through the 

transmitter drive unit, producing a sound pressure in the coupler. This sound pressure 

produces an output voltage signal from the receiver, which is connected to an insert voltage 

preamplifier. At the output of this preamplifier, the signal is routed through a measuring 

amplifier and band-pass filter. The amplified and filtered signal voltage produces a reading 

on an expanded scale meter capable of resolving 0.01 dB increments in voltage level. The 

meter reading is noted since it must be matched during the next step of the measurement. 

Without changing any settings of the signal generator, amplifier, and filter, the 

sound/attenuator switch is set to the attenuator position. This setting disconnects the signal 

generator output from the transmitter and grounds the circuit point at which the generator had 

been connected so that the transmitter produces essentially no sound in the coupler. 

Furthermore, this switch setting connects the signal generator output to the input of an 

attenuation network that consists of a variable attenuator and resistors that provide additional, 

fixed attenuation. The signal is electrically applied through the attenuation network and the 

insert voltage preamplifier so that the output of the network acts as a voltage source in series 

with the electrical terminals of the receiver. Following the receiver terminals, the rest of the 

signal path remains unchanged. The voltage supplied by the network is amplified, filtered, 

and produces a meter reading just as the voltage at the output of the receiver had with 

acoustic excitation when the sound/attenuator switch was in the sound position. The variable 

attenuator is adjusted until the reading observed on the meter is the same as the reading noted 

when the switch had been in the sound position. The sound/attenuator switch is then set back 

to the sound position to check whether the meter indication is still the same. The value of 

attenuation to which the variable attenuator is adjusted is then added to the total value of the 

additional fixed attenuation to obtain the total attenuation A. The measured value of U2/U1 is 

found from A using Eq. (1). 

⎟

⎠

⎞ ⎜

⎝

⎛ − = 20

A

UU 12 antilog (1) 

Voltage ratio data were acquired for microphone pairs formed by using one of two 

Bruel and Kjaer model 4160 microphones as the transmitter with one of four Bruel and Kjaer 

model 4192 microphones as the receiver. These two Bruel and Kjaer model 4160 

microphones are periodically calibrated at NIST by the reciprocity method for use as 

reference standard microphones in reciprocity-based comparison calibrations of customer 

microphones. Combining these microphones in this manner produced eight transmitterreceiver pairs for acquisition of the data required to determine the values of W4192. 

4 
Fig. 2. Block diagram of the system used to measure the voltage ratio U2/U1 of receiver 

voltage to transmitter voltage.

Signal

generator

Transmitter 

drive unit 

Attenuator

Sound/attenuator

 switch

Transmitter 

(sound source) 

microphone

Receiver 

microphone

Acoustic 

coupler

Insert voltage

preamplifier 

Attenuation 

network

Fixed

attenuation

(12.47 dB)

Variable

attenuator

Fixed

attenuation

(20.23 dB)

Expanded

scale meter

Sound

Band-pass 

filter 

Measuring 

amplifier 

input stage 

Measuring 

amplifier 

output stage 

5 
3. PROCEDURES FOR ACQUIRING WAVE MOTION CORRECTION DATA

To set up the microphones with the 4 cm3 coupler for voltage ratio measurements, a 

Bruel and Kjaer model 4192 microphone was threaded into the adapter used for IEC Type 

WS2 microphones after petroleum jelly was applied to the O-ring in this adapter. The 

adapter annulus, which had a thin layer of petroleum jelly applied on its surface, was placed 

in contact with one of the plane mating surfaces of the air-filled coupler. One of the Bruel 

and Kjaer model 4160 reference standard microphones, which had a thin layer of petroleum 

jelly applied on its front annulus, was then placed on the opposite side of the coupler so that 

the annulus was in contact with the other plane mating surface of the coupler. 

