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Estimation of the sound pressure ﬁeld of a bafﬂed uniform elliptically shaped transducer Jorge P. Arenas a,*, Jaime Ramis b, Jesus Alba c a

Institute of Acoustics, Univ. Austral de Chile, PO Box 567, Valdivia, Chile DFISTS, Univ. de Alicante, Alicante, Spain c DISAO, Polytechnic Univ. of Valencia, Gandia, Spain b

a r t i c l e

i n f o

Article history: Received 4 May 2009 Received in revised form 10 August 2009 Accepted 17 August 2009 Available online 10 September 2009 PACS: 43.20.Rz 43.40.Rj

a b s t r a c t In this paper an alternative approach to estimate the sound ﬁeld of an elliptically shaped transducer in an inﬁnite bafﬂe is described. The method is based on a singular value decomposition of a propagating matrix which is computed through a division of the vibrating surface into a ﬁnite number of small circular piston sources ﬂush-mounted on the elliptical surface. This decomposition is combined with the volume velocity vector on the discretized surface to obtain the sound pressure ﬁeld. Numerical examples for both on-axis sound pressure and directivity are presented for the uniform elliptical piston transducer and they are in good agreement with the results given by other methods. Ó 2009 Elsevier Ltd. All rights reserved.

Keywords: Sound radiation Piston source Elliptical piston SVD Transducer

1. Introduction Sound radiation from plane piston sources is of great practical importance in acoustical engineering and it has been an active research subject for many years. In particular, for a rigid piston source the amplitude and phase of vibration are considered constant across the face of the source and many transducer radiating sources can be represented by this simple model. Although many practical examples of elliptical pistons can be found as loudspeakers, ultrasonic radiators, and in sonar applications, there is a relative scarcity of information in the literature for elliptical, as opposed to circular and rectangular pistons. In general, the sound pressure ﬁeld of a piston can be found as integral expressions, such as Rayleigh, Fredholm, Bouwkamp, King, and Fresnel integrals, depending on the assumptions used to simplify the mathematical problem [1–3]. However, it is difﬁcult to ﬁnd a closed-form solution for general pistons. The Rayleigh integral can be solved exactly either for some particular geometry or under certain assumptions. Consequently, numerical integration is usually required to determine the sound pressure ﬁeld, resulting * Corresponding author. Tel.: +56 63 221012; fax: +56 63 221013. E-mail address: [email protected] (J.P. Arenas). 0003-682X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2009.08.003

in signiﬁcant computing time and memory usage. An alternative numerical approach to compute the sound pressure radiated from a vibrating structure is the boundary element method [4]. In a seminal paper, Borgiotti [5] presented a study on the singular function analysis of the radiator operator and the singular value decomposition (SVD) of its matrix discrete representation. His analysis leads to the decomposition of the boundary normal velocity into efﬁcient and inefﬁcient components, where the inefﬁcient components are associated with very weakly radiating evanescent ﬁelds. Thus, source modes that are eigenstates of radiated power are identiﬁed and later applied to radiation ﬁltering. In addition, Photiadis [6] has discussed the close relationship between SVD ﬁltering and the well-known wave vector ﬁltering. This idea was used by Bai and Tsao [7] to estimate the far-ﬁeld sound pressure of a rectangular vibrating plate using a propagation matrix based on a set of monopole sources. Analytical expressions for the radiation of elliptical piston transducers were derived by Thompson et al. [8] within the Fresnel approximation. They reported that, for the axial ﬁelds, the result takes the form of either a simple analytical expression (small perturbations from the circular piston case), a single quadrature in a real variable (unfocused and some focused cases), or a single quadrature in a complex variable (near focal points). In a subsequent

