Estimation of scattered sound field via nearfield measurement by source methods

Applied Acoustics 58 (1999) 261±281 www.elsevier.com/locate/apacoust

Estimation of scattered sound ®eld via near®eld measurement by source methods Gee-Pinn James Too*, Tien-Kuei Su Department of Naval Architecture and Marine Engineering, National Cheng Kung University, Tainan 70101, Taiwan Received 24 September 1997; received in revised form 21 September 1998; accepted 23 November 1998

Abstract Scattered near®eld measurements are utilized to evaluate a source strength distribution of an arbitrary-shaped body. Then, scattered holographs are obtained by source methods. Two approaches are discussed: (1) scattered pressure measurement approach and (2) particle velocity measurement approach. Three alternative source methods: internal source method (ISM), internal parallel source method (IPSM) and similar source method (SSM), are used in this study. Numerical evaluations are performed in order to verify these source methods. Three alternative source methods for the estimation of the scattered sound ®eld and the problem of non-uniqueness are also discussed. # 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction The motivation of the present study is to propose alternative source methods to describe the sound ®eld scattered from an arbitrary-shaped scatterer. The computation of the scattered sound ®eld has been extensively developed using the Helmholtz integral equation. The well-known combined Helmholtz integral equation formula (CHIEF) [1,2] which takes advantage of boundary element method (BEM) describes a three-dimensional sound ®eld by a two-dimensional approach. There were some diculties in the formula, such as, there is singularity in kernel function and there may be non-unique solutions in exterior boundary value problems. The non-uniqueness problem is avoided in the Helmholtz integral equation formula by selecting some interior points in the scatterer. Burton and Miller

* Corresponding author. Tel.: +886-6-274-7018; fax: +886-6-274-7019; e-mail: [email protected]. ncku.edu.tw 0003-682X/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S000 3-682X(98)0007 7-2

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[3] developed a linear combination of Helmholtz integral equation and its di€erential integral equation. By solving a set of these combination equations, the nonuniqueness in an exterior boundary value problem is avoided. Applications of source methods have been extensively developed in the problem of potential ¯ow. An interior source distribution was introduced by Karman [4] to solve the potential ¯ow problem. Also, sources with a surface distribution on the boundary surface [5,6] are used to describe the ¯ow ®eld. The existence and uniqueness of this approach is discussed in detail in [6]. Many researchers have been engaged in solving potential ¯ow problems by surface source distribution methods [7±10]. Alternative computer programs have been developed for two-dimensional and threedimensional boundaries. In recent studies, source methods have been developed to describe radiation and scattering of sound ®elds. Stepanishen et al. [11,12] developed the internal source method to study radiation and scattering. Their development emphasizes that the symmetric and anti-symmetric sound ®eld can be simulated by putting a line source of monopoles and dipoles, respectively. Least square error method is used by comparing the known normal velocity with the estimated normal velocity to determine the strength of the sources. Boag et al. [13] used a multi®lament source model to estimate acoustic scattered ®eld from ¯uid cylinders. Koopmann et al. [14] used a source method which is called the principle of wave superposition to compute acoustic ®elds. These authors also discussed the numerical errors associated with the method of superposition [15]. Near®eld acoustic holography technique developed by Maynard [16] and Williams et al. [17] allows one to describe three-dimensional holograph by measuring two-dimensional data. Bai [18] combined holography technique and boundary element method to describe radiation holograph generated by an arbitrary-shaped body. In the present studies, the source methods and the near®eld holography technique are used together to describe the scattered holograph from an arbitrary-shaped scatterer. The approach, developed earlier by Stepanishen et al., which compares the measured normal particle velocity with the estimated value, is modi®ed. The new approach compares the measured scattered pressure with the estimated scattered pressure. This approach allows one to apply these source methods to another measurement application. Several numerical evaluations are performed in order to verify of these source methods. In addition, the problem of non-uniqueness in an exterior boundary value problem can be avoided by using the source methods. An example is shown to demonstrate this advantage of the source methods. 2. Basic theory The theory is developed by measuring the scattered pressure or the particle velocity on the imaginary surface. The distribution of imaginary source strengths is determined by an inverse procedure of the least square error method. After the distribution of the source strengths is evaluated, the sound ®eld can be described. Two di€erent approaches, which are the pressure measurement approach and the velocity

