- Email: [email protected]
Measurement of sound intensity using a single moving microphone
Applied Acoustics 66 (2005) 579–589 www.elsevier.com/locate/apacoust
Technical note
Measurement of sound intensity using a single moving microphone Takaaki Musha a, Jun-ichi Taniguchi b
b,*
a 3-11-7-601, Namiki, Kanazawa-ku, Yokohama 236-0005, Japan JRC Tokki Co. Ltd., Shin-yosida-cho, Kohoku-ku, Yokohama 223-8572, Japan
Received 9 April 2004; received in revised form 4 August 2004; accepted 15 September 2004 Available online 25 November 2004
Abstract A method for measuring sound intensity by using a single moving microphone is proposed. Experiments that confirm the validity of this method are reported and the advantages and disadvantages of the method are discussed. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Intensity measurement; Moving single microphone
1. Introduction The conventional sound intensity probe consists of two microphones mounted face-to-face with a solid spacer in-between. For the calculation of sound intensity, the pressure gradient is measured to approximate the particle velocity of sound [1]. But approximation of the pressure gradient by a straight line between two points causes an error if the wavelength is small compared with the microphone separation [2]. Thus, the upper frequency for measuring the accurate sound intensity is limited
*
Corresponding author. E-mail addresses: [email protected], [email protected] (T. Musha), [email protected] (J. Taniguchi). 0003-682X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2004.09.003
580
T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
by the microphone separation. Furthermore the two microphones system requires accurate phase-matched microphones to avoid phase mismatch error [3]. To overcome these limitations, we propose a single microphone method, which utilizes the modulation of sound induced by the movement of the measuring probe.
2. Theoretical background According to the work of Euler, the particle velocity of sound can be given by [4] Z 1 op dt: ð1Þ u¼ q ox For the harmonic plane wave with sound pressure given by p = A cos(kx x0t), where k is a wave number and x0 is an angular frequency of sound, the particle velocity follows from Eq. (1) as: u¼
Ak cosðkx x0 tÞ : qx0
ð2Þ
From the definition of the sound intensity I given by I ¼ p u,
ð3Þ
the sound intensity for a plane progressive wave becomes Z 1 T A2 kcos2 ðkx x0 tÞ A2 k A2 cos h: I¼ dt ¼ ¼ T 0 qx0 2qx0 2qc
ð4Þ
Considering the case shown in Fig. 1, where the propagating sound arrives at an angle of the elevation h above the x-axis, the sound pressure can be shown as p(t) = A cos(k 0 x(t) x0t /), when we let k 0 = k cos h. If the movement of the microphone along the x-axis is represented by x(t) = d sin xmt, where xm is an angular frequency of the movement of a probe, the received sound is modulated by the movement of the microphone shown as follows:
Fig. 1. Schematic diagram of the single moving microphone system.
T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
581
pðtÞ ¼ A cosðk 0 d sin xm t x0 t /Þ ¼ A ½cosðk 0 d sin xm tÞ cosðx0 t þ /Þ þ sinðk 0 d sin xm tÞ sinðx0 t þ /Þ: 0
0
When satisfying k d 1, we have cos (k d sinxmt) 1 sinxmt) k 0 d Æ sinxmt, then Eq. (5) can be rewritten as
and
ð5Þ sin (k 0 d
pðtÞ A ½cosðx0 t þ /Þ þ k 0 d sin xm t sinðx0 t þ /Þ k0d ¼ A cosðx0 t þ /Þ þ ½cosðfx0 xm gt þ /Þ cosðfx0 þ xm gt þ /Þ : 2 ð6Þ Let p_ o ðtÞ ¼ Ax0 sinðx0 t þ /Þ be the differential output of p(t) after rejecting side band frequency components through the narrow band filter, the product of p(t) and p_ o ðtÞ becomes pðtÞ p_ o ðtÞ ¼ A2 fx0 sinðx0 t þ /Þ cosðx0 t þ /Þ þ cosðfx0 xm gt þ /Þ