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article, Newberry and Rose presented an improved analytical approximation for the same case [9]. Gaussian-beam expansion technique has been probably the most widely used technique to calculate the Fresnel sound ﬁeld integral. Zhang et al. [10] have presented a method based on a straightforward extension of Gaussian-beam expansion for calculation of the Fresnel ﬁeld integral for uniform elliptical piston transducers. The source distribution function is expanded into the superposition of a series of two-dimensional Gaussian functions. Then, the corresponding radiation ﬁeld is expressed as the superposition of these two-dimensional Gaussian beams and is then reduced to the computation of these simple functions. Additional work on the beam ﬁeld modeling using the superposition of Gaussian beams has been presented in several papers dealing with ultrasonic piston transducers [11–14]. On the other hand, elliptical pistons are suitable to be used in some specialized acoustic devices since, unlike the circular piston, elliptical shaped transducers have the property to produce a fantype acoustic beam. Wu and Zielinski [15] introduced a concept of angle mapping which transforms a two-dimensional radiation pattern generated by a circular piston to a three-dimensional radiation pattern generated by an elliptical piston. They used the method to propose a novel array, consisting of several concentric elliptical ring radiators, which is capable of generating a superior radiation pattern [15,16]. In this paper, an alternative to estimate the sound pressure ﬁeld of a bafﬂed elliptically shaped transducer is presented. An appropriate discretization of the elliptical vibrating plane surface is used to deﬁne a propagating matrix, whose elements are computed from a set of small equivalent elementary pistons. Then, Singular Value Decomposition allows one to represent the propagating process by considering only a few dominant modes, which reduces the computation time.

2. Theory The sound waves radiated from a piston are primarily created by the interaction of its vibrating surface and a surrounding ﬂuid medium. The vibration causes the particles in the medium in the immediate vicinity to also vibrate thus generating propagating waves. Consider a piston-like vibrating surface of elliptical shape which is mounted in the x–y plane, centered at x; y ¼ 0, and is radiating

sound waves into a ﬂuid in the half space z > 0, as indicated in Fig. 1. The piston is speciﬁed to vibrate with a uniform harmonic velocity in contrast to the remaining portion of the x–y plane which is speciﬁed to be rigid (bafﬂed piston). It is well-known that the harmonic sound pressure in the half space z > 0, can be expressed through the Rayleigh surface integral as [1]

p¼j

kqc 2p

Z Z

VðSÞ

S

ejkr dS; r

ð1Þ

where k is the wavenumber, q is the ﬂuid density, c is the speed of sound, VðSÞ is the normal velocity distribution on surface S, and r is the distance between dS and the ﬁeld point. In Eq. (1) the harmonic time dependence has been dropped. Except for a few problems that can be done analytically, exact solution of Eq. (1) is difﬁcult to carry out. Therefore, in many practical situations, the Rayleigh surface integral is approximated through the evaluation of the Fresnel ﬁeld integral. In the Fresnel approximation, one considers a region that is far enough from the source to be outside the geometric near ﬁeld, but not far enough for the Fresnel correction to be neglected. For example, Thompson et al. [8] have shown that the pressure ﬁeld on the axis of a bafﬂed elliptical piston radiator vibrating harmonically with a peak velocity amplitude V 0 can be written, within the Fresnel approximation, as

pðzÞ ¼ qcV 0 ejkz

Z 2p dh jp 1 exp ; 2p s þ Ds cos 2h 0 2

ð2Þ

2

where s ¼ 12 ðzk=a2 þ zk=b Þ; Ds ¼ 12 ðzk=a2 zk=b Þ, and k is the wavelength; a and b are the semi-major and semi-minor axes of the elliptical piston, respectively. Therefore, the aspect ratio b=a 6 1. It is important to note that s represents the normalized ax2 ial distance, where zk=a2 and zk=b are the propagation distances 2 measured in units of the nearﬁeld transition a2 =k and b =k, for a circular transducer of radius a and b, respectively. Then, the parameter s as deﬁned in Ref. [8] for the elliptical case is the average of these normalized distances computed from the major and minor axes. Thus, jDs=sj 6 1. Now, consider that the vibrating plane surface is divided into N small elements of area Si , with i ¼ 1; 2; . . . ; N. Assuming that the characteristic length of the surface elements is small compared to a typical acoustic wavelength, the radiation integral can be represented by a linear transformation of the volume velocity vectors through a propagation matrix operator. Using this approach, the volume velocity of the ith surface element is V i Si , where V i is the normal velocity at the center point of element i. If the receiving ﬁeld space is divided into M discrete points, we can write

p ¼ Gu;

