Acta Polytechnica https://doi.org/10.14311/AP.2024.64.0470 Acta Polytechnica 64(5):470–486, 2024 © 2024 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague ACOUSTIC RADIATION BEAM PATTERNS OF DIFFERENT POLYGONAL PISTON SURFACES Taofeek Ayotunde Yusufa,∗, Sheriff Abiodun Aodub, Abeeb Opeyemi Alabib a Joseph Sarwuan Tarka University, Mechanical Engineering Department, P. M. B. 2373, 970101 Makurdi, Nigeria b Kyungpook National University, School of Mechanical Engineering, Daehak-ro 80, 41566 Daegu, South Korea ∗ corresponding author: yustaofeekay@yahoo.com Abstract. Transducers are used in acoustic remote sensing and several other modern applications. The radiating surface geometry of a single acoustic transducer affects its beam pattern and has a cumulative effect on the overall performance of an array configuration. In addition, the transducers with polygon radiating surfaces provide more compact structure in an array than the circular pistons due to their flat surface edges. However, until now, the radiation patterns of acoustic polygonal pistons have not been exclusively studied and documented. In this study, the directional factors of acoustic pistons with polygonal surfaces of three to ten sides were derived analytically from the first principle using the Rayleigh integral. These directional factors were used to synthesise and characterise the radiation patterns of the piston sources in comparison with the baffled circular piston of an equivalent surface area. The rectangular, the triangular, and the rest of the polygons have the same performance with the circular piston when the surface diameter is not higher than 0.5λ, 0.28λ and 0.43λ, respectively. The main lobe of the acoustic emissions from the square is very close to that of the circular piston whose diameter is greater than a wavelength while that of the rectangle is wider. The results show that the radiating surface perimeter and symmetry are the two most critical factors affecting the beam characteristics of a piston source rather than the surface area or the number of sides. The theoretical results were validated using the finite element method with an excellent compliance. Keywords: Polygonal piston source, acoustic transducer, Rayleigh integral, radiation beam, finite element analysis. 1. Introduction The acoustic transducer technology is relevant in many areas of contemporary applications, such as underwater sonar [1], biomedical diagnosis [2], fluid flow and heat transfers [3], energy harvesting [4], non-destructive testing [5], and enhanced oil recovery [6]. In underwater applications, acoustic transducers are used as sensors through the technology of transmission of sound waves for the purpose of remote communication, imaging, or detection [7, 8]. One of the output characteristics for determining the performance of such devices is a beam pattern. The beam pattern is a logarithmic function (decibel scale) of an angle-dependent characteristic called the directivity factor. The directional factor is strongly influenced by the combined effect of the area of its radiating surface and the frequency of operation. This makes the geometric dimensions of an acoustic element a critical aspect of an effective design [9]. In addition, the surface geometry of individual component transducer elements affects both the overall performance and the optimisation of inter-element spacing in an array transducer structure. For the beam pattern synthesis, a single transducer is considered as an elastic membrane vibrating like an acoustic piston. The directional factor of a circular piston on a rigid infinite baffle is well known and has been used in several studies [10–16]. However, circular pistons are unsuitable for composing an array transducer of densely packed elements. They typically leave inevitable gaps between their edges contributing to poor space conservation [17, 18] and a considerable loss in the radiated power [19]. Meanwhile, acoustic pistons with flat surfaces such as polygons have more prospects in providing a more compact transducer structure [20]. In addition, their side lengths can be varied independently to increase the radiating surface area without increasing the size in all the directions. This offers a more efficient way to control the overall size of the transducer than using the circular pistons without the liability of an added cost [21]. While the square and rectangular pistons are also being fairly used in few instances [22, 23], the higher-sided polygons such as the hexagon have more elements in their primary neighbourhood, which enhances their radiation characteristics in a staggered array configuration [20]. 