The noise source identification technology based on acoustic radiation modes could identify the noise source under less measuring points, while the sensor placement has great influence on the result of noise source identification. Under the prescribed number of sensors, a method to configure the measuring points is proposed. By the means of successive removal, select a set of measuring points in the candidate field points, which make the orthogonality of the acoustic field distribution modes matrix achieve the maximum value. The results of numerical simulation and sound box experiment have shown that when sensors are arranged in accordance with the measuring points selected by this method, the noise source could be identified effectively, and the identification result is better than that when sensors are arranged evenly

Chang-Wei Su

Hai-Chao Zhu

Jun-Bo Su

Liang Guo

Rong-Fu Mao

English

Vibroengineering PROCEDIA

  © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 235 
Sensor placement in the noise source identification based 
on acoustic radiation modes 
Jun-Bo Su1, Hai-Chao Zhu2, Rong-Fu Mao3, Chang-Wei Su4, Liang Guo5 
National Key Laboratory on Ship Vibration and Noise, Naval University of Engineering, Wuhan, China 
2Corresponding author 
E-mail: 1sujunbo0618@163.com, 2haiczhu@163.com, 3maorfu@163.com, 4scwtz@163.com, 
5hgliangg@163.com 
(Accepted 31 August 2015) 
Abstract. The noise source identification technology based on acoustic radiation modes could 
identify the noise source under less measuring points, while the sensor placement has great 
influence on the result of noise source identification. Under the prescribed number of sensors, a 
method to configure the measuring points is proposed. By the means of successive removal, select 
a set of measuring points in the candidate field points, which make the orthogonality of the 
acoustic field distribution modes matrix achieve the maximum value. The results of numerical 
simulation and sound box experiment have shown that when sensors are arranged in accordance 
with the measuring points selected by this method, the noise source could be identified effectively, 
and the identification result is better than that when sensors are arranged evenly. 
Keywords: sensor placement, noise source identification, acoustic radiation modes. 
1. Introduction 
The near-field acoustic holography(NAH) proposed in the 1980s has obtained great success in 
noise source identification and localization, but the NAH requires a large number of sensors on 
the holographic plane, especially when applied to large structures. This limits NAH’s application 
in engineering practice. The acoustic radiation modes was proposed by Borgiotti etc. [1-3] in the 
1980s, and have been applied in active structural acoustic control and sound power calculation 
[4, 5]. Jiang Zhe [6] and Nie Yong-Fa [7, 8] have proved that the acoustic radiation modes could 
be applied in the noise source identification with small amount of sensors. When the number of 
sensors is small, the arrangement method of sensors has important influence on the reconstruction 
result of surface’s velocities. If the sensor placement is illogicality, the reconstruction result would 
become worse due to the correlation between the data of sensors. A method to select measuring 
points in the acoustic field is proposed in this paper. The measuring points are selected from a set 
of candidate field points to make the vectors that comprise the acoustic field distribution modes 
matrix achieve the highest orthogonality. The effectiveness of the sensor placement method is 
verified by numerical simulation and sound box experiment. 
2. Sensor placement in the noise source identification based on acoustic radiation modes 
2.1. The theory of noise source identification based on acoustic radiation modes 
The source’s surface can be decomposed a number of elements and the acoustic power radiated 
by the acoustic source may be written: 
܅ = ቀݏ2ቁ ܴ݁[ܞ
ு܈ܞ] = ܞு܀ܞ, (1)
in which ܀ = (ݏ 2⁄ )Re[܈] is the acoustic radiation resistance matrix; ܈ is the acoustic radiation 
impedance matrix; ݏ  is the area of the radiating element; ܞ is the vector of velocities on the 
source’s surface. Upon an eigen-decomposition of matrix, the matrix ܀ can be written as: 
܀ = ઴઩઴ு, (2)
SENSOR PLACEMENT IN THE NOISE SOURCE IDENTIFICATION BASED ON ACOUSTIC RADIATION MODES.  
JUN-BO SU, HAI-CHAO ZHU, RONG-FU MAO, CHANG-WEI SU, LIANG GUO 
236 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
in which ઴ is an ܰ×ܰ orthogonal matrix including a acoustic radiation mode in each column. ઩ 
(઩ = diag[ߣଵ, ߣଶ, . . . , ߣ௜, . . . ]) is a diagonal matrix of which diagonal elements are the eigenvalues 
of ܀. The vector of velocities on the source’s surface may be represented as: 
ܞ = ઴܋, (3)
