- Email: [email protected]
Spherical acoustical holography of low-frequency noise sources
Appkd
Acoustics, Vo1.48, No. 1, pp. 8595, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0003-682X/96/$15.00+0.00
0003-682X(95)00068-2
ELSEVIER
Technical Note Spherical Acoustical Holography of Low-Frequency Noise Sources J. Department
C.Lee
of Marine Engineering Technology, National Taiwan Ocean University, 2 Pei Ning Road, Keelung, Taiwan (Received 4 September 1995; accepted 3 November 1995)
ABSTRACT The performance of near-jield reconstruction at lowfrequency using spherical holographic techniques is investigated. The spherical Hankel function with small argument (near-jield and low frequency) has divergent behaviour in the near-field. This divergent phenomenon influences the precision of holographic reconstruction. Two computer simulations based on the derived equations are used to demonstrate the capability of holography to construct the source surface pressure and particle velocity for noise source iden@cation. The results of multipole source simulation show that the precision oj backward holography to the near-jield depends on using the optimum order of holographic wave. Analysis of another simulation shows that more precise reconstruction of source surface pressure and particle velocity can be obtained from original pressure data at a smaller radius. Keywords: Acoustics,
spherical
holography.
1 INTRODUCTION
Acoustic field data in the vicinity of a vibrating structure can be used to evaluate either the acoustic far-field or the source characteristics. The prediction of acoustic far-field from near-field data is a special case of the forward holography of acoustic fields. Alternately, the identification of source characteristics on a vibrating structure is a special case of the backward holography of acoustic fields. Acoustical holography for the purpose of noise source identification has most often been applied in planar geometries. However, many small 85
86
J. C. Lee
machines have shapes that make planar holography inconvenient. In these cases, analysis based on spherical or cylindrical geometries may yield better results. The spherical holography technique has been applied in cases where the sound source may be entirely enclosed by a sphere on which sound pressure is measured. DeVries et al.’ applied the technique in both simulation and experiment to confirm the accuracy of spherical holography and to identify the noise source for small machines, especially refrigeration compressors. Zhukov et al.* identify multipole sound sources by deriving the general theoretical expressions of multipole source sound field energy characteristics in spherical coordinates. Weinreich and Arnold” have designed an experiment in which a boom system was employed to measure the complex sound pressure on two concentric spheres. They expanded the solution of the wave equation in spherical coordinates with the spherical harmonics calculated from the measurements; their method can be used to characterize the sound field when both incoming and outgoing waves are present. Laville et aZ.4have developed a spherical acoustical holography technique that uses sound intensity measurements to project the sound field for outgoing waves only. The acoustical holography for the identification of noise sources is a promising direction in aeroacoustics and underwater acoustics. However, for low-frequency noise source identification, the spherical Hankel function with small argument is divergent for backward holography. In previous work, mentioned above, the influence of the near-field divergent phenomenon on the source identification by backward holography has not been discussed. To investigate the performance and capability of lowfrequency spherical holography, two kinds of source are used in numerical simulation. One source consists of any number of multipole sources and the other is a point source on the surface of a sphere. This paper expands the solution of the wave equation in spherical coordinates in spherical harmonics, and determines the spherical harmonic coefficients by using the theoretical sphere pressure data from numerical calculation. These coefficients are then used to predict the sound field in either forward and backward holography; forward, to visualize the sound field, and backward to offer noise source identification. The acoustic particle velocity distribution over the surface of the source can also be reproduced by backward holography from the theoretical sphere pressure data.
2 THEORY
OF SPHERICAL
HOLOGRAPHY
The wave equation in spherical coordinates can be solved using Fourier’s method of separating the variables. The radial component of the sound
Spherical acoustical holography of low-frequency noise sources
87
field is described by spherical Hankel functions, and the polar and azimuthal components are described by Legendre functions, through which the spherical harmonics can be calculated. We take the acoustic pressure as the primary field quantity and represent the general form as
Here p(M,w) is the complex pressure at angular frequency w and locations M on any spherical surface. Spherical Hankel functions, h,(kr), can be calculated using a linear combination of the spherical Bessel and Neumann functions.5 The spherical surface harmonics Yim(O,4) and Yrr,(O,4) are defined by
where 8 is the polar angle measured from the position z-axis, and I$ is the azimuthal angle measured from the x-axis. P,(cosO) is the nth order Legendre function expanded in powers of cos8. The radial component of acoustic particle velocity can be defined as: u,(M, w) = -
-1 ~P(WW>
jwpo at-
(3)
The radial direction derivative of the Hankel function can be expressed in a recursive form dh,(kr)
~
d(kr)
n
=
-&#r)
- hn+l(kr)
(5)
where k is the wave number. The harmonic coefficients A,, and B,, can be derived by using the orthogonality of the spherical harmonics4
J. C. Lee
88
B nrnFpE _2n-1
4lr
(n-m)!