ð3Þ

where p is the M 1 vector of sound pressure, G is the M N propagation matrix and u is the N 1 volume velocity vector. We can notice that matrix G is complex, independent of the vibration distribution on the surface of the piston, and it depends only on the geometry of the piston and the frequency. Moreover, if we assume that each surface element on the elliptic piston’s surface can be represented by a circular piston having an area equal to that of the corresponding element [17], the entries of the propagating matrix can be obtained from the farﬁeld sound pressure due to a full circular piston [1]

Gik ¼ j

qc J1 ðkra sin hÞ expðjkrik Þ; pra rik sin h

ð4Þ

where r ik is the distance between element i on the piston’s surface to ﬁeld point ra is the radius of the elementary equivalent circupk; ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ lar piston Si =p ; J 1 is the ﬁrst-order Bessel function of the ﬁrst Fig. 1. Geometry of elliptical piston source.

kind, and h can be calculated as (see Fig. 1)

J.P. Arenas et al. / Applied Acoustics 71 (2010) 128–133

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sin h ¼ 1 ðz0 =r ik Þ2 :

ð5Þ

The computation of Eq. (3) can be further reduced using the Singular Value Decomposition (SVD) of the propagating matrix G. Given the matrix G, there are two unitary matrices V and B, such that we may write [18]

BH GV ¼

R 0 0

;

0

ð6Þ

where R is a diagonal matrix of singular values R ¼ diagðr1 ; r2 ; . . . ; rL Þ; L is the rank of matrix G, and 0 is the null matrix. The singular values are ordered as r1 P r2 P rL > 0. The columns of the unitary matrix V, that is, v 1 ; v 2 ; . . . ; v N , are the right singular vectors of G, and the columns of the second unitary matrix B, that is, b1 ; b2 ; . . . ; bM are the left singular vectors of G. In addition, the right singular vectors are eigenvectors of GH G, whereas the left singular vectors are eigenvectors of GGH . Since BBH equals the identity matrix, we ﬁnd from Eq. (6) that

R 0 GV ¼ B : 0 0

ð7Þ

Therefore, it follows that

Gv i ¼

ri bi ; i ¼ 1; 2; . . . ; L; 0;

i ¼ L þ 1; . . . ; M:

ð8Þ

Then, we can express the matrix G in the expanded form

G¼

L X

ri bi v Hi :

ð9Þ

i¼1

Therefore, we see that considering the most important singular values of matrix G, Eq. (9) can be used in Eq. (3) to obtain the vector of sound pressure p. Since computer resources are becoming less expensive and more readily available, implementation of Eq. (9) into computer codes results quite practical in predicting the sound pressure ﬁeld of a vibrating piston. In addition, some functions provided in modern numerical programs allow the efﬁcient computation of a few singular values and singular vectors of the propagating matrix. One way to enhance computation performance is to save memory space by using compact data types. The direct computation of the sound pressure in Eq. (3) for each value of frequency, involves storage of an M N matrix of complex data. Using Eq. (9), the amount of data storage required to compute G is M þ N þ 1 times the number of singular values used. For example, if M ¼ N ¼ 1000, Eq. (3) requires provision for storing and accessing 1 M of complex data. On the other hand, if 20 singular values are used in Eq. (9), computation involves storage of roughly 40 k of complex data. Thus, for large arrays, the method may show a signiﬁcant advantage in saving memory space.

elements, the center coordinates of each element can easily be obtained. This discretization produces a 4n2 1 vector u. In the following numerical examples, n was chosen to be the largest integer greater than or equal to 3a=k. Now, since we consider a uniform plane piston as a benchmark case, the phase angles and the normal component of velocity are constant over the entire vibrating surface. Then, we can consider the volume velocity vector as a constant vector u ¼ V 0 ½1; 1; . . . ; 1T , where V 0 is some constant real number representing the normal peak velocity amplitude. As customary, the sound pressure amplitude is normalized by qcV 0 . In the following examples, the minimum number of singular values was chosen to produce an error less than 0.1% of the exact value predicted by Eq. (3). 3.1. Pressure ﬁeld on the axis In general, the surface of the piston is divided into a number of small elements each of which is considered as an equivalent simple source (monopole) vibrating in phase with all the other elements [7]. However, the use of equivalent circular pistons seems to be more physically suitable than the model of monopole sources to describe the vibration of a planar elementary source. Fig. 2 shows the results of normalized sound pressure on the axis of a circular piston as a function of the dimensionless Fresnel distance zk=a2 . It can be seen that the computation of the propagating matrix using the monopole approach results in an overestimation of the sound pressure ﬁeld close to the piston surface ðzk=a2 < 0:4Þ. On the contrary, an excellent agreement is observed between the theoretical sound pressure [1] and the numerical results estimated through Eq. (4). Fig. 3 shows the on-axis normalized sound pressure ﬁeld of a circular piston as a function of the dimensionless Fresnel distance for

