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Sound field separating on arbitrary surfaces enclosing a sound scatterer based on combined integral equations
Ultrasonics 54 (2014) 2169–2177
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Sound field separating on arbitrary surfaces enclosing a sound scatterer based on combined integral equations Zongwei Fan, Deqing Mei ⇑, Keji Yang, Zichen Chen The State Key Laboratory of Fluid Power Transmission and Control, Department of Mechanical Engineering, Zhejiang University, Hangzhou 310027, PR China
a r t i c l e
i n f o
Article history: Received 7 February 2013 Received in revised form 8 February 2014 Accepted 11 June 2014 Available online 20 June 2014 Keywords: Sound field separation Arbitrary spatial surfaces Sound scatterer Combined integral equations Boundary element discretization
a b s t r a c t To eliminate the limitations of the conventional sound field separation methods which are only applicable to regular surfaces, a sound field separation method based on combined integral equations is proposed to separate sound fields directly in the spatial domain. In virtue of the Helmholtz integral equations for the incident and scattering fields outside a sound scatterer, combined integral equations are derived for sound field separation, which build the quantitative relationship between the sound fields on two arbitrary separation surfaces enclosing the sound scatterer. Through boundary element discretization of the two surfaces, corresponding systems of linear equations are obtained for practical application. Numerical simulations are performed for sound field separation on different shaped surfaces. The influences induced by the aspect ratio of the separation surfaces and the signal noise in the measurement data are also investigated. The separated incident and scattering sound fields agree well with the original corresponding fields described by analytical expressions, which validates the effectiveness and accuracy of the combined integral equations based separation method. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Sound field separation, which decomposes the total composite sound field to the incident and scattering partial fields on a surface enclosing a sound scatterer, is an important subject in many acoustic applications. For near-field acoustic holography (NAH) of vibrating structures mounted in non-anechoic environment, sound fields propagating inward to the measurement surfaces should be extracted and removed from the total sound field [1,2]. Besides, in contactless micro-manipulation by means of acoustic radiation force, sound field synthesis technology [3] is applied to produce the desired sound field that corresponds with the specific distribution of acoustic radiation force [4,5]. However, obstacles often exist, which make the sound field synthesis domain a non-free space. To guarantee the applicability of the sound field synthesis, sound field separation technique is used to overcome adverse disturbances in the propagation operator caused by the obstacle’s scattering field. Other applications necessitating sound field separation include characteristic parameter measurement for acoustic material [6], acoustic target strength characterization in underwater acoustics [7], etc.
⇑ Corresponding author. Tel.: +86 571 87951906; fax: +86 571 87951145. E-mail address: [email protected] (D. Mei). http://dx.doi.org/10.1016/j.ultras.2014.06.007 0041-624X/Ó 2014 Elsevier B.V. All rights reserved.
In conventional sound field separation methods, spatial Fourier transform (SFT) is applied to convert the total sound field from spatial domain to wave-number domain, in which subsequent sound field separating operation is accomplished. Weinreich and Arnold [8] proposed a method to decompose double layer sound fields with spherical harmonic functions. Frisk et al. [9] derived a SFT-based method for measuring the reflection coefficient of the seafloor. Tamura et al. [10,11] developed a method to measure the plane-wave reflection coefficient at oblique incidence. Cheng et al. [12] extended that method, by which the scattering field from a complex shaped object is separated with the incident field in Cartesian and cylindrical coordinates. This method was further applied by Yu et al. [13] to achieve sound field separation in spherical coordinates. The major advantage of the SFT-based methods mentioned above is their simplicity and efficiency in application. However, due to the inherent characteristics of SFT, these methods can only accomplish sound field separation for three types of regular surfaces: those with planar, cylindrical, and spherical geometries. In many other situations, irregularly shaped separation surfaces are often involved. For example, to obtain near-field information on the scattering field from an irregular sound scatterer, the separation surfaces have to be conformal with the physical frontier of the scatterer and then become irregular accordingly. In addition, to enlarge the applicable region dimension of sound field synthesis,
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the spatial points on which the propagation operator values are measured should be distributed over a surface as close to the obstacle as possible. Therefore, if the obstacle is irregular, sound field separation on irregularly shaped surfaces will be required. Besides, in SFT-based separation process, wave-number spectrum of the sound pressure on the double measurement surfaces should be truncated at the same order. This requirement leads to amplification of the measurement error due to the ill-posedness effect of inverse problems. Up to now, there have been developed several methods for sound field separation on irregular surfaces. Bi et al. [14,15] proposed a technique based on the equivalent source method. In its application, sound pressure fields are measured and separated on two closely-spaced and parallel surfaces. Based on the fundamental solution of the Helmholtz equation and the wave superposition principle, Fernandez-Grande et al. [16,17] developed a method to deal with sound field separation on non-separable geometries. The common idea of the methods in Ref. [14–17] is to approximate the original sound fields by linear combination of the component fields from the equivalent sources, then solve the weight coefficients by matching the assumed-form solution to the measurement data on the separation surfaces. Since the separating operation is performed directly in the spatial domain instead of the wave-number domain, these techniques are applicable to arbitrary surfaces, and the limitations (such as window effects) due to SFT can be avoided. A difficulty in using these equivalent source based methods is that there exists little theoretical guidance to determine the proper location, order (or type) and number of the equivalent sources, although separation effect is closely related with these characteristics. Based on the Helmholtz integral equation, Langrenne et al. [18,19] developed a technique for sound field separation on general surfaces. For its implementation, normal velocity on the objective separation surface is obtained indirectly by finite difference of sound pressure, which are measured on two adjacent and conformal surfaces. Owing to the ill-posedness of the finite difference operation, small noise in the measurement data might induce large error in the approximated normal velocity [20], and therefore make the separation results depart from the corresponding original fields. To overcome the limitations of these existing approaches, a sound field separation method is proposed in this study based on combined integral equations. In Section 2, three separation methods in the form of combined integral equations are established theoretically. In Section 3, three corresponding systems of linear equations are obtained through boundary element discretization of the separation surfaces. Numerical simulations on sound field separation under different conditions are carried out in Section 4. Conclusions and suggestions for future research are given in Section 5.
2. Sound field separation based on combined integral equations In the exterior domain of an arbitrarily shaped sound scatterer, the total sound pressure pt(r, t) consists of two parts, i.e., the original incident sound pressure pin(r, t) and the scattering sound pressure psc(r, t). The geometry of interest is depicted in Fig. 1. The domain occupied by the scatterer is denoted by X and its physical frontier by S0. S1 and S2 are two arbitrarily shaped surfaces enclosing the scatterer, on which the total sound fields are measured and then separated. For a time-harmonic exp(ixt) disturbance of angular frequency x, the spatial part of sound pressure outside the scatterer satisfies the Helmholtz equation Dp + k2p = 0, in which k is the wave number. When subjected to the Sommerfeld radiation condition and the boundary condition on the scatterer surface, boundary
integral equations can be found. These boundary integral equations are often classified as direct or indirect, where direct corresponds to the Helmholtz integral equation and the numerous indirect equations are based on layer potentials. The Helmholtz integral equation satisfied by the spatial part pin(r) of the incident sound pressure is
a pin ðrÞ ¼ tS pin ðr0 Þ
@ 0 0 dSðr0 Þ; Gðr; r0 Þ iq0 xv in ðr ÞGðr; r Þ n @n
ð1Þ
where the integral surface S is an arbitrary smooth surface enclosing the scatterer, G is the free space Green function, q0 is the equilibrium density of the sound medium, n is the unit interior normal to S, v in n is the incident normal velocity along the normal direction n [21]. The value of the coefficient a varies as
a¼
8 > <0
if r is outside S : if r is on S : 1 if r is inside S 1 >2
According to Eq. (1) for a = 0, the incident sound pressure and normal velocity on S1 can be applied to compute the incident sound pressure on S2, i.e.,
@ 0 ¼ tS1 pin ðr1 Þ Gðr2 ; r1 Þ iq0 xv in n ðr1 ÞGðr2 ; r1 Þ dSðr1 Þ: @n
ð2Þ
Similarly, the incident sound pressure and normal velocity on S2 can also be applied to compute the incident sound pressure on S1
@ pin ðr1 Þ ¼ tS2 pin ðr2 Þ Gðr1 ; r2 Þ iq0 xv in n ðr2 ÞGðr1 ; r2 Þ dSðr2 Þ: @n
ð3Þ
The Helmholtz integral equation satisfied by the spatial part pt ðrÞ of the total sound pressure is
a pt ðrÞ ¼ pin ðrÞ þ tS pt ðr0 Þ
@ Gðr; r0 Þ iq0 xv tn ðr0 ÞGðr; r0 Þ dSðr0 Þ; @n
ð4Þ where v tn is the total normal velocity along the normal direction n [21]. The coefficient
8 > < 1 if r is outside S a ¼ 12 if r is on S : > : 0 if r is inside S In Ref. [18,19], the total sound pressure pt(r) and the total normal velocity v tn ðrÞ on S are both known. Hence the incident sound pressure pin(r) on S can be computed based on Eq. (4), while the scattering sound field psc(r) on S equals pt(r) pin(r). In this study, we consider the application conditions where only one type of sound field data is known by measurement, either the total sound pressure or the total normal velocity. When the field point r is outside the integral surface S, Eq. (4) could be written as
@ 0 0 0 pt ðrÞ ¼ pin ðrÞ þ tS pin ðr0 Þ Gðr; r0 Þ iq0 xv in n ðr ÞGðr; r Þ dSðr Þ @n @ 0 0 0 þ tS psc ðr0 Þ Gðr; r0 Þ iq0 xv sc n ðr ÞGðr; r Þ dSðr Þ @n @ 0 0 dSðr0 Þ: ðr ÞGðr; r Þ ¼ pin ðrÞ þ tS psc ðr0 Þ Gðr; r0 Þ iq0 xv sc n @n Since pt(r) = pin(r) + psc(r), it can be derived from the above formula that in the domain outside S,
@ 0 0 0 psc ðrÞ ¼ tS psc ðr0 Þ Gðr; r0 Þ iq0 xv sc n ðr ÞGðr; r Þ dSðr Þ: @n
ð5Þ
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Fig. 1. Geometry configuration of sound field separation.