1 -
“(n+m)!h,(kr)
2?T 71 d$ p(M,w) Yirn(O,4)sin 8 d/3 I s 0
(7)
0
Here if m = 0, cm= 1 and if rn> 1, t, = 2. Once the harmonic coefficients are found, the pressure or particle velocity fields may be projected away from or toward the source, as needed, by using the eqns (l)-(4).
3 THE PERFORMANCES
OF SPHERICAL
HOLOGRAPHY
As shown in eqn (1) the wave equation separates in spherical coordinates into a product of spherical surface harmonics Y,,(Q, 4) and spherical Hankel functions h&r). The pressure difference between near-field and far-field are due to the influence of the radial functions h,(kr) only. At large kr, the function h,(kr) for far-field can be expressed as5 exp(jkr - jn i K) , jkr
h,(kr) ---f
kr+cc
(8)
here j = a, and the influence of y1on h,(kr) is only in characteristic phase shifts. For small values of kr, near-field and low frequency, h,(kr) can be described by
hdkr) +
(2n)!
kr -+ 0
j(n)!2”(kr)“+’ ’
When kr becomes smaller than 12,the near-field divergent behaviour will be found in this equation, and the near-field extended further and further out for high ~1.Therefore, the convergence of eqn (1) requires that the coefficients A,, and B,, become small at high K If the sound field is due to a source of approximate extent D located at the origin, it may be assumed that the projected wave will not contain appreciable contributions for n larger than kD, as long as the coefficients A,, and B,, do not increase for high n. On the other hand, if the sound source is modelled by a number of point monopoles situated in a region having a maximum distance R from the origin, the significant partial projected wave component will contribute up to about n = kR. Thus, for low frequency and small radius, eqn (1) should cut-off at an y1value at which the agreement with reality begins to become worse rather than better. As mentioned above, the lower-order ingoing or outgoing wave is significant for spherical holography. For a monopole placed at the origin, the
Spherical acoustical holography of low-frequency noise sources
89
sound field is a zero-order projected wave (n = 0) with radial factor h&r) and only the A00 coefficient is non-zero. The first-order projected waves (n = 1) with radial factor hi(kr) have three possible solutions for the three orientations of a dipole. If a dipole is oriented along the x-axis, the Ai0 is non-zero. When the monopole or dipole is moved off the coordinate axis, the sound field loses its symmetry, the orders of the coefficients change and the coefficient B,,, becomes significant. Similarly, the set of secondorder projected waves (n= 2) with its radial factor h&r) have five independent quadrupole waves. In general, the source order may be characterized by checking how the magnitudes of the harmonic coefficients fall off with increasing order.
4 MONOPOLE
SOURCES
ACOUSTIC
FIELD SIMULATION
The pressure field at a given radius due to any number of monopole sources can be predicted by using the Fourier transform pressure equation6
where p(Mi, w) is the Fourier transform pressure at point Mi due to a monopole source located at a distance ri. s is the number of monopole sources. Qdw) is the strength of source and p is the density of the medium. A,, and B,, may be estimated using discretized versions of eqns (6)--(s). The resulting harmonic coefficients could then be used to reconstruct the complex pressure field, which can be compared to the predicted values at any radius to check accuracy. The strength, phases and locations of sources are variable so that higher-order wave patterns may be simulated. To check the accuracy of numerical calculation, we first choose a set of five monopole sources (listed in Table 1) with the same location and TABLE 1
Source information Source number
1 2 3 4 5
of Section 4
Coordinates (X, Y, 2)
(0.000, 0.000, 0.000) (-0.201, -0.201, 0.101) (-0.201, 0.201, 0.001) (0.200, 0.100, 0.100) (0.300, -0.300, -0.200)
Strength Q (m’/s) 2.0 + J2.0
3.O-tjl.O 3.O+jl.O l.O+jl.O 2.0-j1.0
90
J. C. Lee
strength used elsewhere. 4 The performance and capabilities of lowfrequency spherical holography are studied through numerical simulation. In Figs l-6 the pressure predicted along a meridian (4=0) to compared with the complex pressure reconstructed from the spherical holography. The dotted lines represent the predicted, and the solid lines indicate the reconstruction using spherical holography. These line types are used in all figures unless otherwise stated. The curves of the complex pressure at a radius of 1 m and frequency of 200 Hz resulting from theoretical prediction and spherical holography are shown in Fig. l(a). With the same condition, Fig. l(b) shows results obtained elsewhere.4 The agreement of these two parts of Fig. 1 is excellent, and the accuracy of numerical simulation in this paper is reliable.