x0 k 0 d sinðx0 t þ /Þ 2
x0 k 0 d sinðx0 t þ /Þ cosðfx0 þ xm gt þ /Þg: 2 ð7Þ
Rearranging Eq. (7), we have
x0 k 0 d 2 x0 sinðxm tÞ A sinð2x0 t þ 2/Þ pðtÞ p_ o ðtÞ ¼ A 2 2 x0 k 0 d fsinðf2x0 xm gt þ 2/Þ sinðf2xo þ xm gt þ 2/Þg þ 4 2
ð8Þ
As the output through the low pass filter satisfying x0 > xm becomes 2 A2 x0 k 0 d sin xm t, we obtain the following result by integrating the product of the filter output and x(t) over a time period T Z 1 T A2 J¼ x0 k 0 d sin xm t xðtÞ dt T 0 2 Z 2 A 1 T A2 sin2 xm t x0 k 0 d 2 : ð9Þ ¼ x0 k 0 d 2 T 0 2 4 Then, the sound intensity for a plane progressive wave can be given by I ¼
2 J, qx20 d 2
ð10Þ
the calculation process of which is shown in Fig. 2. For the case of a static sound field, the sound pressure can be expressed as a combination of a positive-going and a negative-going plane waves shown as pðtÞ ¼ A cosðk 0 xÞ cosðx0 t þ /Þ A A ¼ cosðk 0 x x0 t /Þ þ cosðk 0 x þ x0 t þ /Þ: 2 2
ð11Þ
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T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
Fig. 2. Block diagram of intensity measurement using a single moving microphone.
From which, we have A k0d pðtÞ ½cosðfx0 xm gt þ /Þ cosðfx0 þ xm gt þ /Þ cosðx0 t þ /Þ þ 2 2 A k0d cosðx0 t þ /Þ ½cosðfx0 xm gt þ /Þ cosðfx0 þ xm gt þ /Þ , þ 2 2 ð12Þ and p_ 0 ðtÞ ¼ Ax0 sinðx0 t þ /Þ: Then, Eq. (9) becomes Z 1 T A2 A2 0 0 J¼ x0 k d sin xm t þ x0 k d sin xm t xðtÞ dt ¼ 0: T 0 4 4
ð13Þ
ð14Þ
It is known that the sound intensity for a static sound field can be obtained by using a single microphone and measuring phase shifts at the two adjacent points [2]. In contrast to this method, it is shown that the sound intensity measured by the moving single microphone method is the time averaged rate of the transferred acoustic energy for the plane wave. 3. Calculation of sound intensity by the Fourier transform When satisfying xm/x0 1, the sideband signals is difficult to be totally rejected by the electrical band-pass filter, and so the Fourier transform method is used for the calculation of sound intensity shown as follows. If we define the narrowband filter function as Pp ðxÞ ¼ 0ð1 < x < p=2Þ, ¼ 1ðp=2 6 x 6 p=2Þ ¼ 0ðp=2 < x < 1Þ,
ð15Þ
T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
583
the output through the narrow band filter in a frequency region can be given by P ðxÞ Pp ðx þ x0 Þ ¼ 2pAdðx þ x0 Þ expðj/Þ,
ð16Þ
where P(x) is the Fourier transform of p(t). When satisfying kd 1 (or d k/2p), P(x) can be written as P ðxÞ ¼ 2pA½dðx x0 Þ expðj/Þ þ dðx þ x0 Þ expðj/Þ k0d fdðx x0 þ xm Þ expðj/Þ 2 þ dðx þ x0 xm Þ expðj/Þ þ
þ dðx x0 xm Þ expðj/Þ dðx þ x0 þ xm Þ expðj/Þg:
ð17Þ
By using the convolution property of the Diracs delta function [5] dðx x1 Þ dðx x2 Þ ¼ dðx x1 x2 Þ,
ð18Þ
the derivation of which is shown in Appendix A, we have Z xm þe P ðxÞ ðP ðxÞ Pp ðx þ x0 ÞÞ dx xm e 2
¼ 4p A
2
Z
xm þe
xm e
k0d dðx xm Þ dx ¼ 2p2 A2 k 0 d, 2
ð19Þ
for e > 0. Hence, the intensity of sound for a sinusoidal sound can be given by Iðx0 Þ ¼
1 A2 Re P ðxÞ cos h: ½ fP ðxÞ P ðx x Þg ¼ p 0 x¼xm 4p2 qx0 d 2qc
ð20Þ
From which, the sound intensity for the single microphone system can be obtained by using the Fourier transform as shown in Fig. 3. The polar plot of the intensity for a plane progressive wave shown in Fig. 4, is obtained by using the computer simulation program, the DADiSP (DSP Development Corporation), by setting d = 1 mm, xm/2p = 100 Hz and x0/2p = 5000 Hz, where the value of intensity is normalized by the maximum value at h = 0°. The data sampling frequency for computation is selected to be 200 kHz. From this figure, it is shown that the simulation result coincides with the theoretical cosine pattern of sound intensity.
