Research on Array Structures of Acoustic Directional Transducer

This paper focuses on the directivity design of array structures of acoustic directional transducers. Based on Huygens principles, the directivity formula of transducer arrays under random distribution in xyz space is derived when the circular piston transducers are used as the array element, which is used to analyze the directivity and acoustic pressure of conical transducer arrangements. In addition, a practical approach to analyze the directivity and acoustic pressure of transducer arrays under random arrangements is proposed. Findings . The conical transducer arrays show side lobes at higher frequency. Below the frequency of 2kHz, array directivity shows rapid changes. Above the frequency of 2 kHz, array directivity varies slowly with frequency. Besides, the beam width is Θ − 3dB ≤ 29 . 85 ° .


Theoretical Calculation of Transducer Array Directivity
For the single transducer, its directivity is decided by the ratio of sound wave length λ to size a. Take the circular piston source on the baffle as an example, as shown Figure 1.
As for the single transducer, its directivity is expressed as the following formula [1][2][3]: In this formula, J 1 is first-order Bessel function; wave number is k � 2π/λ; a represents sound source radius; and d � 2a represents sound source diameter.
For linear arrays of point acoustic sources that consist of multiple transducers, the directivity is not decided by the ratio of sound wave length λ to transducer size a, but by array element arrangement. It is important to obtain the formula of directivity for transducer arrays in any random arrangements.

Comparison of Directivity of Transducer Arrays and Random Array Configurations
According to Huygens principles, the linear array directivity function of n point acoustic sources is derived, as shown in Figure 1. It should be noted that at this point, each unit transducer is regarded as point with no radius, assuming d � 2a. It is possible to derive the directivity function of transducer arrays in N lines and M rows, as shown in Figure 2 [1]: D(α, θ) � sin kMd 1 /2 cos α sin θ M sin kd 1 /2 cos α sin θ · sin kNd 2 /2 sin α sin θ N sin kd 2 /2 sin α sin θ . (2) In this formula, d 1 and d 2 represent line space and space between columns; α is angle between the projection of vector OP on XOY and positive x axis; and θ is the angle between vector OP and z axis. e above formula obtains the following conclusion: for this function, it is necessary to demonstrate equidistant distribution in a certain direction (set the spacing distance in x axis or y axis); it needs to be a rectangular distribution instead of random arrays (such as polygon or circular array). When M or N is 1, this formula can calculate the array directivity. When M � N � 1, the formula can calculate directivity of single transducer [7][8][9].

Theoretical Calculation of Random
Transducer Array Directivity Figure 3 shows the rectangular coordinate system of directional acoustic transducer arrays. e central point of transducer arrays O is the origin of coordinate. Assuming transducer array within the xoy plane of the three-dimensional coordinate system, any single transducer is at the position Q (x 0 , y 0 ). In the sound field, the distance between any single observation point P (x 0 , y 0 , z 0 ) and the origin of coordinate is r, the intersection angle between the point and z axis is θ, and rotation angle is φ. It is likely to obtain normalized directivity function of n circular piston transducer arrays with the radius a and random placements: From formula (3), it can be concluded that, for this function, there is no need to set the distance between transducer array elements or rectangular arrangements to obtain the directivity of random array arrangements. It is only necessary to identify the coordinate of each array element. When P (x 0 , y 0 ) is determined, the directivity of random transducer plane layouts can be obtained. Besides, this formula is relatively complex and difficult to obtain directivity patterns of arrays by conducting research on twodimensional directivity. Hence, a research method of threedimensional directivity is used to identify directivity [4][5][6].