-31 0
300 100 200 (pressure) angle-->
Fig. 1. (a) The complex pressure at meridian holographicAvalue). (b) The complex theoretical data, __
0
I
2
3
4
S
6
7
B-
(c$= 0) from this paper (0, theoretical data, pressure at meridian (q5=0) from Ref.4 (09 holographic value).
81
8
A
6
ii -4
300 200 100 (pressure) angle-->
200 300 100 (pressure) angle-->
Fig. 2. (a) The amplitude of pressure at meridian (q5= 0) for order n = 2 (0, holographic value). (b) The amplitude of pressure at meridian data, __ holographic value). order n = 3 (0, theoretical data, ~
theoretical (d= 0) for
Spherical acoustical holography of low-frequency noise sources
91
The maximum distance R from the origin of these monopole sources is 0.5 m; the spherical harmonic coefficients A,, and B,, evaluated from the predicted pressure at 200 Hz and radius R are shown in Table 2. The amplitudes of the spherical harmonic coefficients A,, do not increase with n. According to the analysis in Section 3, the appreciable partial backward projected wave due to multipole sources radiating at low frequency will contribute up to order 12= kR. The value of kR is about 2. Therefore, for these multipole sources, the order n required for spherical backward
0
100 200 (pressure) angle-b
300
-2. 0
100 200 300 (pressure) angle-->
Fig. 3. (a) The real part of pressure at meridian (C#J = 0) for backward holography at 100 Hz from a radius of 2 m to a radius of 2 m (r = 2), 1 m (r = 1) and 0.5 m (r = 0.5) (0, theoretical
data, __ holographic value). (b) The imaginary part of pressure at meridian (Q,= 0) for backward holography at 100 Hz from a radius of 2 m to a radius of 2 m (r = 2) 1 m (r = 1) and 0.5 m (r = 0.5) (0, theoretical data, __ holographic value).
2
-1 0
100
200
(pressure) angle->
300
100
200
300
(pressure) angle-->
Fig. 4. (a) The real part of pressure at meridian (4 = 0) for forward holography at 100 Hz from a radius of 0.5 m to a radius of 0.5 m (r = 0.5) 1 m (r = 1) and 2 m (r = 2) (0, theoretical data, __ holographic value). (b) The imaginary part of pressure at meridian (4 = 0) for forward holography at 100 Hz from of 0.5 m to a radius of 0.5 m (r = 0.5) 1 m (r = 1) and 2 m (r = 2) (a, theoretical data, _ holographic value).
92
J. C. Lee
holography to radius R from any far-field is 2. The results of backward spherical holography from a radius of 1 m to a radius of R with the order of n = 2 and 12= 3 are shown in Fig. 2(a) and (b), respectively. It is obvious that the agreement between the two curves in Fig. 2(a) (n=2) is better than that in Fig. 2(b) (n = 3). Thus, in this case, inclusion of partial waves of order 2 is enough for backward holography to radius R. In Fig. 3(a) and (b), the real and imaginary part of the complex pressure, using backward spherical holography with order II= 3 at frequency 100 Hz from a
100
200 300 (velocity) angle-->
(pressure) angle-s
Fig. 5. (a) The amplitude of surface pressure at meridian (4 = 0) for backward holography at 100 Hz from a radius of 0.5 m to the surface of sphere with a source at north pole (0, holographic value). (b) The amplitude of particle velocity at mertheoretical data, ~ idian (4=0) for backward holography at 100 Hz from a radius of 0.5 m to the surface of holographic value). snhere with a source at north pole (0, theoretical data, ~
100
200
(pressure) angle->
300
100
200
300
(velocity) an{Jle->
Fig. 6. (a) The amplitude of surface pressure at meridian (4 = 0) for backward holography at 100 Hz from a radius of 1.0 m to the surface of sphere with a source at north pole holographic value). (b) The amplitude of particle velocity at (0, theoretical data, meridian (4 = 0) for backward holography at 100 Hz from a radius of 1.O m to the surface holographic value). of sphere with a source at north pole (0, theoretical data, __
Spherical acoustical holography of low-frequency noise sources
93
TABLE 2 The values of A,, and B,, for five monopole sources Mode (n. m)
Bnm
AlITI
@,O)
U,O) (171) (290) ‘(z; (310) (391)
9.838 3.156 2.256 0.937
0.00 0.00 1.74 0.00
0.656 0.047 0.236 0.053 0.003 0.001
0.345 0.375 0.00 0.017 0.075 0.014
_
radius of 2 m to a radius of 2 m (r = 2.0), 1 m (r = 1.O) and 0.5 m (r = OS), are compared with the predicted values for the same conditions. Except for the results of backward holography at the radius of 0.5 m, the results of holography at the other two radii are excellent. Therefore, the effect of spherical Hankel functions on the low-frequency near-field acoustical pressure reconstruction from backward holography is significant; this effect can be improved by using the optimum lower-order n=2 from eqn (1). However, comparison of the shapes of dotted and solid lines in Fig. 3 shows that the pressure distribution, even at near-field (r = OS), can be characterized by using backward spherical holography. For forward spherical holography from a radius of 0.5 m to a radius of 0.5 m (r = 0.5), 1 m (r = 1.O) and 2 m (r = 2.0), Fig. 4(a) and (b) shows only that at r = 0.5 m in the near-field complex pressure (r= 0.5) does it not match the theoretical data. Thus, to investigate the far-field pressure character, the results of forward spherical holography can be trusted in the near-field.