in which ܋ is the expansion coefficient vector of acoustic radiation modes. Because the radiation 
efficiency of each acoustic radiation mode is proportional to the corresponding eigenvalue, the 
acoustic radiation modes are arranged in the matrix ઴ as the order of radiation efficiencies from 
high to low. Therefore vector of velocities on the source’s surface can be represented by the front 
parts of acoustic radiation modes. Assuming that the cutoff order number of acoustic radiation 
modes is ଵܰ, Eq. (3) becomes: 
ܞ ≈ ઴ଵ܋ଵ, (4)
in which ઴ଵ is a ܰ× ଵܰ matrix; ܋ଵ is a ଵܰ×1 vector.  
In the acoustic near-field, M field points are selected to measure acoustic pressures. The vector 
of the pressures of the ܯ field points can be presented by acoustic radiation modes [8]: 
۾ = ݆ߩܿ଴݇۵઴ଵcଵ, (5)
in which ߩ is fluid density; ܿ଴ is the sound speed in the fluid; ݇ is the wave number; ۵ is a ܯ× ଵܰ 
matrix whose elements are ܩ௜௝ = ݁ି௝௞௥೔ೕ 4ߨݎ௜௝ൗ , ݎ௜௝ is the distance between the field point and the 
point on the source’s surface. Let ૐ = ݆ߩܿ଴݇۵઴ଵ, the Eq. (5) becomes: 
۾ = ૐ܋ଵ. (6)
We define the column vectors of ૐ as the acoustic field distribution modes. The expansion 
coefficients of acoustic radiation modes ܋ଵ can be solved by Eq. (6), then the vector of velocities 
on the source’s surface can be calculated by Eq. (4). The noise sources could be identified by the 
reconstructed velocities finally. 
2.2. Sensor placement 
Because the location of the sensors impacts the selection of the vectors that constitute acoustic 
field distribution modes matrix, the sensor placement has great influence on the solution of 
expansion coefficients of acoustic radiation modes. If the sensor placement is illogicality, the 
column vectors of the acoustic field distribution modes matrix may have high correlation property 
that make the condition number of the acoustic field distribution modes matrix too large. Large 
condition number would make the reconstruction result worse, due to the error produced in the 
process of solving expansion coefficients of acoustic radiation modes. Therefore, in this paper, 
the measuring points are selected by the criterion achieving the highest possible orthogonality 
property of the vectors that constitute acoustic field distribution modes matrix. 
The same as Section 2.1, there are ܯ field points in the acoustic field, and the ܯଵ points of 
them are selected to solve the expansion coefficients. To make the equations well posed, we should 
select parts of the acoustic radiation modes. Assuming the cutoff order number of the acoustic 
radiation modes is ଵܰ  (ܯଵ > ଵܰ ), the vector of the pressures of the ܯ  field points can be 
reconstructed: 
ܘ௥(ெ×ଵ) = ૐ(ெ×ேభ)܋ଵ(ேభ×ଵ) = ૐૐାܘ଴(ெ×ଵ) = ܃܁܄ு܄܁ା܃ுܘ଴(ெ×ଵ) = ܃܁܁ା܃ுܘ଴(ெ×ଵ). (7)
Eq. (7) shows that the reconstructed pressure vector is reconstructed by itself, so the matrix ܂ 
SENSOR PLACEMENT IN THE NOISE SOURCE IDENTIFICATION BASED ON ACOUSTIC RADIATION MODES.  
JUN-BO SU, HAI-CHAO ZHU, RONG-FU MAO, CHANG-WEI SU, LIANG GUO 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 237 
(܂ = ܃܁܁ା܃ு) must be a unit matrix. The diagonal elements of matrix ܂ and the elements of the 
reconstructed pressure vector are one-to-one correspondence. In order to eliminate the field point 
that is the most correlative to the smallest singular value, the last diagonal element of matrix ܁ 
and matrix ܁ା in Eq. (7) are replaced by zero. Then the matrix ܂ is recalculated, and the field 
point that corresponds with the smallest diagonal element of the matrix ܂ is the most correlative 
to the smallest singular value. Therefore, an iterative process could be established for selecting 
the optimum sensor placement for a prescribed number of field points from a set of candidate 
points. Within each iteration, the matrix ܂ is calculated, and the field point that corresponds to the 
smallest diagonal value is eliminated from the set of candidate points. For the remaining points 
the matrices ܃, ܄ and ܁ are evaluated again, and the product ܂ is computed. The elimination 
process continues until the number of the remaining field points is equal to the prescribed number. 
The whole iteration process would ensure that the condition number of the final acoustic field 
distribution modes matrix is the smallest. 
3. Numerical simulation 
The validity of the proposed method in this paper is certified by a numerical simulation of a 
plate with infinite baffle. A single point-driven steel plate is selected, the dimensions of the plate 
are 0.5 m×0.5 m×0.008 m. The baffled plate is modeled in air with 16×16 constant boundary 
elements, and speed of sound of 340 m/s is assumed for air. When the plate is motivated by an 
exciting force with a frequency of 380 Hz, an amplitude of 10 N, the vibration mode of the plate 
is (1, 2). Fig. 1 shows the given particle velocity pattern of the plate. 
x/m
y /m
 