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measurement approach, are discussed. The derivation of the velocity measurement approach is based on the earlier development by Stepanishen et al. The derivation of the pressure measurement approach is similar to the velocity measurement approach except that the measured data are the pressures instead of the normal particle velocities. Although the derivations of the two approaches are similar, the measurement data are di€erent. These di€erences result in the di€erent applications. The pressure measurement approach needs to measure the pressure on the imaginary surface by use of hydrophones or microphones. However, the velocity measurement approach needs to measure the normal particle velocity on the real body by use of accelerometers or other devices. 2.1. Approach via the measurement of scattered pressure A distribution of imaginary acoustic sources is considered inside an arbitraryshaped body. The scattered pressure Ps can be expressed as … Ps …x† ˆ ik0 C0 G…x; †Q…†d

…1†

s

where Q…† is the unknown source strength to be determined. G…x; † is the free ®eld Green function which is ( G…x; † ˆ

ÿ 4i H…1† 0 …kr† for two dimensional sound field eikr for three dimensional sound field r

…2†

where H10 …kr† is the zero order Hankel function of the ®rst kind. By measuring the scattered pressure Ps(s) and evaluating the scattered pressure ^ P…s† on the imaginary surface, one can utilize least square error method to obtain the imaginary source strength. The least square error function is de®ned as …

J ˆ W…s†jPs …s† ÿ P^ s …s†j2 ds

…3†

s

where W…s† is the weighting function. From Eq. (1), P^ s …s† can be expressed as P^ s …s† ˆ

N X

KP …s; xi †Q…xi †

iˆ1

where Kp…s; xi † is kernel function which is

…4†

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KP …s; xi † ˆ

… xi ‡
xi 2

xi ÿ

kp …s; x†dx

…5†


xi 2

In the present study, kp …s; x† is expressed as kp …s; x† ˆ ik0 c0 G…s; x†

…6†

Substituting Eq. (4) into Eq. (3) leads to … J ˆ W…s†jPs …s† ÿ

N X

kp …S; Xi †Q…Xi †j2 ds

…7†

iˆ1

Eq. (7) is expanded and is expressed as " # … n N N X N X X X KPi Qi ÿ Ps KPj Qj ‡ KPi KPj Qi Qj ds J ˆ W…s† jPs …s†j2 ÿ Ps iˆ1

jˆ1

iˆ1 jˆ1

…8† where * represents complex conjugate of the variables. When the error function J has a minimum value, then … N … X @J  ˆ ÿ KPk Ps Wds ‡ KPk KPj WdsQj ˆ 0 @Qk jˆ1

…9a†

… N … X @J  ˆ ÿ KP1 Ps Wds ‡ KP1 KPi WdsQi ˆ 0 @Q1 iˆ1

…9b†

s

s

s

s

Eq. (9b) can be expressed as N X iˆ1

Ali Qi ˆ B1

…10†

where  Ali ˆM mˆ1 Wm KP …Sm ; X1 †Kp …Sm ; X1 †

B1 ˆ

M X mˆ1

Wm KP …Sm ; X1 †Ps …Sm †

M represents the number of the measuring points.

…10a†

…10b†

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2.2. Approach via measurement of the particle normal velocity of the scattered wave From Euler equation and Eq. (1), one can express the particle velocity of the scattered wave as … V~s …x† ˆ ÿr G…x; †Q…†d

…11†

s

Therefore, the normal particle velocity on the imaginary surface is: … n
r G…x; †Q…†d n~
V~s …x† ˆ ÿ~ s

…

ÿ kv …x; †Q…†d

…12†

s

ˆ Vs By measuring the normal particle velocity Vs of the scattered wave and evaluating the normal particle velocity V^ on the imaginary surface, one can utilize least square error method to obtain the imaginary source strength. The least square error function is de®ned as … J ˆ W…s†jVs …s† ÿ V^ s …s†j2 ds

…13†

s

where W…s† is the weighting function and V^ s …s† can be expressed as V^ s …s† ˆ

N X

Kv …s; xi †Q…xi †

…14†

iˆ1

where Kv …s; xi †

… xi ‡
xi 2

xi ÿ

kv …s; x†dx


xi 2

kv …s; x† is the kernel function for the approach of the velocity measurement and can be expressed as kv …s; xi † ˆ n~
rG…x; †