Through a similar derivation, we can derive that when the field point r locates on S,
1 @ 0 0 dSðr0 Þ; psc ðrÞ ¼ tS psc ðr0 Þ Gðr; r0 Þ iq0 xv sc ðr ÞGðr; r Þ n 2 @n
velocity on the same surface could be derived. When the field point is on S1,
(
ð6Þ
@ 0 0 0 12 pin ðr1 Þ ¼ tS1 pin ðr01 Þ @n Gðr1 ; r01 Þ iq0 xv in n ðr1 ÞGðr1 ; r1 Þ dSðr1 Þ : 1 @ 0 0 0 p ðr Þ ¼ tS1 psc ðr01 Þ @n Gðr1 ; r01 Þ iq0 xv sc n ðr1 ÞGðr1 ; r1 Þ dSðr1 Þ 2 sc 1 ð12Þ
and when the field point r is inside S,
@ 0 0 0 0 psc ðrÞ ¼ tS psc ðr0 Þ Gðr; r0 Þ iq0 xv sc n ðr ÞGðr; r Þ dSðr Þ: @n
ð7Þ
According to Eq. (5), the scattering sound pressure and normal velocity on S1 can be applied to compute the scattering sound pressure on S2, i.e.,
@ psc ðr2 Þ ¼ tS1 psc ðr1 Þ Gðr2 ; r1 Þ iq0 xv sc n ðr1 ÞGðr2 ; r1 Þ dSðr1 Þ: @n
ð8Þ
According to Eq. (7), by using the scattering sound pressure and normal velocity on S2 to compute the scattering sound pressure on S1, we have
@ 0 ¼ tS2 psc ðr2 Þ Gðr1 ; r2 Þ iq0 xv sc n ðr2 ÞGðr1 ; r2 Þ dSðr2 Þ: @n
ð9Þ
By summing Eqs. (2) and (8), we have @ sc psc ðr2 Þ ¼ tS1 pt ðr1 Þ Gðr2 ; r1 Þ iq0 x½v in n ðr1 Þ þ v n ðr1 ÞGðr2 ; r1 Þ dSðr1 Þ: @n
ð10Þ
while the summation of Eqs. (3) and (9) gives @ sc pin ðr1 Þ ¼ tS2 pt ðr2 Þ Gðr1 ; r2 Þ iq0 x½v in ðr Þ þ v ðr ÞGðr ;r Þ dSðr2 Þ: 2 2 1 2 n n @n
ð11Þ
In Eqs. (10) and (11), there are six unknown quantities, i.e., pin(r1), sc in sc psc(r2), v in n ðr1 Þ; v n ðr1 Þ; v n ðr2 Þ and v n ðr2 Þ. To achieve sound field separation, four more equations about these six unknown quantities have to be found for solving the sound pressure pin(r1) on S1 and psc(r2) on S2. Based on Eqs. (1) and (6) for the field points on S1 and S2, quantitative relationship between the sound pressure and normal
According to the first formula in Eq. (12), the incident normal velocity v in n ðr1 Þ can be determined by the incident sound pressure pin(r1). Since the scattering sound pressure psc(r1) = pt(r1) pin(r1), the scattering normal velocity v sc n ðr1 Þ can also be determined by the incident sound pressure pin ðr1 Þ based on the second formula in Eq. (12). Similarly, when the field point is on S2, we have
(
q xv q xv
1 @ in 0 0 0 p ðr Þ ¼ tS2 pin ðr02 Þ @n Gðr2 ; r02 Þ i 0 n ðr2 ÞGðr2 ; r2 Þ dSðr2 Þ 2 in 2 : 1 @ 0 0 sc 0 0 0 2 psc ðr2 Þ ¼ tS2 psc ðr2 Þ @n Gðr2 ; r2 Þ i 0 n ðr2 ÞGðr2 ; r2 Þ dSðr2 Þ
ð13Þ According to the second formula in Eq. (13), the scattering normal velocity v sc n ðr2 Þ can be determined by the scattering sound pressure psc ðr2 Þ. Since the incident sound pressure pin(r2) = pt(r2) psc(r2), the incident normal velocity v in n ðr2 Þ can also be determined by the scattering sound pressure psc(r2) based on the first formula. So far, the required four equations mentioned above have been obtained. sc Through expressing v in n ðr1 Þ and v n ðr1 Þ by pin ðr1 Þ and then substituting them into Eq. (10), we get one integral equation describing the relationship between the incident sound pressure pin(r1) on S1 and the scattering sound pressure psc(r2) on S2. Simisc larly, through expressing v in n ðr2 Þ and v n ðr2 Þ by psc(r2) and then substituting them into Eq. (11), the second integral equation about pin(r1) and psc(r2) is obtained. By combining these two integral equations and then solving the unknown quantities pin(r1) and psc(r2), sound field separation on the measurement surfaces S1 and S2 is consequently realized. We further derive the second system of combined integral equations with the total normal velocity v tn ðr1 Þ and v tn ðr2 Þ as the
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unsolved quantities. On summing the two integral equations in Eq. (12), we have
1 @ pin ðr1 Þ ¼ pt ðr1 Þ tS1 pt ðr01 Þ Gðr1 ; r01 Þ iq0 xv tn ðr01 ÞGðr1 ;r01 Þ dSðr01 Þ: 2 @n
ð14Þ
Substituting Eq. (14) into Eq. (11), we can get one integral equation which describes the relationship about the total sound pressure pt(r1), pt(r2) and the total normal velocity v tn ðr1 Þ; v tn ðr2 Þ. Similarly, on summing the two integral equations in Eq. (13), we have
psc ðr2 Þ ¼
1 p ðr2 Þ 2 t @ tS2 pt ðr02 Þ Gðr2 ; r02 Þ iq0 xv tn ðr02 ÞGðr2 ; r02 Þ dSðr02 Þ: @n ð15Þ
By substituting Eq. (15) into Eq. (10), we get the second equation about pt(r1), pt(r2), v tn ðr1 Þ and v tn ðr2 Þ. Therefore, the second system of combined integral equations are established with the total normal velocity v tn ðr1 Þ and v tn ðr2 Þ as the unknown quantities. On substituting the solution of the total normal velocity v tn ðr1 Þ and v tn ðr2 Þ into Eqs. (10) and (11), the incident sound pressure pin(r1) on S1 and the scattering sound pressure psc(r2) on S2 will be solved, then sound field separation on S1 and S2 is achieved. Furthermore, if the statuses of the total sound pressure and the total normal velocity in the second system of combined integral equations are exchanged, the third system can be derived, in which the total sound pressure pt(r1) and pt(r2) are turned to be the unsolved quantities, while the normal velocity v tn ðr1 Þ and v tn ðr2 Þ are obtained by measurement. 3. Boundary element discretization of the combined integral equations Through boundary element discretization of the surfaces S1 and S2 involved in above integral equations [22], three systems of combined integral equations developed in the previous section are transformed to three corresponding systems of linear equations for practical applications. The matrix formulae of Eqs. (10) and (11) are
(
sc psc ðr2 Þ ¼ D21 pt ðr1 Þ M21 v in n ðr1 Þ M21 v n ðr1 Þ;
ð16Þ
sc pin ðr1 Þ ¼ D12 pt ðr2 Þ M12 v in n ðr2 Þ M12 v n ðr2 Þ;
in which matrices Dij and Mij imply the effects on a field point on Si by the dipoles and monopoles on Sj, respectively. The matrix formulae of Eqs. (12) and (13) are (
v
12 pin ðr1 Þ ¼ D11 pin ðr1 Þ M11 in n ðr1 Þ; 1 p ðr Þ ¼ D11 psc ðr1 Þ M11 sc n ðr1 Þ; 2 in 1
v
(
v
1 p ðr Þ ¼ D22 pin ðr2 Þ M22 in n ðr2 Þ; 2 in 2 12 psc ðr2 Þ ¼ D22 psc ðr2 Þ M22 sc n ðr2 Þ;
v
ð17Þ
in which matrices Dii and Mii imply the effects on a surface point on Si by dipoles and monopoles on the same surface Si. Substituting Eq. (17) into Eq. (16), we have
8 h i 1 1 > < psc ðr2 Þ ¼ D21 M21 M1 11 D11 2 I11 pt ðr1 Þ M21 M11 pin ðr1 Þ; h i 1 > 1 : pin ðr1 Þ ¼ D12 M12 M1 22 D22 2 I22 pt ðr2 Þ M12 M22 psc ðr2 Þ; ð18Þ where I11 and I22 are identity matrices. By reorganizing above two equations, the following system of linear equations could be obtained:
F1
pin ðr1 Þ psc ðr2 Þ
¼ b1 ;
ð19Þ
where " F1 ¼
I11
M12 M1 22
M21 M1 11
I22
# ;
0h 1 i 1 D M12 M1 22 D22 2 I22 pt ðr2 Þ C B 12 i b1 ¼ @ h A: 1 D21 M21 M1 11 ðD11 2 I11 Þ pt ðr1 Þ
On solving Eq. (19), pin(r1) and psc(r2) are separated from the total sound pressure pt(r1) on S1 and pt(r2) on S2, while the scattering sound pressure psc(r1) = pt(r1) pin(r1) and the incident sound pressure pin(r2) = pt(r2) psc(r2). Thus, sound field separation on the surfaces S1 and S2 is accomplished. Similarly, we can also derive the discretization formulae for the second system of combined integral equations described in the former section. The matrix formulae of Eqs. (14) and (15) are
1 pin ðr1 Þ ¼ D11 I11 pt ðr1 Þ þ M11 v tn ðr1 Þ; 2
ð20Þ
1 psc ðr2 Þ ¼ D22 I22 pt ðr2 Þ þ M22 v tn ðr2 Þ: 2
ð21Þ
Substituting Eqs. (20) and (21) into Eq. (16), we have
1 M11 v tn ðr1 Þ þ M12 v tn ðr2 Þ ¼ D11 I11 pt ðr1 Þ þ D12 pt ðr2 Þ; 2
ð22Þ
1 M21 v tn ðr1 Þ þ M22 v tn ðr2 Þ ¼ D21 pt ðr1 Þ þ D22 I22 pt ðr2 Þ: 2
ð23Þ
By reorganizing the above two equations, the second system of linear equations is obtained:
F2
v tn ðr1 Þ v tn ðr2 Þ
¼ b2 ;
ð24Þ
in which
F2 ¼
M11
M12
M21
M22
;
" b2 ¼
D11 12 I11
D12
D21
D22 12 I22
#
pt ðr1 Þ pt ðr2 Þ
:
For cases where the total sound pressure are quantities to be solved on basis of the total normal velocity data, the third system of linear equations can be derived through exchanging the statuses of the total sound pressure and the total normal velocity in Eqs. (22) and (23):