2 Amplitude

130

1.5 1 0.5 0

0

0.2

0.4 0.6 2 zλ/a

0.8

1.0

Fig. 2. Comparison of the normalized sound pressure ﬁeld on the axis of a circular piston as a function of the Fresnel distance. - - - - Eq. (3) with equivalent point sources, — Eq. (3) with equivalent piston sources, —— exact solution.

3. Numerical examples In this section, some numerical examples are given as evidence of the applicability of the method. To simplify the implementation of the computational codes, the elliptical surface with semi-major and semi-minor axes of a and b, respectively, is divided into small elements of equal area. Transformation to a classical elliptical coordinate system will produce surface elements of different area. The concept described in Ref. [19] for a circular disk is extended here for this purpose. Therefore, if we divide the surface of an ellipse into n concentric, contiguous elliptical rings, it is possible to divide each ring into elements of equal area S ¼ ab=ð4n2 Þ. Since the ith ring will have 4ð2i 1Þ

Amplitude

2 1.5 1 0.5 0

0

0.2

0.4 0.6 2 zλ/a

0.8

1.0

Fig. 3. The on-axis normalized sound pressure ﬁeld of a circular piston as a function of the dimensionless Fresnel distance for several numbers of singular values. - - Eq. (9) with two terms, - - - - Eq. (9) with ﬁve terms, —h— Eq. (9) with nine terms, — exact solution.

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several numbers of singular values. We can observe how the estimation improves as more terms are added to the approximation. Fig. 4 shows the normalized sound pressure ﬁeld on the axis of the uniform elliptical pistons for various ratios of b=a. The results are plotted as a function of the dimensionless distance s. Included are the results of a numerical integration of Eq. (2). For all the cases, a number of 14 terms in Eq. (9) was found enough to perfectly match the results predicted by the full propagating matrix. It is observed a good agreement with the results reported in Refs. [8,9] within the Fresnel approximation. It can also be seen that, as expected, the results using this matrix approach are better in

the region close to the surface of the piston than the ones predicted by Eq. (2). There is also good agreement with the results obtained through the Gaussian expansion technique presented by Zhang et al. [10].

3.2. Directivity The directivity pattern of a piston source is typically obtained from the sound pressure computed at a distance sufﬁciently large (farﬁeld) so that the sound pressure decreases linearly with dis-

1.5

2

Amplitude

Amplitude

1.5 1

0.5

1

0.5 b/a=0.5 0 0

0.2

0.4

0.6

0.8

b/a=0.7738 0 0

1

2

0.6

0.8

1

2 Amplitude

Amplitude

0.4

2.5

1.5

1

0.5

0.2

0.4 0.6 0.8 Normalized distance S

1.5 1 0.5

b/a=0.9045 0 0

0.2

0 0

1

b/a=1.0 0.2

0.4 0.6 0.8 Normalized distance S

1

Fig. 4. The on-axis normalized sound pressure ﬁeld with different values of b=a as a function of the normalized distance s. — Eq. (2), Eq. (9) with 14 terms.

a=λ/2

−5

−10

−15 0

50

100

0 Relative power (dB)

Relative power (dB)

0

a=2λ −10 −20 −30 −40

Relative power (dB)

Relative power (dB)

−20 −30

50

100

150

0

0

−50 0

−10

−40 0

150

a=λ

a=5λ

−10 −20 −30 −40 −50

50

100 Angle (degrees)

150

−60 0

50

100 Angle (degrees)

150

Fig. 5. Sound radiation patterns in plane x–z (h2 ¼ 0) of an elliptical piston transducer with b ¼ 0:175a for several values of a=k. –– Exact solution from Eq. (10), — Eq. (9). The number of terms used in Eq. (9) are: a ¼ k=2: 6 terms, a ¼ k: 9 terms, a ¼ 2k: 14 terms, a ¼ 5k: 28 terms.