5 SOURCE
IDENTIFICATION
SIMULATION
We represent a point source of sound on the surface of a sphere by a small circular area of radius 6, having a velocity ua situated at the north pole (0 = 0). The velocity amplitude U(0) is defined as: U(0) = uo, 0 6 0 < S/a and U(0) = 0, S/a < 8< T, where a is the radius of sphere. The sphere surface vibrates with a velocity U(t9)e-jut . In term of a series of Legendre functions, the velocity amplitude can be expressed as5 u(e) = 2 U,P,(cos e> n=O
(11)
J. C. Lee
94
TABLE3 The values of A,,
and B,,, for a point source on the surface of a sphere
Mode (n, m)
A nm
(W (l>O) (171) (UY
B “t?Z 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.0067 0.0151 0.0000
0.0028 0.0000 0.0000 0.0020 0.0000 0.0000 0.0002
(231) Gs) (3>0) (3>1) (3,2) (353)
77
U(@P,(cos
(14
B)sin 8 d0
0
corresponding to velocity expressed in a series
U(@)e-@‘, and
p = fl: A&( n=O
the
cos @h,(kr)e+’
theoretical
pressure
is
(13)
where the values of the coefficients A, must be determined with the known coefficient U,. Using the numerical values a = 0.2, u. = 1, S = 0.01 and frequencyf= 100 Hz, Figs 5 and 6 and Table 3 can be obtained. As shown in Table 3 the spherical harmonic coefficient A,, increases until IZis larger than 2, and the optimum order y1= 3 in this simulation is determined by trial and error. The amplitude of sphere surface pressure and velocity reconstructed from the theoretical pressure at a radius of 0.5 m are shown in Fig. 5(a) and (b), respectively. Although the curves of surface velocity distribution in Fig. 5(b) do not agree as well as those for pressure distribution in Fig. 5(a), the surface velocity amplitude distribution can be investigated clearly by using spherical holography. The curves in Fig. 6, with the same conditions as in Fig. 5 except that they are reconstructed from a radius of 1.0 m, show that the agreement between dotted and solid lines is worse than that of Fig. 5. Therefore, it is believed that more precise reconstructions of pressure and velocity distribution at the sphere surface will be obtained with measurement of the original pressure field at a smaller radius.
Spherical acoustical holography of low-frequency noise sources
95
6 CONCLUSION The analysis described here has ascertained the capabilities of spherical acoustical holography for low-frequency near-field reconstruction. The precision of reconstruction is influenced by the near-field divergent behaviour, nevertheless it has been shown that the distribution of source surface pressure and particle velocity can still be identified by using holographic techniques. The results show that the effects of near-field divergent behaviour on the near-field holographic reconstruction can be improved by using the optimum order n in the spherical pressure equation. It has also been found that the results of backward holography to source surface from a smaller radius are more accurate than that of reconstruction from a larger radius.
REFERENCES 1. DeVries, L.A., Bolton, J.S. & Lee, J.C., Acoustical holography in spherical coordinates for noise source identification. Proceedings of NOISE-CON #94, 1944, pp. 935-940.
2. Zhukov, A.N., Ivannikov, A.N. & Pavlov, V.I., Identification of multipole sound sources. Soviet Physics and Acoustics, 363 (1990) 249-252. 3. Weinreich, G. & Arnold, E.B., Method for measuring acoustic radiation fields. Journal of the Acoustical Society of America, 682 (1980) 404-411. 4. Laville, F., Sidki, M. & Nicolas, J., Spherical acoustical holography using sound intensity measurements: theory and simulation. Acustica, 76 (1992) 193-198. 5. Morse, P.M. & Ingard, K.U., Theoretical Acoustics. McGraw-Hill, New York, 1968. 6. Laville, F. & Nicolas, J., A computer simulation of sound power determination using two-microphone sound intensity measurements. Journal qf the Acoustical Society of America, 91 (1992) 2042-2055.