F3
ptn ðr1 Þ ptn ðr2 Þ
¼ b3 ;
ð25Þ
where
" F3 ¼
D11 12 I11
D12
D21
D22 12 I22
# ;
b3 ¼
M11
M12
M21
M22
v tn ðr1 Þ v tn ðr2 Þ
:
In Eq. (19), the computation of the left-hand-side matrix F1 and the right-hand-side vector b1 relies on the inverse matrices M1 11 and M1 22 . When the scale of sound field separation problem becomes quite large, such as high frequency sound field or large sized, three dimensional sound scatterer, inversion operation of the matrices M11 and M22 will consume much computation time and memory. In contrast, explicit inversion of matrices is inherently avoided in Eqs. (24) and (25), so the latter two ways for sound field separation will be better choices for large scale problems. 4. Numerical simulations To validate the effectiveness and accuracy of the sound field separation method proposed in the former sections, numerical study is carried out in this section. In the following numerical
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Fig. 2. Geometry configuration for numerical simulation. (a) Elliptic cylinder with elliptical and rectangular measurement contours and (b) circular cylinder with circle measurement contours.
simulations, the measurement contours are divided by a set of straight linear segments with the well-known one-sixth of the wavelength criterion satisfied. Constant elements are adopted for boundary element discretization. For quantifying the accuracy of the separated results compared with the corresponding analytic values, relative error in percent is defined as
PM E¼
2 i¼1 kpex ðiÞ psfs ðiÞk2 PM 2 i¼1 kpex ðiÞk2
100ð%Þ;
ð26Þ
incident sound field is composed of three plane progressive waves propagating in the xoy plane with the incident direction angles u1in ¼ 30 , u2in ¼ 150 and u3in ¼ 270 . For air medium, the sound speed is 344 m/s. The frequency of the incident sound field is set to be 1.5 kHz. In addition, random Gaussian noise is imposed with predetermined variance and 30 dB SNR (Signal/Noise Ratio). The Cartesian coordinates (x, y, z) could be expressed by elliptic cylindrical coordinates (u, v, z) as follows,
x ¼ f cosh u cos v ; y ¼ f sinh u sin v ; z ¼ z
where M is the number of discretization nodes on the separation surfaces, pex is the analytic value given by theoretical expression and psfs denotes the sound field separation result.
where 0 6 u < 1 and 0 6 v < 2p [23]. The spatial part of the incident sound pressure field described by elliptic cylindrical coordinates (u, v) is
4.1. Sound field separation for an elliptic cylinder
pin ðu; v Þ ¼
In Fig. 2a, we consider an elliptic cylinder with its center at the origin of the xyz coordinates and the central axis coincides with the z axis. The cross section ellipse boundary S0 in the xoy plane is defined by x2 =x20 þ y2 =y20 ¼ 1. The half major axis length x0 and the half minor axis length y0 are set to be 0.5 m and 0.05 m, respecqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tively. The semi-focal distance f of S0 is given by f ¼ x20 y20 . The
n 1 pffiffiffiffiffiffiffiX i Je ðs; uÞSen ðs; v Þ½Sen ðs; u1in Þ þ Sen ðs; u2in Þ 8p Nen n n¼0
þ Sen ðs; u3in Þ þ
n 1 pffiffiffiffiffiffiffiX i Jo ðs; uÞSon ðs; v Þ½Son ðs; u1in Þ 8p Non n n¼0
þ Son ðs; u2in Þ þ Son ðs; u3in Þ; in which s = (kf)2, k is the wave number, Sen and Son are angular Mathieu functions, Jen and Jon are radial Mathieu functions of the
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first kind, Nen and Non are normalization constants [24,25]. For an elliptic cylinder with a sound rigid boundary, the spatial part of the scattering sound pressure field is n 1 pffiffiffiffiffiffiffiX i Je0n ðs; uÞ 1 psc ðu; v Þ ¼ 8p Heð1Þ n ðs; uÞSen ðs; v Þ½Sen ðs; uin Þ Nen Heð1Þ0 n¼0 n ðs; uÞ n 1 pffiffiffiffiffiffiffiX i þ Sen ðs; u2in Þ þ Sen ðs; u3in Þ 8p Non n¼0
Jo0n ðs; uÞ Honð1Þ0 ðs; uÞ
the incident and scattering sound pressure fields described by cylindrical coordinates (r, u) are
pin ðr; uÞ ¼ expðikr cos uÞ; psc ðr; uÞ " # 1 X J 0n ðkaÞ ¼ en ð1Þ0 Hnð1Þ ðkrÞ cosðnuÞ; Hn ðkaÞ n¼0
1 2 Hoð1Þ n Son ðs; v Þ½Son ðs; uin Þ þ Son ðs; uin Þ
þ Son ðs; u3in Þ; where Henð1Þ and Honð1Þ are radial Mathieu functions of the third kind. We first consider the measurement contours S1 and S2 to be two ellipses which are confocal with S0. The half major axis length of S1 and S2 are respectively 0.55 m and 0.60 m. Since S1 and S2 are not circular, the conventional SFT-based separation methods cannot apply. With the combined integral equations based method developed in the former sections, sound field separation on ellipses S1 and S2 is achieved and the results are shown in Fig. 3. Then, the measurement contours are turned to be two rectangles with their sides parallel with the x or y axis. This type of separation boundaries could make the sensor placement much easy and precise in practical implementations. The side lengths of S1 and S2 are 1.1 0.2 m and 1.2 0.3 m, respectively. The separation results are shown in Fig. 4. From Figs. 3 and 4, it can be seen that the separated sound fields obtained by Eqs. (19), (24), and (25) coincide with the corresponding original fields given by analytical expressions very well. The results of Eqs. (19) and (24) are identical, and have a little difference with that obtained by Eq. (25). 4.2. Geometrical influence induced by sound field separation surfaces Due to the singular kernel of the Green function, Helmholtz integral equation has a singularity problem when it is applied to surfaces of thin shape or regular surfaces with thin appendages [26]. Consequently, research on the geometrical influence induced by the aspect ratio of the separation surfaces is necessary. We consider an elliptic cylinder as the sound scatterer which has an extremely high aspect ratio 1000. The half major axis length x0 and the half minor axis length y0 of the cross section S0 are set to be 0.23 m and 0.23 mm, respectively. The incident sound field and the signal noise are the same as those in the former subsection. The measurement contours S1 and S2 are two ellipses which are confocal with S0. Aspect ratio f (i.e. the ratio between the major and the minor axis length) of S1 changes from 2 to 50, while the half major and minor axis length of S2 are fixed and set to be 0.275 m and 0.152 m, respectively. For different f, relative error E of the separated sound pressure field on S1 is calculated and shown in Fig. 5. The results of Eqs. (19) and (24) are identical, and differ with that obtained by Eq. (25) slightly. From Fig. 5, it can be found that the relative error E increases along with the increase of f. Therefore, measurement surfaces on which sound field separation is implemented should have a moderate aspect ratio. 4.3. Effects of signal noise on sound field separation To test the robustness of the sound field separation method, relative error induced by signal noise is investigated in this subsection. As depicted in Fig. 2b, we consider a circular cylinder with its cross section enclosed by two concentric circles for eliminating the influence of geometric shape. The radius a of the cross section circle S0 is 0.5 m, while the radius of S1 and S2 are 0.55 m and 0.65 m, respectively. The incident sound field is a plane progressive wave with the incident direction angle uin = 0°. The spatial part of
Fig. 3. Sound separation results on elliptical contours S1 and S2 enclosing an elliptic cylinder. (a) Amplitude of sound pressure field on S1, (b) amplitude of sound pressure field on S2, (c) phase angle of sound pressure field on S1 and (d) phase angle of sound pressure field on S2. Solid line: original field given by analytical expressions. Dot line: separation result by Eqs. (19) and (24). Dot dash line: separation result by Eq. (25). Blue line: incident sound pressure field. Red line: scattering sound pressure field. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Z. Fan et al. / Ultrasonics 54 (2014) 2169–2177