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tance along a radial line connecting with the source. Thus, the calculation of the directivity function of an arbitrary piston can be obtained from the farﬁeld approximation of Eq. (1). In particular, the directivity function of an elliptical piston is given by [15]

Dðh1 ; h2 Þ ¼ 2pab

J 1 ðkc sin h1 Þ ; kc sin h1

ð10Þ

where

c¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 ða2 cos2 h2 þ b sin h2 Þ;

h1 is the angle between the normal to the surface of the piston and the projection of the line joining the middle of the surface and the observation point on the plane normal to the surface and parallel to b, and h2 is the same as h1 , with b substituted for a. To illustrate the manner in which the major and minor axes affect the sound radiation pattern of an elliptic piston, several numerical examples are presented in Figs. 5 and 6. Fig. 5 shows the radiation patterns of an elliptical piston transducer with b ¼ 0:175a for several values of a=k in a plane perpendicular to its surface and passing through its major axis, i.e. plane x–z in Fig. 1. Also shown are the exact calculations given by Eq. (10). Fig. 6 shows the radiation patterns of an elliptical piston transducer with a ¼ 5k in a plane passing through its minor axis (plane y–z) for various ratios of b=a. The number of singular values used to compute Eq. (9) is also indicated in both ﬁgures. These are the minimum numbers required to obtain the same radiation pattern as with the full propagating matrix. Again, we observe that the predictions are in very good agreement with theoretical results, in particular for angles close to points on the z-axis (90 ). In addition, the results are in agreement with those presented by Wu and Zielinski [15]. It is a well-known fact that an elliptical piston generates a fantype beam, which is relatively broad in one direction and narrow in the direction orthogonal to the broad beam direction. This is because in the elliptical transducer problem, axial symmetry does not occur. It should also be noted that the maximum sidelobe level associated with an elliptical piston is 17.6 dB below the main lobe maximum, the same as that of a circular piston [15]. Both facts are

clearly observed in Figs. 5 and 6. Moreover, it is seen that as the ratio between b and a increases, the number of zeros and lobes appearing in the radiation pattern increases, and this in turn results in a narrower main lobe in the radiation pattern. As expected, when the size of the piston is very small compared with the wavelength of the sound radiated, the elliptical piston behaves essentially like a point source. On the contrary, an elliptical piston becomes very directional at higher frequencies. We note that when the directivity of the elliptical piston increases, the number of terms required in Eq. (9) to accurately predict the sound pressure ﬁeld also increases.

4. Concluding remarks An alternative approach to estimate the sound ﬁeld of an elliptically shaped transducer using the concept of the propagating matrix and its singular value decomposition has been presented. It is noted that, through an appropriate discretization of the elliptical surface of the transducer and its transformation to a set of small equivalent circular pistons, better results than the classical point sources model are obtained. Numerical examples show that the method performs quite well when compared to the results reported in the literature. Evidently, the method is limited by the number of elementary radiators which would make the process very expensive in computation time. The method may be extended, in principle, to irregular shapes and nonuniform pistons commonly designed for speciﬁc applications [20]. Theoretically, sound radiation from a nonuniform piston may be calculated by retaining the piston velocity amplitude function inside the integral in Eq. (1). However, the radiation problem becomes much more complex and Eq. (1) is usually solved numerically. Since the propagating matrix depends only on the geometry of the piston and it is independent of the velocity distribution on the surface of the transducer, the information of the propagating matrix (including its singular values and vectors) may be stored in a computer for reusing with a different velocity distribution. This could be the case of focusing by a lens, which is approximately

0 b/a=1

−10 −20 −30 −40

Relative power (dB)

Relative power (dB)

0

−50 −60 0

50

100

−30 −40

−60 0

150

0

50

100

150

0 b/a=0.5

−20 −30 −40 −50 50

100 Angle (degrees)

150

Relative power (dB)

Relative power (dB)

−20

−50

−10

−60 0

b/a=0.7738

−10

b/a=0.2294 −10 −20 −30 −40 −50 0

50

100 Angle (degrees)

150

Fig. 6. Sound radiation patterns in plane y–z ðh2 ¼ 90 Þ of an elliptical piston transducer with a ¼ 5k for several values of b=a. –– Exact solution from Eq. (10), — Eq. (9). The number of terms used in Eq. (9) are: b=a ¼ 1 ðjDs=sj ¼ 0Þ: 28 terms, b=a ¼ 0:7738 ðjDs=sj ¼ 0:3Þ: 23 terms, b=a ¼ 0:5 ðjDs=sj ¼ 0:6Þ: 17 terms, b=a ¼ 0:2294 ðjDs=sj ¼ 0:9Þ: 10 terms.