4. Experimental validation An experiment to validate the single microphone method was conducted using the set-up shown in Fig. 5. The condenser microphone, 6 mm long and 10 mm in diameter, was attached to the center of the vibrating plate oscillated by the electromagnetic exciter with the frequency of xm/2p = 100 Hz. The sinusoidal sound is generated by the speaker contained in a plastic enclosure, the frequency of which is x0/ 2p = 5000 Hz, driven by the oscillator. The displacement at the centre of the vibrating plate was monitored by the gap sensor, which was connected to the FFT
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T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
Fig. 3. Block diagram of intensity measurement by Fourier transform.
Fig. 4. Polar plot of the acoustic intensity from the numerical simulation.
T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
Measurement devise
585
FFT
Holder
Noise Source
Gap sensor
Microphone
Oscillator
Amplifier
Exciter
Fig. 5. Experimental setup to validate the single moving microphone method.
analyzer. The output signal from the microphone was processed by the FFT analyzer. A photo of the experimental set-up is shown in Fig. 6. During the experiment, the sound intensity was measured at intervals of 10° from h = 0° to 80° by shifting the position of the sound source as shown in Fig. 7. The displacement of the plate measured by the gap sensor was about 0.3 mm, from which we have k 0 d 6 0.03 that satisfies k 0 d 1, which is the requirement of the intensity calculation. Fig. 8 shows the FFT processed result of the data at h = 0°, where the horizontal axis is for frequencies ranging from 4500 and 5500 Hz and the vertical axis is
Fig. 6. Photo of the measuring device.
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T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
Holder
FFT
Noise Source
Microphone
ω0
Fig. 7. Schematic diagram of the experiment.
Fig. 8. Result of FFT analysis measured at h = 0°.
for the amplitude of a signal in dB. From Eq. (17), the intensity level of sound can be obtained by the side band spectrum at x0 + xm, shown as follows: 2 A DL , ð21Þ 10 log jIðx0 Þj ¼ 10 log þ 2 qx0 d where DL ¼ 20 log jP ðx0 Þj 20 log jP ðx0 þ xm Þj ¼ 20 log
k0d : 2
From which, the intensity level of sound can be calculated as shown in Fig. 9.
ð22Þ
T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
587
Fig. 9. Comparison of the simulation result and the data for the intensity of sound (a), errors of the measurement (b) and the influence on the sound pressure at the surface of a spherical housing (c).
The upper figure (a) in Fig. 9 shows both of the simulation result of sound intensity and the normalized measured results by the amplitude measured at h = 0°, where the horizontal axis is for the incident angle of sound and the vertical axis is for the sound intensity level.
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T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
The lower figure (b) shows the errors of the measurement. It is seen that errors are not exceeding 2 dB when the angle of the incident sound is less than 70°. On the other hand the error increases to 2.5 dB at the angle of 80°. As the radius of the microphone housing is a = 5 mm and the frequency of sound is f = 5 kHz, we have ka 0.5. From the diagram of the influence on the sound pressure at the surface of a spherical housing [6] shown in Fig. 9(c), the maximum error due to diffraction effect given by 20 logðP h =P Þ will reach 2 dB at ka = 0.5, hence, it is considered that errors of sound intensity by the single microphone method might be due to the diffraction effect of sound by the microphone housing attached to the vibrating plate. From the theoretical analysis and the experimental result, the advantages and disadvantages of this method compared with the two microphone method may be deduced.