…15†

Similarly, the least square error function J is expanded and di€erentiated with respect to Q(xi). Then, the equation is discretized and rewritten as

266

Gee-Pinn J. Too, Tien-Kuei Su / Applied Acoustics 58 (1999) 261±281 N X iˆ1

Ali Qi ˆ B1

…16†

where Ali ˆ

B1 ˆ

M X mˆ1

M X mˆ1

Wm KV …Sm ; X1 †KV …Sm ; Xi †

…16a†

Wm KV …Sm ; X1 †Vs …Sm †

…16b†

From these two approaches, one obtains the unknown source strengths Q…s†s by solving either Eqs. (10) or (16). Then, by evaluating Eqs. (1) and (11), the scattered pressure and the particle velocity of the scattered wave can be evaluated in any position in the sound ®eld. An alternative approach to obtain scattered holograph is to solve Eqs. (10) and (16). Then, from Eqs. (1) and (12), the scattered pressure and the normal particle velocity on the imaginary surface is evaluated. Finally, the scattered holograph can be obtained by evaluating exterior Helmholtz integral equation formula.

Fig. 1. Arrangement of sources in internal source method.

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Fig. 2. Arrangement of sources in Internal Parallel Source Method for (a) two±dimensional problems and (b) three±dimensional problems.

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3. Alternative source methods Three alternative source methods are used in this study. They are internal source method (ISM), internal parallel source method (IPSM) and similar source method (SSM). They can be developed by either matching the normal particle velocity or pressure on the imaginary surface. The main di€erence between these methods is the arrangement of the imaginary sources. Due to the alternative arrangements of sources, the applications and the advantages of these methods are di€erent. The arrangement of sources in ISM is shown in Fig. 1 The sources are lined up in a straight line. The sources can be monopoles for a symmetric sound ®eld or dipoles for an anti-symmetric sound ®eld or combination of monopoles and dipoles for an arbitrary two-dimensional sound ®eld. The arrangements of sources in the IPSM are shown in Fig. 2(a) and (b) which are the arrangements for 2-D and 3-D problems, respectively. The sources are lined up parallell with each other and only monopoles are used in the simulation. In Fig. 2(b) there are six di€erent line sources with two line sources oriented along each orthogonal axis. The arrangement of sources in SSM can be either on the imaginary surface around the body or on the real body surface as shown in Fig. 3. In the numerical evaluation, the unknown of all the three methods are source strengths Q(s)s which are obtained by solving Eq. (10) or (16). Eqs. (10) and (16) are solved by calling IMSL DLSLHF subroutine. A diculty in this evaluation for SSM is the singularity problem if the source is too close to the imaginary surface. However, the singularity can be avoided by positioning the sources not being too close to the imaginary surface.

Fig. 3. Arrangement of sources in similar source method.

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Fig. 4. Variations of scattered pressure on axial axis (=0 ) with ka=1 evaluated by velocity measurement approach.

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Fig. 5. Variations of normal particle velocity in the far ®eld (r=10 a) with ka=1 evaluated by pressure measurement approach.

4. Results and discussion 4.1. Comparison of pressure measurement approach and velocity measurement approach In the following, an incident planar wave on an in®nite rigid circular cylinder is considered. The analytical solution of the scattered pressure is: [19,20]

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Fig. 6. Amplitudes of the scattered pressure on axial axis (=0 ) with ka=15.

Fig. 7. The e€ects of source number used in Internal Source Method with ka=15.

271

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Ps …r; † ˆ ÿ

1 X

"n …ÿi†n

nˆ0

J0n …ka†

H…n1† 0 …ka†

H…n1† …kr† cos…n†

…17†

where 

"n ˆ 1 when n ˆ 1 "n ˆ 2 when n > 1

…1†

Hn is the nth order Hankel function, and Jn(ka) is the nth order of Bessel function. From Euler equation and Eq. (17), one obtains the radial component of the particle velocity of the scattered wave ( 1 X i J1 …ka† 1=2"n …ÿi†n H…11† …kr† ‡ ÿ "0 Vs …r; † ˆ  0 c0 J1 …ka† ‡ iY1 …ka† nˆ1 ) i ‰Jnÿ1 …ka† ÿ Jn‡1 …ka†Š h …1† …1 † h i Hnÿ1 …kr† ÿ Hn‡1 …kr† cos…n† 1† 1† H…nÿ1 …ka† ÿ H…n‡1 …ka†

Fig. 8. The e€ects of source number used in internal parallel source method with ka=15.