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equivalent to a spatial modulation of the plane piston distribution [10,21]. This aspect clearly needs further study. Acknowledgments This work has been supported by CONICYT-FONDECYT No 1060117 and 7060073, which is gratefully acknowledged. References [1] Morse PM, Ingard KU. Theoretical acoustics. Princeton: Princeton University Press; 1986. [2] Kirkup SM. Computational solution of the acoustic ﬁeld surrounding a bafﬂed panel by the Rayleigh integral method. Appl Math Model 1994;18:403–7. [3] Herrin DW, Martinus F, Wu TW, Seybert AF. An assessment of the high frequency boundary element and Rayleigh integral approximations. Appl Acoust 2006;67:819–33. [4] Marburg S, Nolte B. Computational acoustics of noise propagation in ﬂuids: ﬁnite and boundary element methods. Berlin: Springer; 2008. [5] Borgiotti GV. The power radiated by a vibrating body in an acoustic ﬂuid and its determination from boundary measurements. J Acoust Soc Am 1990;88:1884–93. [6] Photiadis DM. The relationship of singular value decomposition to wave-vector ﬁltering in sound radiation problems. J Acoust Soc Am 1990;88:1152–9. [7] Bai MR, Tsao M. Estimation of sound power of bafﬂed planar sources using radiation matrices. J Acoust Soc Am 2002;112:876–83. [8] Thompson RB, Gray TA, Rose JH, Kogan VG, Lopes EF. The radiation of elliptical and bicylindrically focused piston transducers. J Acoust Soc Am 1987;82: 1818–28.

133

[9] Newberry BP, Rose JH. An approximation for the on-axis ﬁeld of an elliptical piston transducer. J Acoust Soc Am 1989;85:1357–9. [10] Zhang Y, Liu J, Ding D. Sound ﬁeld calculations of elliptical pistons by the superposition of two-dimensional Gaussian beams. Chin Phys Lett 2002;19:1825–7. [11] Ding D, Zhang Y, Liu J. Some extensions of the Gaussian beam expansion: radiation ﬁelds of rectangular and elliptical transducers. J Acoust Soc Am 2003;113:3043–8. [12] Ding D, Xu J. The Gaussian beam expansion applied to Fresnel ﬁeld integrals. IEEE Trans Ultrason Ferroelectr Freq Contr 2006;53:246–50. [13] Kim HJ, Schmerr LW, Sedov A. Generation of the basis sets for multiGaussian ultrasonic beam models – an overview. J Acoust Soc Am 2006;119:1971–8. [14] Spies M. Ultrasonic ﬁeld modeling for immersed components using Gaussian beam superposition. Ultrasonics 2007;46:138–47. [15] Wu L, Zielinski A. A novel array of elliptic ring radiators. IEEE J Ocean Eng 1993;18:271–9. [16] Wu L, Zielinski A, Bird JS. Acoustic transmitting/receiving elliptic ring arrays. In: Proc Ocean’93. MTS-IEEE conference; 1993. p. 409–14. [17] Hashimoto N. Measurement of sound radiation efﬁciency by the discrete calculation method. Appl Acoust 2001;62:429–46. [18] Horn RA, Johnson CR. Matrix analysis. Cambridge: Cambridge Univ. Press; 1996. [19] Arenas JP. Numerical computation of the sound radiation from a planar bafﬂed vibrating surface. J Comput Acoust 2008;16:321–41. [20] Mitra R. Effect of diameter-to-thickness ratio of crystal disks on the vibrational characteristics of ultrasonic ceramic transducers. Appl Acoust 1996;48:1–13. [21] Hsu DK, Margetan FJ, Thompson DO. Bessel beam ultrasonic transducers: fabrication method and experimental results. Appl Phys Lett 1989;55:2066–8.