4.1. Advantages Theoretically there are no frequency limitations due to the pressure gradient approximation. Accurate phase matching of microphones is not required.
4.2. Disadvantages The influence of the diffraction of sound by the microphone housing cannot be neglected for higher frequencies. It is necessary to use a microphone satisfying ka 1 to avoid the influence of diffraction of sound. Intensity analysis is restricted to a single frequency at a time. A device for oscillating the microphone is required. It is considered that the single moving microphone method can be applied for the measurement of sound intensity at higher frequencies where it is difficult to make accurate measurements of sound intensity by the conventional two microphones method, because the displacement of the moving microphone required is much smaller than the separation required in the two microphones system.
5. Conclusion We have proposed a method for measuring sound intensity by using a single moving microphone. From the theoretical analysis and the experiment conducted to confirm the validity of this method, it is considered that the proposed single moving microphone method can be applied to the measurement of the intensity of sound at higher frequencies where it is difficult to measure accurately with the conventional two microphone method.
T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
589
Appendix A Since the inverse Fourier transform of d(x x0)*d(x x1) is given by [7] 1 jx0 t jx1 t 1 jðx0 þx1 Þt e e ¼ e , 2p 2p
ðA:1Þ
the Fourier transform of the right term of Eq. (A.1) becomes d(x x0 x1). From which, we have dðx x0 Þ dðx x1 Þ ¼ dðx x1 x2 Þ:
ðA:2Þ
References [1] Fahy FJ. Measurement of acoustic intensity using the cross-spectral density of two microphone signals. J Acoust Soc Am 1977;62(4):1057–9. [2] Fahy FJ. Sound intensity. 2nd ed.. London: E&FN Spon; 1995. [3] Batel M, Marroquin M, Hald J, Christensen JJ, Schumacher AP, Nielsen TG. Noise source location techniques-simple to advanced applications. Sound Vibrat 2003(March):24–38. [4] Rasmussen G. Intensity-its measurement and uses. Sound Vibrat 1989;20:12–21. [5] James JE. A students guide to Fourier transforms with applications in physics and engineering. Cambridge: Cambridge University Press; 2000. [6] Kinsler LE, Frey AR, Coppens AB, Sanders JV. Fundamental of acoustics. 3rd ed.. New York: Wiley; 1982. p. 376. [7] Hsu HP. Applied fourier analysis, Books for professionals. San Diego: Harcourt Brace Jovanovich Publishers; 1984.
Technical note
Measurement of sound intensity using a single moving microphone Takaaki Musha a, Jun-ichi Taniguchi b
b,*
a 3-11-7-601, Namiki, Kanazawa-ku, Yokohama 236-0005, Japan JRC Tokki Co. Ltd., Shin-yosida-cho, Kohoku-ku, Yokohama 223-8572, Japan
Received 9 April 2004; received in revised form 4 August 2004; accepted 15 September 2004 Available online 25 November 2004
Abstract A method for measuring sound intensity by using a single moving microphone is proposed. Experiments that confirm the validity of this method are reported and the advantages and disadvantages of the method are discussed. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Intensity measurement; Moving single microphone
1. Introduction The conventional sound intensity probe consists of two microphones mounted face-to-face with a solid spacer in-between. For the calculation of sound intensity, the pressure gradient is measured to approximate the particle velocity of sound [1]. But approximation of the pressure gradient by a straight line between two points causes an error if the wavelength is small compared with the microphone separation [2]. Thus, the upper frequency for measuring the accurate sound intensity is limited
*
Corresponding author. E-mail addresses: [email protected], [email protected] (T. Musha), [email protected] (J. Taniguchi). 0003-682X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2004.09.003
580
T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
by the microphone separation. Furthermore the two microphones system requires accurate phase-matched microphones to avoid phase mismatch error [3]. To overcome these limitations, we propose a single microphone method, which utilizes the modulation of sound induced by the movement of the measuring probe.