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Fig. 5. Relative error of sound field separation on S1 with different aspect ratio. (a) Relative error of separation result by Eqs. (19) and (24) and (b) relative error of separation result by Eq. (25). Solid line: relative error for separated sound field on S1. Dash line: relative error for separated sound field on S2. Blue line: incident sound pressure field. Red line: scattering sound pressure field. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Sound separation results on rectangular contours S1 and S2 enclosing an elliptic cylinder. (a) Amplitude of sound pressure field on S1, (b) amplitude of sound pressure field on S2, (c) phase angle of sound pressure field on S1 and (d) phase angle of sound pressure field on S2. Solid line: original field given by analytical expressions. Dot line: separation result by Eqs. (19) and (24). Dot dash line: separation result by Eq. (25). Blue line: incident sound pressure field. Red line: scattering sound pressure field. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
in which e0 = 1 and en = 2 for n P 1, Jn and Hnð1Þ are the Bessel and first kind Hankel functions [27]. Meanwhile, random Gaussian noise with different level SNR is added to simulate practical measurement data. Sound field separation method proposed in Ref. [18,19] is also carried out on S1 for comparison. For every node point of S1, sound pressure is measured on two adjacent points, which distribute on both sides of the node point and have a constant 3 cm interval along the normal direction. Based on Euler’s equation, normal
Fig. 6. Relative error of finite difference approximation on S1 under different SNR level noise. (a) Relative error of approximate normal velocity and (b) relative error of approximate sound pressure. Solid line: 25 dB SNR. Dot line: 30 dB SNR. Dot dash line: 35 dB SNR.
velocity can be indirectly obtained by finite difference of sound pressure instead of using velocity probes. By recasting the Euler’s equation into an integral form, i.e.,
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deviation appears in the low frequency region and this error decreases with the increase of the non-dimensional parameter ka. In view of signal processing theory, finite difference operation is equivalent to a high pass filter [30]. So the rapidly varying part in noise is amplified, and this effect behaves more prominent for the relatively large wavelength. Also presented in Fig. 6b is the relative error of sound pressure at the node points on S1, which is approximated by the arithmetical average of sound pressure on the two adjacent points. This error increases in an oscillating manner since the constant interval becomes relatively large in contrast to the shortening wavelength. As shown in Fig. 7, the relative error E of the separated sound pressure field changes as the parameter ka varies from 0.5 to 10. It can be seen that as concerns the overall tendency, the relative error decreases gradually along with the increase of ka for all methods. The separated incident sound pressure fields are always more accurate than the scattering parts. Separation results obtained by Eqs. (19) and (24) are identical, and differ with that of Eq. (25) slightly. With respect of robustness, the sound field separation method developed in this study acquires a more accurate performance comparing with that in Ref. [18,19] especially for the low frequency conditions. 5. Conclusions In this study, a new sound field separation method in the form of combined integral equations has been developed. On basis of Helmholtz integral equations for the incident and scattreing sound fields outside a sound scatterer, three systems of combined integral equations are derived for sound field separation on irregular surfaces enclosing the scatterer. Afterwards, through boundary element discretization of the separation surfaces involved in the combined integral equations, three corresponding systems of linear equations are obtained for practical application. Numerical simulations are carried out under different conditions, and the results show that the effect and accuracy of this combined integral equations based method for sound field separation are satisfactory. In future research work, indirect boundary integral equation instead of the direct one could be anticipated to improve the separation accuracy for extremely thin or flat bodies. We also expect to integrate the discrete sources method to improve the efficiency of the separation process. Acknowledgments The authors would like to acknowledge the support from the National Natural Science Foundation of China (Grant Nos. 51175465 and 51205351), China Postdoctoral Science Foundation (Grant No. 2012M521165) and Program for Zhejiang Leading Team of S&T Innovation (Grant No. 2009R50008). Fig. 7. Relative error of sound field separation on S1 under different SNR level noise. (a) ka = 0.5–2.5, 25 dB SNR, (b) ka = 2.5–10, 25 dB SNR, (c) ka = 0.5–2.5, 35 dB SNR, (d) ka = 2.5–10, 35 dB SNR. Solid line: relative error for Eqs. (19) and (24). Dot line: relative error for Eq. (25). Dot dash line: relative error for finite difference based method. Blue Line: incident sound pressure field. Red Line: scattering sound pressure field. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
v n ðrÞ ¼
1
@pðrÞ ! ðK v n ÞðrÞ ¼ iq0 x iq0 x @n
Z
Ln
v n ðlÞdl ¼ pðrÞ;
0
in which K is an integral operator and Ln is the distance along the normal direction n, we found that finite difference scheme is essentially an ill-posed problem owing to the compact characteristics of K [28]. As a result, even a small noise in the measurement data could induce a large error to the normal velocity [29]. In Fig. 6a, a great
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[18] C. Langrenne, M. Melon, A. Garcia, Boundary element method for the acoustic characterization of a machine in bounded noisy environment, J. Acoust. Soc. Am. 121 (2007) 2750–2757. [19] C. Langrenne, M. Melon, A. Garcia, Measurement of confined acoustic sources using near-field acoustic holography, J. Acoust. Soc. Am. 126 (2009) 1250– 1256. [20] M.R. Bai, C. Wang, S.W. Juan, Optimal two-layer directive microphone array with application in near-field acoustical holography, J. Acoust. Soc. Am. 132 (2012) 862–871. [21] E.G. Williams, Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography, Academic Press, San Diego, 1999. [22] R.D. Ciskowski, C.A. Brebbia, Boundary Element Methods in Acoustics, Springer-Verlag, Berlin, 1991. [23] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1968. [24] J.J. Stamnes, Exact two-dimensional scattering by perfectly reflecting elliptical cylinders, strips and slits, Pure Appl. Opt. 4 (1995) 841–855. [25] J.J. Stamnes, B. Spjelkavik, New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders, Pure Appl. Opt. 4 (1995) 251–262. [26] R. Martinez, The thin-shape breakdown (TSB) of the Helmholtz integral equation, J. Acoust. Soc. Am. 90 (1991) 2728–2738. [27] P.M. Morse, K.U. Ingard, Theoretical Acoustics, Princeton University Press, Princeton, 1968. [28] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer-Verlag, New York, 2011. [29] D. Stanzial, G. Sacchi, G. Schiffrer, Calibration of pressure-velocity probes using a progressive plane wave reference field and comparison with nominal calibration filters, J. Acoust. Soc. Am. 129 (2011) 3745–3755. [30] F. Jauberteau, J.L. Jauberteau, Numerical differentiation with noisy signal, Appl. Math. Comput. 215 (2009) 2283–2297.