…18†

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where Y is Neumann function. When r=a, one obtains the radial particle velocity on the imaginary surface which is also the surface of the circular cylinder. Figs. 4±9 show several evaluations by the three alternative source methods. The parameters used in the evaluation are: radius of the cylinder a=1 m; sound speed c0=1500 m/s; 0=1000 kg/m3. The measured points are all selected on the surface of the cylinder. The number of the measured points used in the evaluation is set to be 360. For SSM, the source points are oriented at a smaller cylinder inside the in®nite cylinder with the radius of the smaller cylinder being 0.9 a. Fig. 4 shows the variations of the scattered pressure on axial axis evaluated by velocity measurement approach with ka=1. The results are compared with the analytical solution evaluated from Eq. (17). The results of all three source methods show excellent agreement with the analytical solution. Fig. 5 shows the variations of the normal particle velocity of the scattered wave in the far ®eld (r=10 a) by pressure measurement approach with ka=1. The results are compared with the analytical solution evaluated from Eq. (18). The results also show good agreements with the analytical solution. From Figs. 4 and 5, it is clear that at low ka (such as ka=1) the pressure measurement approach and velocity measurement approach both result in good estimation. Also, three alternative source methods all give excellent agreement with the analytical solution.

Fig. 9. The e€ects of source number used in similar source method with ka=15.

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Fig. 10. Variations of normal particle velocity in the far ®eld (r=10 a) with ka=1 for the two incident waves case.

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Fig. 11. Variations of scattered pressure in the far ®eld (r=10 a) with ka=15 for the two incident waves case.

Fig. 6 shows the variations of the scattered pressure on axial axis evaluated by the velocity measurement approach with ka=15. Three alternative source methods all give excellent agreement with the analytical solution. Figs. 7±9 show the e€ects of source number used in the evaluation. The results show that ISM and IPSM are both sensitive to the source number used in the evaluation. It is found in Fig. 7 that using ISM with 40 source number will give better results than using the same method with 80 source number. Also, it is found in Fig. 8 that using IPSM with 80

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source number will give better results than using the same method with 92 source number. On the other hand, it is shown in Fig. 9 that SSM gives a convergent solution as the source number increases. 4.2. Two planar waves incident on a circular in®nite cylinder Two planar waves incident on a circular in®nite cylinder are considered. One incident angle is set to be 0 while the other incident angle is set to be 90 from the x axis. The exact solution is obtained by a superposition of the solution of each incident wave evaluated by Eqs. (17) and (18). Fig. 10 shows the variations of normal particle velocity with ka=1 in the far ®eld (r=10 a). In the ®gure, all three methods provide good estimations in the far ®eld. Fig. 11 shows the variations of scattered pressure with ka=15 in the far ®eld (r=10 a). It is clear that the estimation of ISM will have some small discrepancies in some areas. However, the other two methods still give very good estimation. 4.3. Three dimensional application In order to show the application of SSM in a three-dimensional case, one shows an example in which a planar wave incident on a rigid sphere is considered. The analytical solution of the scattered pressure of this example is derived in details in Ref. [20]. In the numerical evaluation of SSM of this case, the number of the measured points is chosen to be 162 points which are distributed uniformly on the surface of the rigid sphere. The diagram of the sphere is shown in Fig. 12. The source points are chosen on a smaller sphere inside the rigid sphere. The radius, A, of the smaller sphere is selected to be A=0.9a, where a is the radius of the rigid sphere. Also, the number of source points is selected to be the same as the number of the measured points. The parameters used in the evaluation are: radius of the rigid sphere a=1 m; sound speed c0=1500 m/s; 0=1000 kg/m3. A convergent test is

Fig. 12. Diagram of a planar wave incident on a rigid sphere.

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Fig. 13. Numerical evaluation of scattered pressure on axial axis in the propagation direction of the planar wave: ÐÐ, Real part (exact); Ð-, Imag part (exact); *, real part (SSM); ~, Imag part (SSM).