Directivity Analysis of Three-Dimensional Transducer Arrangement
As shown in Figure 4, the three-dimensional space is established in the three-dimensional coordinate system with random transducers. O is the central point of the threedimensional coordinate system. Assume transducer arrays are within the xyz plane of the three-dimensional coordinate system. If the circular piston transducer with the radius a vibrates at the velocity, u � u 0 e jωt , where u 0 is the velocity amplitude. Assume it is right at the origin of coordinate O.
en, its sound pressure is generated at observation point P (x 0 , y 0 , z 0 ) (the distance from origin of coordinate is r. e intersection angle with z axis is θ. e rotation angle is the position of φ ). e resulting sound pressure is In this formula, ρ 0 is the density of medium. k � 2π/λ is the wave number (λ is the sound wave length). Frequency f � 10000 Hz. Sound speed is When circular piston transducer is at any point of xyz space, set r 1 as the sonic path distance between Q and P. Likewise, for at any point of xyz space Q, the sound pressure generated by circular piston transducer with the radium a at point P is In this formula, θ Q is the included angle between vector QP  2 Mathematical Problems in Engineering In this formula, r Q is the module of vector OQ ⇀ and θ 1 is the included angle between vectors OQ ⇀ and OP ⇀ . It should be noted that θ 1 may be an acute angle or an obtuse angle. e rectangular coordinate of point P is (x 0 , y 0 , z 0 ). e rectangular coordinate of point Q is (x 1 , y 1 , z 1 ). Vectors OP ⇀ and OQ ⇀ are represented as x 0 , y 0 , z 0 and x 1 , y 1 , z 1 . en, included angle cosine in formula (4) is Formulas (4) and (5) are combined and arranged to obtain where, x 0 , y 0 , and z 0 in formula (6) are converted to circular cylindrical coordinates as the following: x 0 � r sin θ sin ϕ, Formula (7) is substituted to formula (6): In the far field, the amplitude of formula (2) r 1 ≈ r, θ Q ≈ θ. Formula (8) is substituted to formula (2) to obtain the following: the sound pressure generated by any point Q at point P in the xyz space is 2J 1 (ka sin θ) ka sin θ e j ωt− kr+k x 1 sin θ sin ϕ+y 1 sin θ cos ϕ+z 1 cos θ ( ) [ ] . (11) If planar transducer arrays consist of n circular piston transducers, the i transducer is in position (x i , y i , z i ) of the plane. Each transducer vibrates at the velocity u � u a e jωt . e sound pressure generated by n circular piston transducers at point P in the xyz space is shown in the following: 2J 1 (ka sin θ) ka sin θ e j(ωt− kr) n i�1 e jk x 1 sin θ sin ϕ+y 1 sin θ cos ϕ+z 1 cos θ ( ) .
According to formula (11), it is feasible to calculate the feasibility of circular piston transducers with random arrangement in three-dimensional space. However, the above formula is a function concerned with variables θ and φ, which are hard to identify the directivity of the array. Hence, three-dimensional directivity research method is used in the process. To be more specific, conversion of coordinates is carried out in formula (11). Figure 4: e three-dimensional coordinate system of transducer arrays.
x 0 � sin θ sin ϕ · D( θ, ϕ ), Assume the most central transducer is origin of coordinate O (0, 0), then the position coordinate of acoustic directional dispersion system is shown in Figure 4. e rectangular coordinates of acoustic directional transducer array are substituted in formula (11) to calculate the directivity angle of transducer. By changing the frequency, it is feasible to get the result shown in Figure 3.
It is known from Figure 5 that acoustic directional transducer array shows weak directivity at the sound wave frequency level of 500 Hz. If the sound wave frequency level increases, the level of directivity also rises gradually. When the frequency level reaches 4 kHz, apparent sidelobe shows up. When the frequency level reaches 6 kHz, directivity becomes favorable, but sidelobe becomes more apparent as well.
e figure shows that beam width Θ −3 dB gradually narrows as frequency increases. Below the beam width of 2 kHz, frequency change is apparent. Above the beam width of 2 kHz, frequency change is slow. e frequency of dispersion sound wave ranges between 2.1 and 3.4 kHz. When it is above 2 kHz, the beam width of acoustic directional transducer is Θ −3 dB ≤ 29.85°, directional acute angle (Θ −3 dB /2) ≤ ± 14.925°. Favorable directivity is shown.

Conclusion
Based on the Huygens principle of sound waves, this work has derived the formula for the directivity of transducer arrays in random arrangement when circular piston transducers are used as array elements. Based on this formula, it studied the directivity and sound pressure of conical transducer array arrangements. e work provided a way to analyze directivity and sound pressure of transducer arrays in random arrangements for conical transducer and acoustic directional transducers. Findings: conical transducer arrays demonstrate sidelobe at high-frequency levels, but it can be overlooked compared to the main lobe. Below the frequency level of 2 kHz, array directivity changes rapidly. Above the frequency level of 2 kHz, array directivity changes more slowly, and the beam width is Θ −3 dB ≤ 29.85°. e work expands the formula of calculating the directivity of circular piston transducers with random array arrangements. Based on the digital simulation of computers, it resolves the difficult issues in the directivity design of threedimensional arrays of acoustic directional transducers, providing positive significance for designing acoustic directional transducer arrays.

Data Availability
No data used to support the study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Acknowledgments
is work was financially supported by the Civil Aviation Flight University of China fund project: fault diagnosis of the operating system of the certificate simulator (XM2803).  Mathematical Problems in Engineering