Contents lists available at ScienceDirect
Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Sound field separating on arbitrary surfaces enclosing a sound scatterer based on combined integral equations Zongwei Fan, Deqing Mei ⇑, Keji Yang, Zichen Chen The State Key Laboratory of Fluid Power Transmission and Control, Department of Mechanical Engineering, Zhejiang University, Hangzhou 310027, PR China
a r t i c l e
i n f o
Article history: Received 7 February 2013 Received in revised form 8 February 2014 Accepted 11 June 2014 Available online 20 June 2014 Keywords: Sound field separation Arbitrary spatial surfaces Sound scatterer Combined integral equations Boundary element discretization
a b s t r a c t To eliminate the limitations of the conventional sound field separation methods which are only applicable to regular surfaces, a sound field separation method based on combined integral equations is proposed to separate sound fields directly in the spatial domain. In virtue of the Helmholtz integral equations for the incident and scattering fields outside a sound scatterer, combined integral equations are derived for sound field separation, which build the quantitative relationship between the sound fields on two arbitrary separation surfaces enclosing the sound scatterer. Through boundary element discretization of the two surfaces, corresponding systems of linear equations are obtained for practical application. Numerical simulations are performed for sound field separation on different shaped surfaces. The influences induced by the aspect ratio of the separation surfaces and the signal noise in the measurement data are also investigated. The separated incident and scattering sound fields agree well with the original corresponding fields described by analytical expressions, which validates the effectiveness and accuracy of the combined integral equations based separation method. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Sound field separation, which decomposes the total composite sound field to the incident and scattering partial fields on a surface enclosing a sound scatterer, is an important subject in many acoustic applications. For near-field acoustic holography (NAH) of vibrating structures mounted in non-anechoic environment, sound fields propagating inward to the measurement surfaces should be extracted and removed from the total sound field [1,2]. Besides, in contactless micro-manipulation by means of acoustic radiation force, sound field synthesis technology [3] is applied to produce the desired sound field that corresponds with the specific distribution of acoustic radiation force [4,5]. However, obstacles often exist, which make the sound field synthesis domain a non-free space. To guarantee the applicability of the sound field synthesis, sound field separation technique is used to overcome adverse disturbances in the propagation operator caused by the obstacle’s scattering field. Other applications necessitating sound field separation include characteristic parameter measurement for acoustic material [6], acoustic target strength characterization in underwater acoustics [7], etc.
⇑ Corresponding author. Tel.: +86 571 87951906; fax: +86 571 87951145. E-mail address: [email protected] (D. Mei). http://dx.doi.org/10.1016/j.ultras.2014.06.007 0041-624X/Ó 2014 Elsevier B.V. All rights reserved.
In conventional sound field separation methods, spatial Fourier transform (SFT) is applied to convert the total sound field from spatial domain to wave-number domain, in which subsequent sound field separating operation is accomplished. Weinreich and Arnold [8] proposed a method to decompose double layer sound fields with spherical harmonic functions. Frisk et al. [9] derived a SFT-based method for measuring the reflection coefficient of the seafloor. Tamura et al. [10,11] developed a method to measure the plane-wave reflection coefficient at oblique incidence. Cheng et al. [12] extended that method, by which the scattering field from a complex shaped object is separated with the incident field in Cartesian and cylindrical coordinates. This method was further applied by Yu et al. [13] to achieve sound field separation in spherical coordinates. The major advantage of the SFT-based methods mentioned above is their simplicity and efficiency in application. However, due to the inherent characteristics of SFT, these methods can only accomplish sound field separation for three types of regular surfaces: those with planar, cylindrical, and spherical geometries. In many other situations, irregularly shaped separation surfaces are often involved. For example, to obtain near-field information on the scattering field from an irregular sound scatterer, the separation surfaces have to be conformal with the physical frontier of the scatterer and then become irregular accordingly. In addition, to enlarge the applicable region dimension of sound field synthesis,
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the spatial points on which the propagation operator values are measured should be distributed over a surface as close to the obstacle as possible. Therefore, if the obstacle is irregular, sound field separation on irregularly shaped surfaces will be required. Besides, in SFT-based separation process, wave-number spectrum of the sound pressure on the double measurement surfaces should be truncated at the same order. This requirement leads to amplification of the measurement error due to the ill-posedness effect of inverse problems. Up to now, there have been developed several methods for sound field separation on irregular surfaces. Bi et al. [14,15] proposed a technique based on the equivalent source method. In its application, sound pressure fields are measured and separated on two closely-spaced and parallel surfaces. Based on the fundamental solution of the Helmholtz equation and the wave superposition principle, Fernandez-Grande et al. [16,17] developed a method to deal with sound field separation on non-separable geometries. The common idea of the methods in Ref. [14–17] is to approximate the original sound fields by linear combination of the component fields from the equivalent sources, then solve the weight coefficients by matching the assumed-form solution to the measurement data on the separation surfaces. Since the separating operation is performed directly in the spatial domain instead of the wave-number domain, these techniques are applicable to arbitrary surfaces, and the limitations (such as window effects) due to SFT can be avoided. A difficulty in using these equivalent source based methods is that there exists little theoretical guidance to determine the proper location, order (or type) and number of the equivalent sources, although separation effect is closely related with these characteristics. Based on the Helmholtz integral equation, Langrenne et al. [18,19] developed a technique for sound field separation on general surfaces. For its implementation, normal velocity on the objective separation surface is obtained indirectly by finite difference of sound pressure, which are measured on two adjacent and conformal surfaces. Owing to the ill-posedness of the finite difference operation, small noise in the measurement data might induce large error in the approximated normal velocity [20], and therefore make the separation results depart from the corresponding original fields. To overcome the limitations of these existing approaches, a sound field separation method is proposed in this study based on combined integral equations. In Section 2, three separation methods in the form of combined integral equations are established theoretically. In Section 3, three corresponding systems of linear equations are obtained through boundary element discretization of the separation surfaces. Numerical simulations on sound field separation under different conditions are carried out in Section 4. Conclusions and suggestions for future research are given in Section 5.