2. Theoretical background According to the work of Euler, the particle velocity of sound can be given by [4] Z 1 op dt: ð1Þ u¼ q ox For the harmonic plane wave with sound pressure given by p = A cos(kx x0t), where k is a wave number and x0 is an angular frequency of sound, the particle velocity follows from Eq. (1) as: u¼
Ak cosðkx x0 tÞ : qx0
ð2Þ
From the definition of the sound intensity I given by I ¼ p u,
ð3Þ
the sound intensity for a plane progressive wave becomes Z 1 T A2 kcos2 ðkx x0 tÞ A2 k A2 cos h: I¼ dt ¼ ¼ T 0 qx0 2qx0 2qc
ð4Þ
Considering the case shown in Fig. 1, where the propagating sound arrives at an angle of the elevation h above the x-axis, the sound pressure can be shown as p(t) = A cos(k 0 x(t) x0t /), when we let k 0 = k cos h. If the movement of the microphone along the x-axis is represented by x(t) = d sin xmt, where xm is an angular frequency of the movement of a probe, the received sound is modulated by the movement of the microphone shown as follows:
Fig. 1. Schematic diagram of the single moving microphone system.
T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
581
pðtÞ ¼ A cosðk 0 d sin xm t x0 t /Þ ¼ A ½cosðk 0 d sin xm tÞ cosðx0 t þ /Þ þ sinðk 0 d sin xm tÞ sinðx0 t þ /Þ: 0
0
When satisfying k d 1, we have cos (k d sinxmt) 1 sinxmt) k 0 d Æ sinxmt, then Eq. (5) can be rewritten as
and
ð5Þ sin (k 0 d
pðtÞ A ½cosðx0 t þ /Þ þ k 0 d sin xm t sinðx0 t þ /Þ k0d ¼ A cosðx0 t þ /Þ þ ½cosðfx0 xm gt þ /Þ cosðfx0 þ xm gt þ /Þ : 2 ð6Þ Let p_ o ðtÞ ¼ Ax0 sinðx0 t þ /Þ be the differential output of p(t) after rejecting side band frequency components through the narrow band filter, the product of p(t) and p_ o ðtÞ becomes pðtÞ p_ o ðtÞ ¼ A2 fx0 sinðx0 t þ /Þ cosðx0 t þ /Þ þ cosðfx0 xm gt þ /Þ
x0 k 0 d sinðx0 t þ /Þ 2
x0 k 0 d sinðx0 t þ /Þ cosðfx0 þ xm gt þ /Þg: 2 ð7Þ
Rearranging Eq. (7), we have
x0 k 0 d 2 x0 sinðxm tÞ A sinð2x0 t þ 2/Þ pðtÞ p_ o ðtÞ ¼ A 2 2 x0 k 0 d fsinðf2x0 xm gt þ 2/Þ sinðf2xo þ xm gt þ 2/Þg þ 4 2
ð8Þ
As the output through the low pass filter satisfying x0 > xm becomes 2 A2 x0 k 0 d sin xm t, we obtain the following result by integrating the product of the filter output and x(t) over a time period T Z 1 T A2 J¼ x0 k 0 d sin xm t xðtÞ dt T 0 2 Z 2 A 1 T A2 sin2 xm t x0 k 0 d 2 : ð9Þ ¼ x0 k 0 d 2 T 0 2 4 Then, the sound intensity for a plane progressive wave can be given by I ¼
2 J, qx20 d 2
ð10Þ
the calculation process of which is shown in Fig. 2. For the case of a static sound field, the sound pressure can be expressed as a combination of a positive-going and a negative-going plane waves shown as pðtÞ ¼ A cosðk 0 xÞ cosðx0 t þ /Þ A A ¼ cosðk 0 x x0 t /Þ þ cosðk 0 x þ x0 t þ /Þ: 2 2
ð11Þ
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T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
Fig. 2. Block diagram of intensity measurement using a single moving microphone.