Measurements of U2/U1 were made for a given pair of microphones at selected test 

frequencies in the air-filled coupler by adjusting the variable attenuator to match the 

attenuated insert voltage to the output voltage produced with acoustic excitation of the 

receiver. Table 1a lists the selected test frequencies for the 4 cm3

 coupler, and Table 1b lists 

the selected test frequencies for the 20 cm3

 coupler. These frequencies were determined by 

dividing the midpoint frequencies between each pair of adjacent frequencies used for 

customer calibrations by the ratio of the second radial mode resonance frequency fr

h

 of the 

hydrogen-filled coupler cavity to the second radial mode resonance frequency fr

a

 of the airfilled coupler cavity. A working value of fr

h

/fr

a

 had been determined from preliminary 

measurements for each coupler. This value was 3.60 for the 20 cm3

 coupler, and 3.69 for the 

4 cm3

 coupler. Ideally, this value should be about 3.87, which is the ratio of the speed of 

sound in hydrogen to the speed of sound in air. However, a value of 3.87 is not achieved in 

practice due to impurities in the hydrogen, water vapor, and contamination due to residual air 

trapped in the system [6]. Additional test frequencies were included above and below the 

calculated range of test frequencies to determine the values of W4192 applicable to the upper 

and lower calibration frequencies for both couplers. Measurements were performed in 

descending order of frequency starting at the highest frequency. The barometric pressure, 

temperature, relative humidity, and fr

a

 were measured immediately before and after the 

measurements of U2/U1. 

After the air-filled coupler measurements were completed, the coupler was filled with 

hydrogen. In the case of the 4 cm3

 coupler, the coupler was filled with hydrogen using the 

procedure described in Ref. [8], and fr

h

 was monitored. The value of fr

h

 was measured and 

recorded when it stopped increasing, and the hydrogen flow was turned off. Measurements 

of U2/U1 were then performed for the same microphone pair at the same test frequencies used 

for the measurements done in the coupler when filled with air. These measurements were 

also performed in descending order of frequency starting at the highest frequency. After 

U2/U1 was measured at the lowest frequency, the measurement of U2/U1 was repeated at the 

highest frequency. This value of U2/U1 was always within ± 0.01 dB of the value initially 

measured at the highest frequency. This result is acceptable because the resolution of the 

measurement of U2/U1 is 0.01 dB. At that time, the measurement of fr

h

 was also repeated. 

As was done with the air-filled coupler measurements, the barometric pressure, temperature, 

and relative humidity were measured immediately before and after the measurements of 

U2/U1. 

6 
Table 1a. List of test frequencies at which measurements were made in the 4 cm3

 coupler 

to determine values of W4192, which are values of the wave motion correction applicable to 

reciprocity-based comparison calibrations of a Bruel and Kjaer model 4192 microphone. 

Test frequencies (Hz) for 4 cm3

 coupler 

5556 

5285 

5014 

4743 

4472 

4201 

3930 

3659 

3388 

3117 

2846 

2575 

Table 1b. List of test frequencies at which measurements were made in the 20 cm3 coupler 

to determine values of W4192, which are values of the wave motion correction applicable to 

reciprocity-based comparison calibrations of a Bruel and Kjaer model 4192 microphone. 

Test frequencies (Hz) for 20 cm3

 coupler 

2917 

2639 

2361 

2083 

1806 

1528 

1250 

972 

764 

625 

486 

347 

208 

7 
After completing both the air-filled coupler and hydrogen-filled coupler sets of 

voltage ratio measurements using the 4 cm3

 coupler at the test frequencies listed in Table 1a, 

measurements were made using the 20 cm3 coupler at the test frequencies listed in Table 1b 

using the same procedures except for the hydrogen fill. This coupler was filled by 

connecting a single capillary tube, which was used as a gas inlet, to the hydrogen supply and 

retention system. When fr

h

 stopped increasing, the hydrogen supply was turned off by 

closing a single valve in the hydrogen supply and retention system, and then the 

measurements were performed. 

The entire measurement process in both couplers was repeated with each of the other 

seven microphone pairs formed by using one of the Bruel and Kjaer model 4160 reference 

standard microphones as the transmitter and a Bruel and Kjaer model 4192 microphone as 

the receiver. 