2. Sound field separation based on combined integral equations In the exterior domain of an arbitrarily shaped sound scatterer, the total sound pressure pt(r, t) consists of two parts, i.e., the original incident sound pressure pin(r, t) and the scattering sound pressure psc(r, t). The geometry of interest is depicted in Fig. 1. The domain occupied by the scatterer is denoted by X and its physical frontier by S0. S1 and S2 are two arbitrarily shaped surfaces enclosing the scatterer, on which the total sound fields are measured and then separated. For a time-harmonic exp(ixt) disturbance of angular frequency x, the spatial part of sound pressure outside the scatterer satisfies the Helmholtz equation Dp + k2p = 0, in which k is the wave number. When subjected to the Sommerfeld radiation condition and the boundary condition on the scatterer surface, boundary
integral equations can be found. These boundary integral equations are often classified as direct or indirect, where direct corresponds to the Helmholtz integral equation and the numerous indirect equations are based on layer potentials. The Helmholtz integral equation satisfied by the spatial part pin(r) of the incident sound pressure is
a pin ðrÞ ¼ tS pin ðr0 Þ
@ 0 0 dSðr0 Þ; Gðr; r0 Þ iq0 xv in ðr ÞGðr; r Þ n @n
ð1Þ
where the integral surface S is an arbitrary smooth surface enclosing the scatterer, G is the free space Green function, q0 is the equilibrium density of the sound medium, n is the unit interior normal to S, v in n is the incident normal velocity along the normal direction n [21]. The value of the coefficient a varies as
a¼
8 > <0
if r is outside S : if r is on S : 1 if r is inside S 1 >2
According to Eq. (1) for a = 0, the incident sound pressure and normal velocity on S1 can be applied to compute the incident sound pressure on S2, i.e.,
@ 0 ¼ tS1 pin ðr1 Þ Gðr2 ; r1 Þ iq0 xv in n ðr1 ÞGðr2 ; r1 Þ dSðr1 Þ: @n
ð2Þ
Similarly, the incident sound pressure and normal velocity on S2 can also be applied to compute the incident sound pressure on S1
@ pin ðr1 Þ ¼ tS2 pin ðr2 Þ Gðr1 ; r2 Þ iq0 xv in n ðr2 ÞGðr1 ; r2 Þ dSðr2 Þ: @n
ð3Þ
The Helmholtz integral equation satisfied by the spatial part pt ðrÞ of the total sound pressure is
a pt ðrÞ ¼ pin ðrÞ þ tS pt ðr0 Þ
@ Gðr; r0 Þ iq0 xv tn ðr0 ÞGðr; r0 Þ dSðr0 Þ; @n
ð4Þ where v tn is the total normal velocity along the normal direction n [21]. The coefficient
8 > < 1 if r is outside S a ¼ 12 if r is on S : > : 0 if r is inside S In Ref. [18,19], the total sound pressure pt(r) and the total normal velocity v tn ðrÞ on S are both known. Hence the incident sound pressure pin(r) on S can be computed based on Eq. (4), while the scattering sound field psc(r) on S equals pt(r) pin(r). In this study, we consider the application conditions where only one type of sound field data is known by measurement, either the total sound pressure or the total normal velocity. When the field point r is outside the integral surface S, Eq. (4) could be written as
@ 0 0 0 pt ðrÞ ¼ pin ðrÞ þ tS pin ðr0 Þ Gðr; r0 Þ iq0 xv in n ðr ÞGðr; r Þ dSðr Þ @n @ 0 0 0 þ tS psc ðr0 Þ Gðr; r0 Þ iq0 xv sc n ðr ÞGðr; r Þ dSðr Þ @n @ 0 0 dSðr0 Þ: ðr ÞGðr; r Þ ¼ pin ðrÞ þ tS psc ðr0 Þ Gðr; r0 Þ iq0 xv sc n @n Since pt(r) = pin(r) + psc(r), it can be derived from the above formula that in the domain outside S,
@ 0 0 0 psc ðrÞ ¼ tS psc ðr0 Þ Gðr; r0 Þ iq0 xv sc n ðr ÞGðr; r Þ dSðr Þ: @n
ð5Þ
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Fig. 1. Geometry configuration of sound field separation.
Through a similar derivation, we can derive that when the field point r locates on S,
1 @ 0 0 dSðr0 Þ; psc ðrÞ ¼ tS psc ðr0 Þ Gðr; r0 Þ iq0 xv sc ðr ÞGðr; r Þ n 2 @n
velocity on the same surface could be derived. When the field point is on S1,
(
ð6Þ
@ 0 0 0 12 pin ðr1 Þ ¼ tS1 pin ðr01 Þ @n Gðr1 ; r01 Þ iq0 xv in n ðr1 ÞGðr1 ; r1 Þ dSðr1 Þ : 1 @ 0 0 0 p ðr Þ ¼ tS1 psc ðr01 Þ @n Gðr1 ; r01 Þ iq0 xv sc n ðr1 ÞGðr1 ; r1 Þ dSðr1 Þ 2 sc 1 ð12Þ
and when the field point r is inside S,
@ 0 0 0 0 psc ðrÞ ¼ tS psc ðr0 Þ Gðr; r0 Þ iq0 xv sc n ðr ÞGðr; r Þ dSðr Þ: @n
ð7Þ
According to Eq. (5), the scattering sound pressure and normal velocity on S1 can be applied to compute the scattering sound pressure on S2, i.e.,
@ psc ðr2 Þ ¼ tS1 psc ðr1 Þ Gðr2 ; r1 Þ iq0 xv sc n ðr1 ÞGðr2 ; r1 Þ dSðr1 Þ: @n
ð8Þ
According to Eq. (7), by using the scattering sound pressure and normal velocity on S2 to compute the scattering sound pressure on S1, we have
@ 0 ¼ tS2 psc ðr2 Þ Gðr1 ; r2 Þ iq0 xv sc n ðr2 ÞGðr1 ; r2 Þ dSðr2 Þ: @n
ð9Þ
By summing Eqs. (2) and (8), we have @ sc psc ðr2 Þ ¼ tS1 pt ðr1 Þ Gðr2 ; r1 Þ iq0 x½v in n ðr1 Þ þ v n ðr1 ÞGðr2 ; r1 Þ dSðr1 Þ: @n
ð10Þ
while the summation of Eqs. (3) and (9) gives @ sc pin ðr1 Þ ¼ tS2 pt ðr2 Þ Gðr1 ; r2 Þ iq0 x½v in ðr Þ þ v ðr ÞGðr ;r Þ dSðr2 Þ: 2 2 1 2 n n @n
ð11Þ
In Eqs. (10) and (11), there are six unknown quantities, i.e., pin(r1), sc in sc psc(r2), v in n ðr1 Þ; v n ðr1 Þ; v n ðr2 Þ and v n ðr2 Þ. To achieve sound field separation, four more equations about these six unknown quantities have to be found for solving the sound pressure pin(r1) on S1 and psc(r2) on S2. Based on Eqs. (1) and (6) for the field points on S1 and S2, quantitative relationship between the sound pressure and normal
According to the first formula in Eq. (12), the incident normal velocity v in n ðr1 Þ can be determined by the incident sound pressure pin(r1). Since the scattering sound pressure psc(r1) = pt(r1) pin(r1), the scattering normal velocity v sc n ðr1 Þ can also be determined by the incident sound pressure pin ðr1 Þ based on the second formula in Eq. (12). Similarly, when the field point is on S2, we have
(
q xv q xv
1 @ in 0 0 0 p ðr Þ ¼ tS2 pin ðr02 Þ @n Gðr2 ; r02 Þ i 0 n ðr2 ÞGðr2 ; r2 Þ dSðr2 Þ 2 in 2 : 1 @ 0 0 sc 0 0 0 2 psc ðr2 Þ ¼ tS2 psc ðr2 Þ @n Gðr2 ; r2 Þ i 0 n ðr2 ÞGðr2 ; r2 Þ dSðr2 Þ
ð13Þ According to the second formula in Eq. (13), the scattering normal velocity v sc n ðr2 Þ can be determined by the scattering sound pressure psc ðr2 Þ. Since the incident sound pressure pin(r2) = pt(r2) psc(r2), the incident normal velocity v in n ðr2 Þ can also be determined by the scattering sound pressure psc(r2) based on the first formula. So far, the required four equations mentioned above have been obtained. sc Through expressing v in n ðr1 Þ and v n ðr1 Þ by pin ðr1 Þ and then substituting them into Eq. (10), we get one integral equation describing the relationship between the incident sound pressure pin(r1) on S1 and the scattering sound pressure psc(r2) on S2. Simisc larly, through expressing v in n ðr2 Þ and v n ðr2 Þ by psc(r2) and then substituting them into Eq. (11), the second integral equation about pin(r1) and psc(r2) is obtained. By combining these two integral equations and then solving the unknown quantities pin(r1) and psc(r2), sound field separation on the measurement surfaces S1 and S2 is consequently realized. We further derive the second system of combined integral equations with the total normal velocity v tn ðr1 Þ and v tn ðr2 Þ as the