Acoustics, Vo1.48, No. 1, pp. 8595, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0003-682X/96/$15.00+0.00
0003-682X(95)00068-2
ELSEVIER
Technical Note Spherical Acoustical Holography of Low-Frequency Noise Sources J. Department
C.Lee
of Marine Engineering Technology, National Taiwan Ocean University, 2 Pei Ning Road, Keelung, Taiwan (Received 4 September 1995; accepted 3 November 1995)
ABSTRACT The performance of near-jield reconstruction at lowfrequency using spherical holographic techniques is investigated. The spherical Hankel function with small argument (near-jield and low frequency) has divergent behaviour in the near-field. This divergent phenomenon influences the precision of holographic reconstruction. Two computer simulations based on the derived equations are used to demonstrate the capability of holography to construct the source surface pressure and particle velocity for noise source iden@cation. The results of multipole source simulation show that the precision oj backward holography to the near-jield depends on using the optimum order of holographic wave. Analysis of another simulation shows that more precise reconstruction of source surface pressure and particle velocity can be obtained from original pressure data at a smaller radius. Keywords: Acoustics,
spherical
holography.
1 INTRODUCTION
Acoustic field data in the vicinity of a vibrating structure can be used to evaluate either the acoustic far-field or the source characteristics. The prediction of acoustic far-field from near-field data is a special case of the forward holography of acoustic fields. Alternately, the identification of source characteristics on a vibrating structure is a special case of the backward holography of acoustic fields. Acoustical holography for the purpose of noise source identification has most often been applied in planar geometries. However, many small 85
86
J. C. Lee
machines have shapes that make planar holography inconvenient. In these cases, analysis based on spherical or cylindrical geometries may yield better results. The spherical holography technique has been applied in cases where the sound source may be entirely enclosed by a sphere on which sound pressure is measured. DeVries et al.’ applied the technique in both simulation and experiment to confirm the accuracy of spherical holography and to identify the noise source for small machines, especially refrigeration compressors. Zhukov et al.* identify multipole sound sources by deriving the general theoretical expressions of multipole source sound field energy characteristics in spherical coordinates. Weinreich and Arnold” have designed an experiment in which a boom system was employed to measure the complex sound pressure on two concentric spheres. They expanded the solution of the wave equation in spherical coordinates with the spherical harmonics calculated from the measurements; their method can be used to characterize the sound field when both incoming and outgoing waves are present. Laville et aZ.4have developed a spherical acoustical holography technique that uses sound intensity measurements to project the sound field for outgoing waves only. The acoustical holography for the identification of noise sources is a promising direction in aeroacoustics and underwater acoustics. However, for low-frequency noise source identification, the spherical Hankel function with small argument is divergent for backward holography. In previous work, mentioned above, the influence of the near-field divergent phenomenon on the source identification by backward holography has not been discussed. To investigate the performance and capability of lowfrequency spherical holography, two kinds of source are used in numerical simulation. One source consists of any number of multipole sources and the other is a point source on the surface of a sphere. This paper expands the solution of the wave equation in spherical coordinates in spherical harmonics, and determines the spherical harmonic coefficients by using the theoretical sphere pressure data from numerical calculation. These coefficients are then used to predict the sound field in either forward and backward holography; forward, to visualize the sound field, and backward to offer noise source identification. The acoustic particle velocity distribution over the surface of the source can also be reproduced by backward holography from the theoretical sphere pressure data.
2 THEORY
OF SPHERICAL
HOLOGRAPHY
The wave equation in spherical coordinates can be solved using Fourier’s method of separating the variables. The radial component of the sound
Spherical acoustical holography of low-frequency noise sources
87
field is described by spherical Hankel functions, and the polar and azimuthal components are described by Legendre functions, through which the spherical harmonics can be calculated. We take the acoustic pressure as the primary field quantity and represent the general form as
Here p(M,w) is the complex pressure at angular frequency w and locations M on any spherical surface. Spherical Hankel functions, h,(kr), can be calculated using a linear combination of the spherical Bessel and Neumann functions.5 The spherical surface harmonics Yim(O,4) and Yrr,(O,4) are defined by
where 8 is the polar angle measured from the position z-axis, and I$ is the azimuthal angle measured from the x-axis. P,(cosO) is the nth order Legendre function expanded in powers of cos8. The radial component of acoustic particle velocity can be defined as: u,(M, w) = -
-1 ~P(WW>
jwpo at-
(3)
The radial direction derivative of the Hankel function can be expressed in a recursive form dh,(kr)
~
d(kr)
n
=
-&#r)
- hn+l(kr)
(5)
where k is the wave number. The harmonic coefficients A,, and B,, can be derived by using the orthogonality of the spherical harmonics4
J. C. Lee
88
B nrnFpE _2n-1
4lr
(n-m)!