4. DATA REDUCTION TO OBTAIN WAVE MOTION CORRECTION VALUES AT 

CALIBRATION FREQUENCIES IN HYDROGEN-FILLED COUPLERS

At each test frequency, the value of W4192 was calculated using 

( )

( ) ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

κ P

V

κ P

V

κ P

V Δ

UU

κ P

V

κ P

V

κ P

V Δ

UU

W

r a

a

e,2

r a

a

e,1

a a

a g H

12

a

r h

h

e,2

r h

h

e,1

h h

h

g H

12

h

4192 (2) 

where Vg is the total geometric volume in cubic meters of the cavity, Ve,1 and Ve,2 are the 

equivalent volumes in cubic meters of the Bruel and Kjaer model 4160 transmitter and the 

Bruel and Kjaer model 4192 receiver respectively, P is the measured barometric pressure in 

pascals, Pr is the reference barometric pressure (101.325 kPa) in pascals, κ is the 

dimensionless ratio of specific heats for the gas in the cavity, and ΔH is the heat conduction 

correction (dimensionless) calculated according to Ref. [4]. The superscripts h and a indicate 

whether the parameter is applicable to a hydrogen-filled or air-filled cavity respectively. The 

volume Vg is the sum of the coupler cavity volume, the geometric front cavity volume of the 

transmitter, and the geometric front cavity volume of the adapter into which the receiver has 

been threaded. This equation was derived from the expression for the product of the 

sensitivities for the transmitter and receiver given in Ref. [4]. Values of κa

 (1.402) and κ

h

(1.405), which historically have been used at NIST, are given in Ref. [6]. No capillary tube 

correction was applied, since previous measurements indicated that this correction is 

negligible for the capillary tubes used. 

To determine the frequency at which the values of W4192 are applicable to 

measurements made in a hydrogen-filled coupler, the test frequency was scaled by the 

measured value of the ratio fr

h

/fr

a

. For each set of measurement data acquired with a given 

pair of microphones and a given gas in the coupler, the resonance frequency used to calculate 

8 
this ratio was determined by averaging the resonance frequencies measured at the start and 

stop of data acquisition for that pair and gas. 

The value of W4192 applicable to a given frequency used for customer calibrations in a 

hydrogen-filled coupler was determined by linear interpolation using the wave motion 

correction values applicable to the two scaled frequencies between which the calibration 

frequency lies. 

 

 For each customer calibration frequency and coupler, the average value of W4192 and 

the population standard deviation were calculated from the values of W4192 determined for 

each of the eight microphone pairs. 

5. RESULTS AND SUMMARY

Values of W4192, which are applicable to NIST fixed-cost comparison calibrations of 

Bruel and Kjaer model 4192 customer microphones performed with a Bruel and Kjaer model 

4160 transmitter, are given in Table 2 and Table 3. Table 2 gives values applicable to such 

calibrations performed in the hydrogen-filled 20 cm3

 coupler, and Table 3 gives values 

applicable to such calibrations performed in the hydrogen-filled 4 cm3

 coupler. 

The frequency-dependent values of W4192 in Table 2 and Table 3 determined from the 

measurement data were used to calculate the values of the frequency-dependent correction 

Bcor,4192 applicable to comparison calibrations of Bruel and Kjaer model 4192 microphones 

with the Bruel and Kjaer model 4160 reference standard microphones. Values of Bcor,4192

were determined at each frequency of calibration applicable to the hydrogen-filled couplers 

using 

W W

V Δ 2V

V Δ VV

B 4192 4160 h

e,1

h

g H

h

e,2

h

e,1

h

g H

cor,4192 log 20 + − log 20 log 20 ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+

++ = (3) 

where W4160 is the wave motion correction applicable to the configuration with a Bruel and 

Kjaer model 4160 transmitter and receiver. When a Bruel and Kjaer model 4192 

microphone is calibrated by reciprocity-based comparison with the Bruel and Kjaer model 

4160 reference standard microphones, the value of Bcor,4192 is added to the value of B4160 to 

determine the sensitivity level of the Bruel and Kjaer model 4192 microphone. 

9 
Table 2. Values of the wave motion correction W4192, and 20 log W4192, applicable to the 

NIST fixed-cost comparison calibrations of a Bruel and Kjaer model 4192 microphone 

performed with the hydrogen-filled 20 cm3

 coupler. 