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unsolved quantities. On summing the two integral equations in Eq. (12), we have
1 @ pin ðr1 Þ ¼ pt ðr1 Þ tS1 pt ðr01 Þ Gðr1 ; r01 Þ iq0 xv tn ðr01 ÞGðr1 ;r01 Þ dSðr01 Þ: 2 @n
ð14Þ
Substituting Eq. (14) into Eq. (11), we can get one integral equation which describes the relationship about the total sound pressure pt(r1), pt(r2) and the total normal velocity v tn ðr1 Þ; v tn ðr2 Þ. Similarly, on summing the two integral equations in Eq. (13), we have
psc ðr2 Þ ¼
1 p ðr2 Þ 2 t @ tS2 pt ðr02 Þ Gðr2 ; r02 Þ iq0 xv tn ðr02 ÞGðr2 ; r02 Þ dSðr02 Þ: @n ð15Þ
By substituting Eq. (15) into Eq. (10), we get the second equation about pt(r1), pt(r2), v tn ðr1 Þ and v tn ðr2 Þ. Therefore, the second system of combined integral equations are established with the total normal velocity v tn ðr1 Þ and v tn ðr2 Þ as the unknown quantities. On substituting the solution of the total normal velocity v tn ðr1 Þ and v tn ðr2 Þ into Eqs. (10) and (11), the incident sound pressure pin(r1) on S1 and the scattering sound pressure psc(r2) on S2 will be solved, then sound field separation on S1 and S2 is achieved. Furthermore, if the statuses of the total sound pressure and the total normal velocity in the second system of combined integral equations are exchanged, the third system can be derived, in which the total sound pressure pt(r1) and pt(r2) are turned to be the unsolved quantities, while the normal velocity v tn ðr1 Þ and v tn ðr2 Þ are obtained by measurement. 3. Boundary element discretization of the combined integral equations Through boundary element discretization of the surfaces S1 and S2 involved in above integral equations [22], three systems of combined integral equations developed in the previous section are transformed to three corresponding systems of linear equations for practical applications. The matrix formulae of Eqs. (10) and (11) are
(
sc psc ðr2 Þ ¼ D21 pt ðr1 Þ M21 v in n ðr1 Þ M21 v n ðr1 Þ;
ð16Þ
sc pin ðr1 Þ ¼ D12 pt ðr2 Þ M12 v in n ðr2 Þ M12 v n ðr2 Þ;
in which matrices Dij and Mij imply the effects on a field point on Si by the dipoles and monopoles on Sj, respectively. The matrix formulae of Eqs. (12) and (13) are (
v
12 pin ðr1 Þ ¼ D11 pin ðr1 Þ M11 in n ðr1 Þ; 1 p ðr Þ ¼ D11 psc ðr1 Þ M11 sc n ðr1 Þ; 2 in 1
v
(
v
1 p ðr Þ ¼ D22 pin ðr2 Þ M22 in n ðr2 Þ; 2 in 2 12 psc ðr2 Þ ¼ D22 psc ðr2 Þ M22 sc n ðr2 Þ;
v
ð17Þ
in which matrices Dii and Mii imply the effects on a surface point on Si by dipoles and monopoles on the same surface Si. Substituting Eq. (17) into Eq. (16), we have
8 h i 1 1 > < psc ðr2 Þ ¼ D21 M21 M1 11 D11 2 I11 pt ðr1 Þ M21 M11 pin ðr1 Þ; h i 1 > 1 : pin ðr1 Þ ¼ D12 M12 M1 22 D22 2 I22 pt ðr2 Þ M12 M22 psc ðr2 Þ; ð18Þ where I11 and I22 are identity matrices. By reorganizing above two equations, the following system of linear equations could be obtained:
F1
pin ðr1 Þ psc ðr2 Þ
¼ b1 ;
ð19Þ
where " F1 ¼
I11
M12 M1 22
M21 M1 11
I22
# ;
0h 1 i 1 D M12 M1 22 D22 2 I22 pt ðr2 Þ C B 12 i b1 ¼ @ h A: 1 D21 M21 M1 11 ðD11 2 I11 Þ pt ðr1 Þ
On solving Eq. (19), pin(r1) and psc(r2) are separated from the total sound pressure pt(r1) on S1 and pt(r2) on S2, while the scattering sound pressure psc(r1) = pt(r1) pin(r1) and the incident sound pressure pin(r2) = pt(r2) psc(r2). Thus, sound field separation on the surfaces S1 and S2 is accomplished. Similarly, we can also derive the discretization formulae for the second system of combined integral equations described in the former section. The matrix formulae of Eqs. (14) and (15) are
1 pin ðr1 Þ ¼ D11 I11 pt ðr1 Þ þ M11 v tn ðr1 Þ; 2
ð20Þ
1 psc ðr2 Þ ¼ D22 I22 pt ðr2 Þ þ M22 v tn ðr2 Þ: 2
ð21Þ
Substituting Eqs. (20) and (21) into Eq. (16), we have
1 M11 v tn ðr1 Þ þ M12 v tn ðr2 Þ ¼ D11 I11 pt ðr1 Þ þ D12 pt ðr2 Þ; 2
ð22Þ
1 M21 v tn ðr1 Þ þ M22 v tn ðr2 Þ ¼ D21 pt ðr1 Þ þ D22 I22 pt ðr2 Þ: 2
ð23Þ
By reorganizing the above two equations, the second system of linear equations is obtained:
F2
v tn ðr1 Þ v tn ðr2 Þ
¼ b2 ;
ð24Þ
in which
F2 ¼
M11
M12
M21
M22
;
" b2 ¼
D11 12 I11
D12
D21
D22 12 I22
#
pt ðr1 Þ pt ðr2 Þ
:
For cases where the total sound pressure are quantities to be solved on basis of the total normal velocity data, the third system of linear equations can be derived through exchanging the statuses of the total sound pressure and the total normal velocity in Eqs. (22) and (23):
F3
ptn ðr1 Þ ptn ðr2 Þ
¼ b3 ;
ð25Þ
where
" F3 ¼
D11 12 I11
D12
D21
D22 12 I22
# ;
b3 ¼
M11
M12
M21
M22
v tn ðr1 Þ v tn ðr2 Þ
:
In Eq. (19), the computation of the left-hand-side matrix F1 and the right-hand-side vector b1 relies on the inverse matrices M1 11 and M1 22 . When the scale of sound field separation problem becomes quite large, such as high frequency sound field or large sized, three dimensional sound scatterer, inversion operation of the matrices M11 and M22 will consume much computation time and memory. In contrast, explicit inversion of matrices is inherently avoided in Eqs. (24) and (25), so the latter two ways for sound field separation will be better choices for large scale problems. 4. Numerical simulations To validate the effectiveness and accuracy of the sound field separation method proposed in the former sections, numerical study is carried out in this section. In the following numerical
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Fig. 2. Geometry configuration for numerical simulation. (a) Elliptic cylinder with elliptical and rectangular measurement contours and (b) circular cylinder with circle measurement contours.
simulations, the measurement contours are divided by a set of straight linear segments with the well-known one-sixth of the wavelength criterion satisfied. Constant elements are adopted for boundary element discretization. For quantifying the accuracy of the separated results compared with the corresponding analytic values, relative error in percent is defined as
PM E¼
2 i¼1 kpex ðiÞ psfs ðiÞk2 PM 2 i¼1 kpex ðiÞk2
100ð%Þ;
ð26Þ
incident sound field is composed of three plane progressive waves propagating in the xoy plane with the incident direction angles u1in ¼ 30 , u2in ¼ 150 and u3in ¼ 270 . For air medium, the sound speed is 344 m/s. The frequency of the incident sound field is set to be 1.5 kHz. In addition, random Gaussian noise is imposed with predetermined variance and 30 dB SNR (Signal/Noise Ratio). The Cartesian coordinates (x, y, z) could be expressed by elliptic cylindrical coordinates (u, v, z) as follows,
x ¼ f cosh u cos v ; y ¼ f sinh u sin v ; z ¼ z
where M is the number of discretization nodes on the separation surfaces, pex is the analytic value given by theoretical expression and psfs denotes the sound field separation result.