1 -
“(n+m)!h,(kr)
2?T 71 d$ p(M,w) Yirn(O,4)sin 8 d/3 I s 0
(7)
0
Here if m = 0, cm= 1 and if rn> 1, t, = 2. Once the harmonic coefficients are found, the pressure or particle velocity fields may be projected away from or toward the source, as needed, by using the eqns (l)-(4).
3 THE PERFORMANCES
OF SPHERICAL
HOLOGRAPHY
As shown in eqn (1) the wave equation separates in spherical coordinates into a product of spherical surface harmonics Y,,(Q, 4) and spherical Hankel functions h&r). The pressure difference between near-field and far-field are due to the influence of the radial functions h,(kr) only. At large kr, the function h,(kr) for far-field can be expressed as5 exp(jkr - jn i K) , jkr
h,(kr) ---f
kr+cc
(8)
here j = a, and the influence of y1on h,(kr) is only in characteristic phase shifts. For small values of kr, near-field and low frequency, h,(kr) can be described by
hdkr) +
(2n)!
kr -+ 0
j(n)!2”(kr)“+’ ’
When kr becomes smaller than 12,the near-field divergent behaviour will be found in this equation, and the near-field extended further and further out for high ~1.Therefore, the convergence of eqn (1) requires that the coefficients A,, and B,, become small at high K If the sound field is due to a source of approximate extent D located at the origin, it may be assumed that the projected wave will not contain appreciable contributions for n larger than kD, as long as the coefficients A,, and B,, do not increase for high n. On the other hand, if the sound source is modelled by a number of point monopoles situated in a region having a maximum distance R from the origin, the significant partial projected wave component will contribute up to about n = kR. Thus, for low frequency and small radius, eqn (1) should cut-off at an y1value at which the agreement with reality begins to become worse rather than better. As mentioned above, the lower-order ingoing or outgoing wave is significant for spherical holography. For a monopole placed at the origin, the
Spherical acoustical holography of low-frequency noise sources
89
sound field is a zero-order projected wave (n = 0) with radial factor h&r) and only the A00 coefficient is non-zero. The first-order projected waves (n = 1) with radial factor hi(kr) have three possible solutions for the three orientations of a dipole. If a dipole is oriented along the x-axis, the Ai0 is non-zero. When the monopole or dipole is moved off the coordinate axis, the sound field loses its symmetry, the orders of the coefficients change and the coefficient B,,, becomes significant. Similarly, the set of secondorder projected waves (n= 2) with its radial factor h&r) have five independent quadrupole waves. In general, the source order may be characterized by checking how the magnitudes of the harmonic coefficients fall off with increasing order.
4 MONOPOLE
SOURCES
ACOUSTIC
FIELD SIMULATION
The pressure field at a given radius due to any number of monopole sources can be predicted by using the Fourier transform pressure equation6
where p(Mi, w) is the Fourier transform pressure at point Mi due to a monopole source located at a distance ri. s is the number of monopole sources. Qdw) is the strength of source and p is the density of the medium. A,, and B,, may be estimated using discretized versions of eqns (6)--(s). The resulting harmonic coefficients could then be used to reconstruct the complex pressure field, which can be compared to the predicted values at any radius to check accuracy. The strength, phases and locations of sources are variable so that higher-order wave patterns may be simulated. To check the accuracy of numerical calculation, we first choose a set of five monopole sources (listed in Table 1) with the same location and TABLE 1
Source information Source number
1 2 3 4 5
of Section 4
Coordinates (X, Y, 2)
(0.000, 0.000, 0.000) (-0.201, -0.201, 0.101) (-0.201, 0.201, 0.001) (0.200, 0.100, 0.100) (0.300, -0.300, -0.200)
Strength Q (m’/s) 2.0 + J2.0
3.O-tjl.O 3.O+jl.O l.O+jl.O 2.0-j1.0
90
J. C. Lee
strength used elsewhere. 4 The performance and capabilities of lowfrequency spherical holography are studied through numerical simulation. In Figs l-6 the pressure predicted along a meridian (4=0) to compared with the complex pressure reconstructed from the spherical holography. The dotted lines represent the predicted, and the solid lines indicate the reconstruction using spherical holography. These line types are used in all figures unless otherwise stated. The curves of the complex pressure at a radius of 1 m and frequency of 200 Hz resulting from theoretical prediction and spherical holography are shown in Fig. l(a). With the same condition, Fig. l(b) shows results obtained elsewhere.4 The agreement of these two parts of Fig. 1 is excellent, and the accuracy of numerical simulation in this paper is reliable.