From which, we have A k0d pðtÞ ½cosðfx0 xm gt þ /Þ cosðfx0 þ xm gt þ /Þ cosðx0 t þ /Þ þ 2 2 A k0d cosðx0 t þ /Þ ½cosðfx0 xm gt þ /Þ cosðfx0 þ xm gt þ /Þ , þ 2 2 ð12Þ and p_ 0 ðtÞ ¼ Ax0 sinðx0 t þ /Þ: Then, Eq. (9) becomes Z 1 T A2 A2 0 0 J¼ x0 k d sin xm t þ x0 k d sin xm t xðtÞ dt ¼ 0: T 0 4 4
ð13Þ
ð14Þ
It is known that the sound intensity for a static sound field can be obtained by using a single microphone and measuring phase shifts at the two adjacent points [2]. In contrast to this method, it is shown that the sound intensity measured by the moving single microphone method is the time averaged rate of the transferred acoustic energy for the plane wave. 3. Calculation of sound intensity by the Fourier transform When satisfying xm/x0 1, the sideband signals is difficult to be totally rejected by the electrical band-pass filter, and so the Fourier transform method is used for the calculation of sound intensity shown as follows. If we define the narrowband filter function as Pp ðxÞ ¼ 0ð1 < x < p=2Þ, ¼ 1ðp=2 6 x 6 p=2Þ ¼ 0ðp=2 < x < 1Þ,
ð15Þ
T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
583
the output through the narrow band filter in a frequency region can be given by P ðxÞ Pp ðx þ x0 Þ ¼ 2pAdðx þ x0 Þ expðj/Þ,
ð16Þ
where P(x) is the Fourier transform of p(t). When satisfying kd 1 (or d k/2p), P(x) can be written as P ðxÞ ¼ 2pA½dðx x0 Þ expðj/Þ þ dðx þ x0 Þ expðj/Þ k0d fdðx x0 þ xm Þ expðj/Þ 2 þ dðx þ x0 xm Þ expðj/Þ þ
þ dðx x0 xm Þ expðj/Þ dðx þ x0 þ xm Þ expðj/Þg:
ð17Þ
By using the convolution property of the Diracs delta function [5] dðx x1 Þ dðx x2 Þ ¼ dðx x1 x2 Þ,
ð18Þ
the derivation of which is shown in Appendix A, we have Z xm þe P ðxÞ ðP ðxÞ Pp ðx þ x0 ÞÞ dx xm e 2
¼ 4p A
2
Z
xm þe
xm e
k0d dðx xm Þ dx ¼ 2p2 A2 k 0 d, 2
ð19Þ
for e > 0. Hence, the intensity of sound for a sinusoidal sound can be given by Iðx0 Þ ¼
1 A2 Re P ðxÞ cos h: ½ fP ðxÞ P ðx x Þg ¼ p 0 x¼xm 4p2 qx0 d 2qc
ð20Þ
From which, the sound intensity for the single microphone system can be obtained by using the Fourier transform as shown in Fig. 3. The polar plot of the intensity for a plane progressive wave shown in Fig. 4, is obtained by using the computer simulation program, the DADiSP (DSP Development Corporation), by setting d = 1 mm, xm/2p = 100 Hz and x0/2p = 5000 Hz, where the value of intensity is normalized by the maximum value at h = 0°. The data sampling frequency for computation is selected to be 200 kHz. From this figure, it is shown that the simulation result coincides with the theoretical cosine pattern of sound intensity.
4. Experimental validation An experiment to validate the single microphone method was conducted using the set-up shown in Fig. 5. The condenser microphone, 6 mm long and 10 mm in diameter, was attached to the center of the vibrating plate oscillated by the electromagnetic exciter with the frequency of xm/2p = 100 Hz. The sinusoidal sound is generated by the speaker contained in a plastic enclosure, the frequency of which is x0/ 2p = 5000 Hz, driven by the oscillator. The displacement at the centre of the vibrating plate was monitored by the gap sensor, which was connected to the FFT
584
T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
Fig. 3. Block diagram of intensity measurement by Fourier transform.
Fig. 4. Polar plot of the acoustic intensity from the numerical simulation.