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Fig. 14. The diagram of an in®nite cylinder with a strip distribution of scattered normal velocities.

Fig. 15. The frequency responses evaluated by the Helmholtz integral equation formula without overdetermined points: Ð-, frequency response of point 1; Ð- - -, frequency response of point 2; - - - - - - , frequency response of point 3.

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performed to check the necessary number of the measured points and the source points. It is found in the test that the number of the measured points and the source points need to be about 162 points in order to obtain a good estimation with the ka values up to 10. The larger the ka values are, the larger the number of the measured points and the source points are needed. Fig. 13 shows the result of the scattered pressure along the axial axis in the propagation direction of the planar wave. Fig. 13 indicates that SSM gives a good agreement with the exact solution [17] along the axial axis and it can be used for the estimations in three-dimensional cases. There is no numerical diculty in the evaluations, but it needs more source points and measured points to obtain good results with the higher ka values. 4.4. Discussion of the problem of non-uniqueness The problem of non-uniqueness is found at certain frequencies as the source points are too close to the measured surface. This problem is also found in CHIEF and is solved by using over-determined points inside the measured surface. In order

Fig. 16. The frequency responses evaluated by the combined Helmholtz integral equation formula (CHIEF) with over-determined points; and by SSM: Ð-, frequency response of point 1 (SSM); - - Ð-, frequency response of point 2 (SSM); - - - - - -, frequency response of point 3 (SSM); *, frequency response of point 1 (CHIEF); ~, frequency response of point 2 (CHIEF); ^, frequency response of point 3 (CHIEF).

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to demonstrate how to avoid the problem in SSM, an in®nite cylinder case with scattered normal velocities shown in Fig. 14 is considered. Point 1, point 2 and point 3 are the ®eld points considered. Fig. 15 shows the frequency response evaluated by Helmholtz integral equation formula where no over-determined point is used. It is shown that there are non-uniqueness solutions at frequencies at 575, 915, 1226, 1318, 1675 and 1812 Hz. Fig. 16 shows the frequency response of the case evaluated respectively by SSM and Helmholtz integral equation formula with over-determined points. In SSM, the source points are oriented at A=0.9 a. It is shown that the nonuniqueness problem can be avoided by selecting the sources not being too close to the measured surface. Also, the accuracy will not be in¯uenced by the positions of source points. 5. Conclusion It is clear from this study that velocity measurement approach and pressure measurement approach both give good estimations. It is shown that the source number used in the evaluation has e€ects on the accuracy of source methods. It is safe to say that all of the three source methods give good estimations of scattered holograph. Also, for three-dimensional cases, SSM gives good estimations in the evaluation of the scattered sound ®eld. In addition, the problem of non-uniqueness at certain frequencies can be avoided by choosing the sources not being too close to the measured surface. This is also the advantage of the source methods. Acknowledgements This study is supported by NCHC under project number NCHC-85-05-023. References [1] Copley LG. Integral equation method for radiation from vibration body. Journal of the Acoustical Society of America 1967;44:807±16. [2] Schenck HA. Improved integral formulation for acoustic radiation problem. Journal of the Acoustical Society of America 1968;44:41±58. [3] Burton AJ, Miller GF. The application of integral equation methods to numerical solution of some exterior boundary-value problems. In: Proc. of the Royal Society, London. 1971; A323:201±10. [4] Von Karman T. Calculation of pressure distribution on airship hulls. NACA TM 574, 1930. [5] Lamb H. Hydrodynamics. London: Cambridge University Press, 1932. [6] Kellogg OD. Foundations of potential theory. New York: Ungar, 1929. [7] Hess JL. Calculation of potential ¯ow about bodies of revolution having axes perpendicular to the free-stream direction. Journal of the Aerospace Sciences 1962;29:726±42. [8] Hess JL, Smith AMO. Calculation of nonlifting potential ¯ow about arbitrary three-dimensional bodies. Journal of Ship Research 1964;8:22±44. [9] Giesing JP. Nonlinear unsteady potential ¯ow with lift. Journal of Aircraft 1968;5:135±43. [10] Hess JL. High-order numerical solution of the integral equation for two- dimensional Neumann problem. Computer Methods in Applied Mechanics and Engineering 1973;2:1±15.

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