 
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Fig. 1. The amplitude of velocity of the plate at the exciting frequency of 380 Hz 
 
Fig. 2. The optimal sensor placement  Fig. 3. The even sensor placement 
The measuring plane is placed in parallel with the plate in the acoustic field. The distance 
between the measuring plane and the plate is 0.1 m. The area of the measuring plane is four times 
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SENSOR PLACEMENT IN THE NOISE SOURCE IDENTIFICATION BASED ON ACOUSTIC RADIATION MODES.  
JUN-BO SU, HAI-CHAO ZHU, RONG-FU MAO, CHANG-WEI SU, LIANG GUO 
238 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
of plate’s area, and there are 32×32 measuring points on the measuring plane. The theoretical 
values of measuring points are obtained by the Rayleigh integral. Supposing 36 measuring sensors 
are placed on the measuring plane, the sensor placement obtained by this paper’s method is shown 
as in Fig. 2. In addition, an even placement is selected for comparison which is as shown in Fig. 3. 
The velocities on the plate surface are reconstructed respectively in the condition of the above 
two sensor placements. The reconstructed result by the sensor placement shown in Fig. 2 is shown 
in Fig. 4; the reconstructed result by the sensor placement shown in Fig. 3 is shown in Fig. 5. The 
reconstruction error of the first even placement is 3.2 %, the reconstruction error of the second 
even placement is 8.7 %. It can be seen from Fig. 4 that the two noise sources on the plate are 
identified accurately, and it can be known that the identification of the sensor placement selected 
by this paper is better than that of the even sensor placement from the comparison between Fig. 4 
and Fig. 5. 
x/m
y /m
 
 
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Fig. 4. Reconstructed velocities on plate’s surface 
(the optimal sensor placement) 
x/m
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Fig. 5. Reconstructed velocities of plate’s surface  
(the even sensor placement) 
4. Sound box experiment 
The sound box experiment is conducted in the open building. In this experiment, two sound 
boxes issue audio signals with the frequency of 100 Hz. The distance between the two sound boxes 
is 12 cm. The size of the measuring plane is 60 cm×50 cm. The measuring points is 13×11. The 
distance between two adjacent measuring points is 5 cm in both two directions. The distance 
between the measuring plane and the plane of sound boxes is 10 cm. The experimental scene is 
shown as Fig. 6. 
 
Fig. 6. The scene of experiment 
In the reconstructing process, the size of source’s plane is equal to that of measuring plane, 
and 42 measuring points are selected in all the 143 measuring points. Two senor placements are 
selected to reconstruct the velocities of source’s plane, one is selected by the paper’s method; the 
other is an even placement, as shown in Fig. 7 and Fig. 8. 
SENSOR PLACEMENT IN THE NOISE SOURCE IDENTIFICATION BASED ON ACOUSTIC RADIATION MODES.  
JUN-BO SU, HAI-CHAO ZHU, RONG-FU MAO, CHANG-WEI SU, LIANG GUO 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 239 
As shown in Fig. 9 and Fig. 10, the velocities of source’s plane are reconstructed by two sensor 
placement respectively. It can be seen from the two figures that the locations of two sound boxes 
are identified accurately in Fig. 9, while only one sound box’s location can be identified in the 
Fig. 10. That is to say, the reconstructed result of this paper’s sensor placement is better than that 
of the even sensor placement. The validity of the method for selecting the optimum sensor 
placement is verified by the experiment.  
 
Fig. 7. The optimal sensor placement  
on measuring plane 
 
Fig. 8. The even sensor placement  
on measuring plane 
 
m/s
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Fig. 9. Reconstructed velocities of measuring plane 
(the optimal sensor placement) 
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Fig. 10. Reconstructed velocities of measuring plane 
(the even sensor placement) 
5. Conclusions 
A method to select the measuring points in the acoustic field is proposed to realize the noise 
source identification with small amount of sensors. The measuring points are selected from a set 
of candidate field points, which make the vectors that comprise the acoustic field distribution 
modes matrix achieve the highest orthogonality. The results of numerical simulation and sound 
box experiment have shown that when sensors are arranged in accordance with the measuring 
points selected by the proposed method, the noise source could be identified effectively, and the 
result is better than that when sensors are arranged evenly. 
References 
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SENSOR PLACEMENT IN THE NOISE SOURCE IDENTIFICATION BASED ON ACOUSTIC RADIATION MODES.  
JUN-BO SU, HAI-CHAO ZHU, RONG-FU MAO, CHANG-WEI SU, LIANG GUO 
240 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
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Sensor placement in the noise source identification based on acoustic radiation modes

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Sensor placement in the noise source identification based on acoustic radiation modes

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