T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
Measurement devise
585
FFT
Holder
Noise Source
Gap sensor
Microphone
Oscillator
Amplifier
Exciter
Fig. 5. Experimental setup to validate the single moving microphone method.
analyzer. The output signal from the microphone was processed by the FFT analyzer. A photo of the experimental set-up is shown in Fig. 6. During the experiment, the sound intensity was measured at intervals of 10° from h = 0° to 80° by shifting the position of the sound source as shown in Fig. 7. The displacement of the plate measured by the gap sensor was about 0.3 mm, from which we have k 0 d 6 0.03 that satisfies k 0 d 1, which is the requirement of the intensity calculation. Fig. 8 shows the FFT processed result of the data at h = 0°, where the horizontal axis is for frequencies ranging from 4500 and 5500 Hz and the vertical axis is
Fig. 6. Photo of the measuring device.
586
T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
Holder
FFT
Noise Source
Microphone
ω0
Fig. 7. Schematic diagram of the experiment.
Fig. 8. Result of FFT analysis measured at h = 0°.
for the amplitude of a signal in dB. From Eq. (17), the intensity level of sound can be obtained by the side band spectrum at x0 + xm, shown as follows: 2 A DL , ð21Þ 10 log jIðx0 Þj ¼ 10 log þ 2 qx0 d where DL ¼ 20 log jP ðx0 Þj 20 log jP ðx0 þ xm Þj ¼ 20 log
k0d : 2
From which, the intensity level of sound can be calculated as shown in Fig. 9.
ð22Þ
T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
587
Fig. 9. Comparison of the simulation result and the data for the intensity of sound (a), errors of the measurement (b) and the influence on the sound pressure at the surface of a spherical housing (c).
The upper figure (a) in Fig. 9 shows both of the simulation result of sound intensity and the normalized measured results by the amplitude measured at h = 0°, where the horizontal axis is for the incident angle of sound and the vertical axis is for the sound intensity level.
588
T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
The lower figure (b) shows the errors of the measurement. It is seen that errors are not exceeding 2 dB when the angle of the incident sound is less than 70°. On the other hand the error increases to 2.5 dB at the angle of 80°. As the radius of the microphone housing is a = 5 mm and the frequency of sound is f = 5 kHz, we have ka 0.5. From the diagram of the influence on the sound pressure at the surface of a spherical housing [6] shown in Fig. 9(c), the maximum error due to diffraction effect given by 20 logðP h =P Þ will reach 2 dB at ka = 0.5, hence, it is considered that errors of sound intensity by the single microphone method might be due to the diffraction effect of sound by the microphone housing attached to the vibrating plate. From the theoretical analysis and the experimental result, the advantages and disadvantages of this method compared with the two microphone method may be deduced.
4.1. Advantages Theoretically there are no frequency limitations due to the pressure gradient approximation. Accurate phase matching of microphones is not required.
4.2. Disadvantages The influence of the diffraction of sound by the microphone housing cannot be neglected for higher frequencies. It is necessary to use a microphone satisfying ka 1 to avoid the influence of diffraction of sound. Intensity analysis is restricted to a single frequency at a time. A device for oscillating the microphone is required. It is considered that the single moving microphone method can be applied for the measurement of sound intensity at higher frequencies where it is difficult to make accurate measurements of sound intensity by the conventional two microphones method, because the displacement of the moving microphone required is much smaller than the separation required in the two microphones system.
5. Conclusion We have proposed a method for measuring sound intensity by using a single moving microphone. From the theoretical analysis and the experiment conducted to confirm the validity of this method, it is considered that the proposed single moving microphone method can be applied to the measurement of the intensity of sound at higher frequencies where it is difficult to measure accurately with the conventional two microphone method.
T. Musha, J. Taniguchi / Applied Acoustics 66 (2005) 579–589
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Appendix A Since the inverse Fourier transform of d(x x0)*d(x x1) is given by [7] 1 jx0 t jx1 t 1 jðx0 þx1 Þt e e ¼ e , 2p 2p
ðA:1Þ
the Fourier transform of the right term of Eq. (A.1) becomes d(x x0 x1). From which, we have dðx x0 Þ dðx x1 Þ ¼ dðx x1 x2 Þ:
ðA:2Þ
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