where 0 6 u < 1 and 0 6 v < 2p [23]. The spatial part of the incident sound pressure field described by elliptic cylindrical coordinates (u, v) is
4.1. Sound field separation for an elliptic cylinder
pin ðu; v Þ ¼
In Fig. 2a, we consider an elliptic cylinder with its center at the origin of the xyz coordinates and the central axis coincides with the z axis. The cross section ellipse boundary S0 in the xoy plane is defined by x2 =x20 þ y2 =y20 ¼ 1. The half major axis length x0 and the half minor axis length y0 are set to be 0.5 m and 0.05 m, respecqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tively. The semi-focal distance f of S0 is given by f ¼ x20 y20 . The
n 1 pffiffiffiffiffiffiffiX i Je ðs; uÞSen ðs; v Þ½Sen ðs; u1in Þ þ Sen ðs; u2in Þ 8p Nen n n¼0
þ Sen ðs; u3in Þ þ
n 1 pffiffiffiffiffiffiffiX i Jo ðs; uÞSon ðs; v Þ½Son ðs; u1in Þ 8p Non n n¼0
þ Son ðs; u2in Þ þ Son ðs; u3in Þ; in which s = (kf)2, k is the wave number, Sen and Son are angular Mathieu functions, Jen and Jon are radial Mathieu functions of the
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first kind, Nen and Non are normalization constants [24,25]. For an elliptic cylinder with a sound rigid boundary, the spatial part of the scattering sound pressure field is n 1 pffiffiffiffiffiffiffiX i Je0n ðs; uÞ 1 psc ðu; v Þ ¼ 8p Heð1Þ n ðs; uÞSen ðs; v Þ½Sen ðs; uin Þ Nen Heð1Þ0 n¼0 n ðs; uÞ n 1 pffiffiffiffiffiffiffiX i þ Sen ðs; u2in Þ þ Sen ðs; u3in Þ 8p Non n¼0
Jo0n ðs; uÞ Honð1Þ0 ðs; uÞ
the incident and scattering sound pressure fields described by cylindrical coordinates (r, u) are
pin ðr; uÞ ¼ expðikr cos uÞ; psc ðr; uÞ " # 1 X J 0n ðkaÞ ¼ en ð1Þ0 Hnð1Þ ðkrÞ cosðnuÞ; Hn ðkaÞ n¼0
1 2 Hoð1Þ n Son ðs; v Þ½Son ðs; uin Þ þ Son ðs; uin Þ
þ Son ðs; u3in Þ; where Henð1Þ and Honð1Þ are radial Mathieu functions of the third kind. We first consider the measurement contours S1 and S2 to be two ellipses which are confocal with S0. The half major axis length of S1 and S2 are respectively 0.55 m and 0.60 m. Since S1 and S2 are not circular, the conventional SFT-based separation methods cannot apply. With the combined integral equations based method developed in the former sections, sound field separation on ellipses S1 and S2 is achieved and the results are shown in Fig. 3. Then, the measurement contours are turned to be two rectangles with their sides parallel with the x or y axis. This type of separation boundaries could make the sensor placement much easy and precise in practical implementations. The side lengths of S1 and S2 are 1.1 0.2 m and 1.2 0.3 m, respectively. The separation results are shown in Fig. 4. From Figs. 3 and 4, it can be seen that the separated sound fields obtained by Eqs. (19), (24), and (25) coincide with the corresponding original fields given by analytical expressions very well. The results of Eqs. (19) and (24) are identical, and have a little difference with that obtained by Eq. (25). 4.2. Geometrical influence induced by sound field separation surfaces Due to the singular kernel of the Green function, Helmholtz integral equation has a singularity problem when it is applied to surfaces of thin shape or regular surfaces with thin appendages [26]. Consequently, research on the geometrical influence induced by the aspect ratio of the separation surfaces is necessary. We consider an elliptic cylinder as the sound scatterer which has an extremely high aspect ratio 1000. The half major axis length x0 and the half minor axis length y0 of the cross section S0 are set to be 0.23 m and 0.23 mm, respectively. The incident sound field and the signal noise are the same as those in the former subsection. The measurement contours S1 and S2 are two ellipses which are confocal with S0. Aspect ratio f (i.e. the ratio between the major and the minor axis length) of S1 changes from 2 to 50, while the half major and minor axis length of S2 are fixed and set to be 0.275 m and 0.152 m, respectively. For different f, relative error E of the separated sound pressure field on S1 is calculated and shown in Fig. 5. The results of Eqs. (19) and (24) are identical, and differ with that obtained by Eq. (25) slightly. From Fig. 5, it can be found that the relative error E increases along with the increase of f. Therefore, measurement surfaces on which sound field separation is implemented should have a moderate aspect ratio. 4.3. Effects of signal noise on sound field separation To test the robustness of the sound field separation method, relative error induced by signal noise is investigated in this subsection. As depicted in Fig. 2b, we consider a circular cylinder with its cross section enclosed by two concentric circles for eliminating the influence of geometric shape. The radius a of the cross section circle S0 is 0.5 m, while the radius of S1 and S2 are 0.55 m and 0.65 m, respectively. The incident sound field is a plane progressive wave with the incident direction angle uin = 0°. The spatial part of
Fig. 3. Sound separation results on elliptical contours S1 and S2 enclosing an elliptic cylinder. (a) Amplitude of sound pressure field on S1, (b) amplitude of sound pressure field on S2, (c) phase angle of sound pressure field on S1 and (d) phase angle of sound pressure field on S2. Solid line: original field given by analytical expressions. Dot line: separation result by Eqs. (19) and (24). Dot dash line: separation result by Eq. (25). Blue line: incident sound pressure field. Red line: scattering sound pressure field. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 5. Relative error of sound field separation on S1 with different aspect ratio. (a) Relative error of separation result by Eqs. (19) and (24) and (b) relative error of separation result by Eq. (25). Solid line: relative error for separated sound field on S1. Dash line: relative error for separated sound field on S2. Blue line: incident sound pressure field. Red line: scattering sound pressure field. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Sound separation results on rectangular contours S1 and S2 enclosing an elliptic cylinder. (a) Amplitude of sound pressure field on S1, (b) amplitude of sound pressure field on S2, (c) phase angle of sound pressure field on S1 and (d) phase angle of sound pressure field on S2. Solid line: original field given by analytical expressions. Dot line: separation result by Eqs. (19) and (24). Dot dash line: separation result by Eq. (25). Blue line: incident sound pressure field. Red line: scattering sound pressure field. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
in which e0 = 1 and en = 2 for n P 1, Jn and Hnð1Þ are the Bessel and first kind Hankel functions [27]. Meanwhile, random Gaussian noise with different level SNR is added to simulate practical measurement data. Sound field separation method proposed in Ref. [18,19] is also carried out on S1 for comparison. For every node point of S1, sound pressure is measured on two adjacent points, which distribute on both sides of the node point and have a constant 3 cm interval along the normal direction. Based on Euler’s equation, normal
Fig. 6. Relative error of finite difference approximation on S1 under different SNR level noise. (a) Relative error of approximate normal velocity and (b) relative error of approximate sound pressure. Solid line: 25 dB SNR. Dot line: 30 dB SNR. Dot dash line: 35 dB SNR.
velocity can be indirectly obtained by finite difference of sound pressure instead of using velocity probes. By recasting the Euler’s equation into an integral form, i.e.,
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deviation appears in the low frequency region and this error decreases with the increase of the non-dimensional parameter ka. In view of signal processing theory, finite difference operation is equivalent to a high pass filter [30]. So the rapidly varying part in noise is amplified, and this effect behaves more prominent for the relatively large wavelength. Also presented in Fig. 6b is the relative error of sound pressure at the node points on S1, which is approximated by the arithmetical average of sound pressure on the two adjacent points. This error increases in an oscillating manner since the constant interval becomes relatively large in contrast to the shortening wavelength. As shown in Fig. 7, the relative error E of the separated sound pressure field changes as the parameter ka varies from 0.5 to 10. It can be seen that as concerns the overall tendency, the relative error decreases gradually along with the increase of ka for all methods. The separated incident sound pressure fields are always more accurate than the scattering parts. Separation results obtained by Eqs. (19) and (24) are identical, and differ with that of Eq. (25) slightly. With respect of robustness, the sound field separation method developed in this study acquires a more accurate performance comparing with that in Ref. [18,19] especially for the low frequency conditions. 5. Conclusions In this study, a new sound field separation method in the form of combined integral equations has been developed. On basis of Helmholtz integral equations for the incident and scattreing sound fields outside a sound scatterer, three systems of combined integral equations are derived for sound field separation on irregular surfaces enclosing the scatterer. Afterwards, through boundary element discretization of the separation surfaces involved in the combined integral equations, three corresponding systems of linear equations are obtained for practical application. Numerical simulations are carried out under different conditions, and the results show that the effect and accuracy of this combined integral equations based method for sound field separation are satisfactory. In future research work, indirect boundary integral equation instead of the direct one could be anticipated to improve the separation accuracy for extremely thin or flat bodies. We also expect to integrate the discrete sources method to improve the efficiency of the separation process. Acknowledgments The authors would like to acknowledge the support from the National Natural Science Foundation of China (Grant Nos. 51175465 and 51205351), China Postdoctoral Science Foundation (Grant No. 2012M521165) and Program for Zhejiang Leading Team of S&T Innovation (Grant No. 2009R50008). Fig. 7. Relative error of sound field separation on S1 under different SNR level noise. (a) ka = 0.5–2.5, 25 dB SNR, (b) ka = 2.5–10, 25 dB SNR, (c) ka = 0.5–2.5, 35 dB SNR, (d) ka = 2.5–10, 35 dB SNR. Solid line: relative error for Eqs. (19) and (24). Dot line: relative error for Eq. (25). Dot dash line: relative error for finite difference based method. Blue Line: incident sound pressure field. Red Line: scattering sound pressure field. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
v n ðrÞ ¼
1
@pðrÞ ! ðK v n ÞðrÞ ¼ iq0 x iq0 x @n
Z
Ln
v n ðlÞdl ¼ pðrÞ;
0
in which K is an integral operator and Ln is the distance along the normal direction n, we found that finite difference scheme is essentially an ill-posed problem owing to the compact characteristics of K [28]. As a result, even a small noise in the measurement data could induce a large error to the normal velocity [29]. In Fig. 6a, a great
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