-31 0
300 100 200 (pressure) angle-->
Fig. 1. (a) The complex pressure at meridian holographicAvalue). (b) The complex theoretical data, __
0
I
2
3
4
S
6
7
B-
(c$= 0) from this paper (0, theoretical data, pressure at meridian (q5=0) from Ref.4 (09 holographic value).
81
8
A
6
ii -4
300 200 100 (pressure) angle-->
200 300 100 (pressure) angle-->
Fig. 2. (a) The amplitude of pressure at meridian (q5= 0) for order n = 2 (0, holographic value). (b) The amplitude of pressure at meridian data, __ holographic value). order n = 3 (0, theoretical data, ~
theoretical (d= 0) for
Spherical acoustical holography of low-frequency noise sources
91
The maximum distance R from the origin of these monopole sources is 0.5 m; the spherical harmonic coefficients A,, and B,, evaluated from the predicted pressure at 200 Hz and radius R are shown in Table 2. The amplitudes of the spherical harmonic coefficients A,, do not increase with n. According to the analysis in Section 3, the appreciable partial backward projected wave due to multipole sources radiating at low frequency will contribute up to order 12= kR. The value of kR is about 2. Therefore, for these multipole sources, the order n required for spherical backward
0
100 200 (pressure) angle-b
300
-2. 0
100 200 300 (pressure) angle-->
Fig. 3. (a) The real part of pressure at meridian (C#J = 0) for backward holography at 100 Hz from a radius of 2 m to a radius of 2 m (r = 2), 1 m (r = 1) and 0.5 m (r = 0.5) (0, theoretical
data, __ holographic value). (b) The imaginary part of pressure at meridian (Q,= 0) for backward holography at 100 Hz from a radius of 2 m to a radius of 2 m (r = 2) 1 m (r = 1) and 0.5 m (r = 0.5) (0, theoretical data, __ holographic value).
2
-1 0
100
200
(pressure) angle->
300
100
200
300
(pressure) angle-->
Fig. 4. (a) The real part of pressure at meridian (4 = 0) for forward holography at 100 Hz from a radius of 0.5 m to a radius of 0.5 m (r = 0.5) 1 m (r = 1) and 2 m (r = 2) (0, theoretical data, __ holographic value). (b) The imaginary part of pressure at meridian (4 = 0) for forward holography at 100 Hz from of 0.5 m to a radius of 0.5 m (r = 0.5) 1 m (r = 1) and 2 m (r = 2) (a, theoretical data, _ holographic value).
92
J. C. Lee
holography to radius R from any far-field is 2. The results of backward spherical holography from a radius of 1 m to a radius of R with the order of n = 2 and 12= 3 are shown in Fig. 2(a) and (b), respectively. It is obvious that the agreement between the two curves in Fig. 2(a) (n=2) is better than that in Fig. 2(b) (n = 3). Thus, in this case, inclusion of partial waves of order 2 is enough for backward holography to radius R. In Fig. 3(a) and (b), the real and imaginary part of the complex pressure, using backward spherical holography with order II= 3 at frequency 100 Hz from a
100
200 300 (velocity) angle-->
(pressure) angle-s
Fig. 5. (a) The amplitude of surface pressure at meridian (4 = 0) for backward holography at 100 Hz from a radius of 0.5 m to the surface of sphere with a source at north pole (0, holographic value). (b) The amplitude of particle velocity at mertheoretical data, ~ idian (4=0) for backward holography at 100 Hz from a radius of 0.5 m to the surface of holographic value). snhere with a source at north pole (0, theoretical data, ~
100
200
(pressure) angle->
300
100
200
300
(velocity) an{Jle->
Fig. 6. (a) The amplitude of surface pressure at meridian (4 = 0) for backward holography at 100 Hz from a radius of 1.0 m to the surface of sphere with a source at north pole holographic value). (b) The amplitude of particle velocity at (0, theoretical data, meridian (4 = 0) for backward holography at 100 Hz from a radius of 1.O m to the surface holographic value). of sphere with a source at north pole (0, theoretical data, __
Spherical acoustical holography of low-frequency noise sources
93
TABLE 2 The values of A,, and B,, for five monopole sources Mode (n. m)
Bnm
AlITI
@,O)
U,O) (171) (290) ‘(z; (310) (391)
9.838 3.156 2.256 0.937
0.00 0.00 1.74 0.00
0.656 0.047 0.236 0.053 0.003 0.001
0.345 0.375 0.00 0.017 0.075 0.014
_
radius of 2 m to a radius of 2 m (r = 2.0), 1 m (r = 1.O) and 0.5 m (r = OS), are compared with the predicted values for the same conditions. Except for the results of backward holography at the radius of 0.5 m, the results of holography at the other two radii are excellent. Therefore, the effect of spherical Hankel functions on the low-frequency near-field acoustical pressure reconstruction from backward holography is significant; this effect can be improved by using the optimum lower-order n=2 from eqn (1). However, comparison of the shapes of dotted and solid lines in Fig. 3 shows that the pressure distribution, even at near-field (r = OS), can be characterized by using backward spherical holography. For forward spherical holography from a radius of 0.5 m to a radius of 0.5 m (r = 0.5), 1 m (r = 1.O) and 2 m (r = 2.0), Fig. 4(a) and (b) shows only that at r = 0.5 m in the near-field complex pressure (r= 0.5) does it not match the theoretical data. Thus, to investigate the far-field pressure character, the results of forward spherical holography can be trusted in the near-field.