470 https://doi.org/10.14311/AP.2024.64.0470 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 64 no. 5/2024 Acoustic radiation beam patterns of different polygonal . . . To date, there has been no independent study that has exclusively analysed these polygonal surfaces and effectively characterised their acoustic radiation performance. Roh et al. made an effort in that direction but apparently made no distinction between the rectangular and square sources while some weird non-symmetrical geometries, such as the triangle and pentagon, were not included [19]. Consequently, this study took a more comprehensive approach on this subject. The directional factors of nine (9) acoustic pistons sources having 3–10 sided polygon surfaces were characterised and compared with that of the circular piston on a rigid baffle, which was used as the standard. Using the finite element commercial software, Pzflex®, Tonpilz transducer structures were modelled and simulated using the geometries of the piston sources for the shapes of the head masses to validate the theoretical results. The comparison between the theoretical and FEM beam patterns showed good agreements. Finally, the expression for the directional factors were standardised to facilitate their adoption in subsequent designs. 2. Equivalent surface area of the piston sources The circular piston of a given radius ac is used as the standard reference as shown in Figure 1. Figure 2 shows the polygonal pistons with three to ten surfaces proposed for this study, respectively. The half-side length (HSL) a of each of them carries the subscripts denoting the first letter of their respective names except for the hexagon defined as ax in Figure 2e. All the polygons are regular except the rectangle where one side is equal to the diameter of the circular piston as shown in Figure 2b. The diameter of the circular surface is specified in terms of the wavelength λ that was calculated using λ = c f , where c = 1 500 m s−1 is the acoustic speed in water and f , is the given frequency of vibration. It is instructive that the radiating surface areas must be the same for all the piston sources to make a realistic comparison of their acoustic beam characteristics. Consequently, the HSL of all polygon surfaces are determined at the equivalent surface area of the circular piston and expressed in terms of ac as shown for the triangle in Equation (1). From this equation, the HSL of the triangle at is determined as given in Equation (2). Similarly, the HSL for the rectangle and the square are derived from the equivalent circular surface area as given in Equations (3) and (4), respectively. 1 2 × 2at × at tan 60 = πac 2, (1) at = (√ π tan π 6 ) ac = ( 4 √ π2 3 ) ac, (2) ar = π 4 ac, (3) as = √ π 2 ac. (4) For the high-sided polygons such as the pentagon shown in Figure 3, the equivalent area is expressed in Equation (5) and the HSL, ap is determined as given in Equation (6). Following the same approach for other polygons, it can be found that the HSL takes the general form of Equation (7), which is true for all the polygons where n is the number of sides except the rectangle, which is non-regular. 5 × 1 2 × 2ap × ap tan 54 = πac 2, (5) ap = (√ π 5 tan π 5 ) ac. (6) Then, generally for all the n-sided regular polygons, the HSL is: ∴ (√ π n tan π n ) ac. (7) 3. Derivation of the directional factors of the acoustic piston surfaces This section describes the procedures for the derivation of the beam patterns for all the sources with the assumption that all the pistons are mounted on rigid baffles. 471 T. A. Yusuf, S. A. Aodu, A. O. Alabi Acta Polytechnica ac Figure 1. The reference circular piston. at at (a). Triangle. ac ar (b). Rectangle. as as (c). Square. ap ap (d). Pentagon. ax ax (e). Hexagon. ah ah (f). Heptagon. ao ao (g). Octagon. an an (h). Nonagon. ad ad (i). Decagon. Figure 2. The polygonal piston surfaces. 472 vol. 64 no. 5/2024 Acoustic radiation beam patterns of different polygonal . . . 2ap 54o 72o Figure 3. The pentagon surface area geometry. 