 

Calibration 

frequency (Hz) 

Wave motion 

correction W4192

(dimensionless) 

20 log W4192

(decibels) 

Expanded (coverage 

factor k=2) 

uncertainty (decibels)

10000 0.9874 -0.110 0.030

9000 0.9874 -0.110 0.026

8000 0.9881 -0.104 0.023

7000 0.9897 -0.090 0.023

6000 0.9917 -0.072 0.023

5000 0.9946 -0.047 0.023

4000 0.9971 -0.025 0.021

3000 0.9989 -0.010 0.021

2500 0.9996 -0.003 0.021

2000 0.9998 -0.002 0.023

1500 0.9997 -0.003 0.022

1000 0.9995 -0.004 0.027

Table 3. Values of the wave motion correction W4192, and 20 log W4192, applicable to the 

NIST fixed-cost comparison calibrations of a Bruel and Kjaer model 4192 microphone 

performed with the hydrogen-filled 4 cm3

 coupler. 

Calibration 

frequency (Hz) 

Wave motion 

correction W4192

(dimensionless) 

20 log W4192

(decibels) 

Expanded (coverage 

factor k=2) 

uncertainty (decibels)

20000 1.0430 0.366 0.087

19000 1.0317 0.271 0.073

18000 1.0230 0.198 0.066

17000 1.0161 0.139 0.063

16000 1.0108 0.093 0.057

15000 1.0065 0.056 0.052

14000 1.0031 0.027 0.050

13000 1.0007 0.006 0.046

12000 0.9992 -0.007 0.039

11000 0.9995 -0.004 0.031

10 
6. UNCERTAINTIES OF WAVE MOTION CORRECTION MEASUREMENT RESULTS

Expanded uncertainties of the measured values of W4192 were estimated by applying 

recognized guidelines for evaluating and expressing uncertainties [9]. For these data, the 

combined standard uncertainty uc, all of the components of uc, and the expanded uncertainty 

(coverage factor k=2), are summarized as a function of frequency in Table 4. A Type A 

standard uncertainty, equal to the standard deviation of the eight values of W4192 measured at 

each frequency, was calculated to indicate the repeatability of the results obtained with 

different microphone pairs. Several additional components, which are Type B standard 

uncertainties, were derived to account for the uncertainties in the values of the capillary tube 

correction, which was assumed to be equal to one (0 dB) regardless of the gas in the 

couplers, and the values of fr

h

, κa and κ

h

. These Type B standard uncertainties were derived 

by assuming a symmetric rectangular probability distribution, and estimating upper and 

lower bounds for the distribution. Given this assumption, the resulting standard uncertainty 

is equal to one-half the width of the distribution multiplied by a factor of 1/√3. 

For the component due to the uncertainty in the capillary tube correction, distribution 

bounds were estimated by calculating capillary tube corrections according to Annex B of 

Ref. [4] and examining the effects of these corrections on the wave motion correction values. 

For the component arising from the uncertainty in the value of fr

h

, distribution bounds were 

estimated by determining the maximum and minimum frequencies measured for both 

hydrogen-filled couplers during the course of all the wave motion correction measurements. 

For the component due to the uncertainty in the value of κa

, distribution bounds were 

estimated based on slight differences found between the value calculated using Annex F of 

Ref. [4] and the value provided in Ref. [6]. For the component arising from the uncertainty 

in the value of κ

h

, the range between estimated bounds was increased over that used for air to 

allow for impurities in the hydrogen, water vapor, and air in the system. 

11 
Table 4. Summary of uncertainties of the measured wave motion correction values. 

Source of 

uncertainty 

Standard 

deviation 

of results 

for eight 

microphone 

pairs 

Capillary 

tube 

correction 

Hydrogenfilled 

cavity 

resonance 

frequency 

κ for 

air 

κ for 

hydrogen 

Combined 

standard 

uncertainty 

uc (%) 

Expanded 

(coverage 

factor k=2) 

uncertainty 

2 uc (%) 

Expanded 

(coverage 

factor k=2) 

uncertainty 

expressed as 

the product 

of 20 and the 

log of 

[1+(2uc/100)] 

(dB) 

Type A B B B B 

Frequency 

(Hz) 

Relative standard uncertainty (%) 