5 SOURCE
IDENTIFICATION
SIMULATION
We represent a point source of sound on the surface of a sphere by a small circular area of radius 6, having a velocity ua situated at the north pole (0 = 0). The velocity amplitude U(0) is defined as: U(0) = uo, 0 6 0 < S/a and U(0) = 0, S/a < 8< T, where a is the radius of sphere. The sphere surface vibrates with a velocity U(t9)e-jut . In term of a series of Legendre functions, the velocity amplitude can be expressed as5 u(e) = 2 U,P,(cos e> n=O
(11)
J. C. Lee
94
TABLE3 The values of A,,
and B,,, for a point source on the surface of a sphere
Mode (n, m)
A nm
(W (l>O) (171) (UY
B “t?Z 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.0067 0.0151 0.0000
0.0028 0.0000 0.0000 0.0020 0.0000 0.0000 0.0002
(231) Gs) (3>0) (3>1) (3,2) (353)
77
U(@P,(cos
(14
B)sin 8 d0
0
corresponding to velocity expressed in a series
U(@)e-@‘, and
p = fl: A&( n=O
the
cos @h,(kr)e+’
theoretical
pressure
is
(13)
where the values of the coefficients A, must be determined with the known coefficient U,. Using the numerical values a = 0.2, u. = 1, S = 0.01 and frequencyf= 100 Hz, Figs 5 and 6 and Table 3 can be obtained. As shown in Table 3 the spherical harmonic coefficient A,, increases until IZis larger than 2, and the optimum order y1= 3 in this simulation is determined by trial and error. The amplitude of sphere surface pressure and velocity reconstructed from the theoretical pressure at a radius of 0.5 m are shown in Fig. 5(a) and (b), respectively. Although the curves of surface velocity distribution in Fig. 5(b) do not agree as well as those for pressure distribution in Fig. 5(a), the surface velocity amplitude distribution can be investigated clearly by using spherical holography. The curves in Fig. 6, with the same conditions as in Fig. 5 except that they are reconstructed from a radius of 1.0 m, show that the agreement between dotted and solid lines is worse than that of Fig. 5. Therefore, it is believed that more precise reconstructions of pressure and velocity distribution at the sphere surface will be obtained with measurement of the original pressure field at a smaller radius.
Spherical acoustical holography of low-frequency noise sources
95
6 CONCLUSION The analysis described here has ascertained the capabilities of spherical acoustical holography for low-frequency near-field reconstruction. The precision of reconstruction is influenced by the near-field divergent behaviour, nevertheless it has been shown that the distribution of source surface pressure and particle velocity can still be identified by using holographic techniques. The results show that the effects of near-field divergent behaviour on the near-field holographic reconstruction can be improved by using the optimum order n in the spherical pressure equation. It has also been found that the results of backward holography to source surface from a smaller radius are more accurate than that of reconstruction from a larger radius.
REFERENCES 1. DeVries, L.A., Bolton, J.S. & Lee, J.C., Acoustical holography in spherical coordinates for noise source identification. Proceedings of NOISE-CON #94, 1944, pp. 935-940.
2. Zhukov, A.N., Ivannikov, A.N. & Pavlov, V.I., Identification of multipole sound sources. Soviet Physics and Acoustics, 363 (1990) 249-252. 3. Weinreich, G. & Arnold, E.B., Method for measuring acoustic radiation fields. Journal of the Acoustical Society of America, 682 (1980) 404-411. 4. Laville, F., Sidki, M. & Nicolas, J., Spherical acoustical holography using sound intensity measurements: theory and simulation. Acustica, 76 (1992) 193-198. 5. Morse, P.M. & Ingard, K.U., Theoretical Acoustics. McGraw-Hill, New York, 1968. 6. Laville, F. & Nicolas, J., A computer simulation of sound power determination using two-microphone sound intensity measurements. Journal qf the Acoustical Society of America, 91 (1992) 2042-2055.