20000 0.450 0.030 0.190 0.040 0.097 0.501 1.002 0.087 

19000 0.381 0.065 0.131 0.040 0.097 0.421 0.842 0.073 

18000 0.340 0.093 0.108 0.040 0.097 0.383 0.766 0.066 

17000 0.324 0.103 0.072 0.039 0.097 0.363 0.726 0.063 

16000 0.289 0.106 0.054 0.039 0.096 0.329 0.658 0.057 

15000 0.252 0.117 0.050 0.039 0.096 0.301 0.602 0.052 

14000 0.235 0.130 0.022 0.039 0.096 0.289 0.578 0.050 

13000 0.199 0.139 0.020 0.039 0.096 0.265 0.530 0.046 

12000 0.153 0.129 0.008 0.039 0.096 0.226 0.452 0.039 

11000 0.118 0.086 0.003 0.039 0.096 0.179 0.358 0.031 

10000 0.135 0.009 0.003 0.041 0.100 0.173 0.346 0.030 

9000 0.101 0.007 0.003 0.041 0.100 0.148 0.296 0.026 

8000 0.081 0.009 0.003 0.041 0.100 0.135 0.270 0.023 

7000 0.073 0.012 0.028 0.041 0.100 0.134 0.268 0.023 

6000 0.077 0.007 0.024 0.041 0.100 0.135 0.270 0.023 

5000 0.062 0.024 0.030 0.040 0.100 0.130 0.260 0.023 

4000 0.026 0.045 0.011 0.040 0.099 0.119 0.238 0.021 

3000 0.044 0.039 0.005 0.040 0.099 0.122 0.244 0.021 

2500 0.051 0.029 0.005 0.040 0.099 0.122 0.244 0.021 

2000 0.079 0.013 0.008 0.040 0.098 0.133 0.266 0.023 

1500 0.066 0.016 0.007 0.040 0.097 0.125 0.250 0.022 

1000 0.093 0.067 0.002 0.040 0.096 0.155 0.310 0.027 

12 
13 

7. REFERENCES

[1] NIST Calibration Services Users Guide, Catalog of NIST Calibration Services, 

Mechanical, Acoustic Measurements, Pressure and Free-Field Responses of Microphones 

(25010C-25050C), NIST Special Publication 250, 

http://ts.nist.gov/MeasurementServices/Calibrations/acoustic.cfm#25010c , [accessed 

November, 2008] 

[2] IEC 61094-1-Ed. 2.0, Measurement Microphones – Part 1: Specifications for Laboratory 

Standard Microphones, International Electrotechnical Commission, Geneva, Switzerland 

(2000). 

[3] ANSI S1.15-1997/Part 1 (R2006), American National Standard Measurement 

Microphones-Part 1: Specifications for Laboratory Standard Microphones, Acoustical 

Society of America, Melville, New York (1997). 

[4] IEC 61094-2, Measurement Microphones – Part 2: Primary Method for Pressure 

Calibration of Laboratory Standard Microphones by the Reciprocity Technique, International 

Electrotechnical Commission, Geneva, Switzerland (1992). 

[5] ANSI S1.15-2005/Part 2, American National Standard Measurement Microphones-Part 2: 

Primary Method for Pressure Calibration of Laboratory Standard Microphones by the 

Reciprocity Technique, Acoustical Society of America, Melville, New York (2005). 

[6] ANSI S1.10-1966 (withdrawn), American National Standard Method for the Calibration 

of Microphones, American National Standards Institute, New York, New York (1997). 

[7] IEC 61094-4, Measurement Microphones – Part 4: Specifications for Working Standard 

Microphones, International Electrotechnical Commission, Geneva, Switzerland (1995). 

[8] W. Koidan, Hydrogen Retention System for Pressure Calibration of Microphones in 

Small Couplers, The Journal of the Acoustical Society of America, 35, No. 4, 614 (1963). 

[9] B. N. Taylor and C. E. Kuyatt, Guidelines for Evaluating and Expressing the Uncertainty 

of NIST Measurement Results, NIST Technical Note 1297-1994 edition, U.S. Government 

Printing Office, Washington DC (1994).