3.1. Circular piston The directional factor of a circular piston on a rigid baffle is well known [24]. However, it would be derived again in this study to set a general background for the approach to be used for the polygons. Consider an acoustic element of a HSL, AB and a small thickness dx whose centre A is located at distance x from the centre C of the circular piston as shown in Figure 4. The piston is considered to be on a rigid baffle such that acoustic radiation is projected in the forward direction from its surface into a point Q at an angle θ from the vertical acoustic axis. Point Q is located at distances R and r from the centres C and A, respectively. Taking AP as a wave front perpendicular to CQ, the relationship between CP and x can be found as expressed in Equation (8). Considering that R and r are distances in the far-field, the line CQ and AQ are parallel resulting into Equation (9). CP = x sin θ, (8) r = R − CP = R − x sin θ. (9) The total acoustic pressure radiated from the acoustic element, pc can be calculated using the Rayleigh integral over the entire region of the piston surface as expressed in Equation (10). In this equation, A is a constant which depends on certain properties of the medium, namely the wave and the acoustic element. ω is the angular frequency, k is the wave constant and t is the time of wave propagation. dS is the cross-sectional surface area of the element calculated as the product of the total length and thickness dx, i.e. 2(AB)dx. Determining AB from Figure 4, dS can be expressed as shown in Equation (11). pc = i ∫ ac −ac A r ei(ωt−kr)dS, (10) dS = 2 × AB × dx = 2 √ ac 2 − x2dx. (11) Substituting Equations (9) and (11) in Equation (10) while applying the condition that point Q is at the far-field distance, i.e. 1 r ≈ 1 R , the constant amplitude is taken out and pc takes the form of Equation (12). Considering only the real part of the exponents under the integral to preserve the pressure as a complex component of the whole expression, Equation (13) is obtained. If x is set to ac cos θ the integral is transformed into Equation (14), which is simplified where J1 is a Bessel’s function of the first kind as shown in [25]. The directional factor Hc is the absolute value of the angular component isolated as expressed in Equation (15). Using the same approach, directional factors of other surfaces would be derived accordingly in their respective sections. The major difference in the analytical procedure is the determination of the HSL, AB for each surface. pc = i A R 2ei(ωt−kR) ∫ ac −ac eikx sin θ √ ac 2 − x2dx, (12) pc = i A R 2ei(ωt−kR) ∫ ac −ac cos (kx sin θ) √ ac 2 − x2dx, (13) pc = i A R 2ac 2ei(ωt−kR) ∫ π 0 cos (kac cos θ sin θ) sin2 θdθ = i A R 2πac 2ei(ωt−kR) J1 (kac sin θ) kac sin θ , (14) Hc (θ) = ∣∣∣∣2J1 (kac sin θ) kac sin θ ∣∣∣∣. (15) 473 T. A. Yusuf, S. A. Aodu, A. O. Alabi Acta Polytechnica Q c A Y X θ ac p B R r Z Figure 4. The surface geometry for the derivation of the directional factor of a circular piston. 3.2. Triangular piston For the triangle piston, the centre C is located from the base and top of the surface as shown in Figure 5. The point C is not located at the exact mid-point, hence the surface is non-symmetry. The total pressure integral for the triangular piston takes the form of Equation (16). The AB of the acoustic element on the surface is determined using the principle of similar triangles as given in Equation (17). Substituting r as previously defined in Equation (9) and dS = 2(AB)dx while retaining only the real component of the exponential as previously done in the case of the circular piston in Section 3.1, Equation (16) becomes Equation (18). Then, the complex component of the pressure, pt is simplified as can be seen from Equations (19)–(21). On further manipulation in Equation (22), pt takes the final form expressed in Equation (23) from which the directional factor is isolated as Equation (24). Pt = i ∫ 2√ 3 at − 1√ 3 at A r ei(ωt−kr)dS, (16) CE CE + x = CD AB = CE tan 30 AB ∴ AB = 2 3at + 1√ 3 x, (17) pt = i A r ei(ωt−kR) ∫ 2√ 3 at − 1√ 3 at · cos (kx sin θ) · 2 ( 2 3at + 1√ 3 x ) dx, (18) = i A R 2ei(ωt−kR) { 2 3at (∣∣∣∣ sin (kx sin θ) k sin θ ∣∣∣∣ 2√ 3 at − 1√ 3 at ) + 1√ 3 (∣∣∣∣x sin (kx sin θ) k sin θ ∣∣∣∣ 2√ 3 at − 1√ 3 at ) + 1√ 3 (∣∣∣∣cos (kx sin θ) k2 sin2 θ ∣∣∣∣ 2√ 3 at − 1√ 3 at )} , (19) = i A R 2ei(ωt−kR) 2 3at  sin ( k 2√ 3 at sin θ ) + sin ( k 1√ 3 at sin θ ) k sin θ  +  2 3 at sin ( k 2√ 3 at sin θ ) − 1 3 at sin ( k 1√ 3 at sin θ ) k sin θ + 1√ 3 cos ( k 2√ 3 at sin θ ) − cos ( k 1√ 3 at sin θ ) k2 sin2 θ  , (20) pt = i A R 2ei(ωt−kR) 4 3at  sin ( k 2√ 3 at sin θ ) k sin θ + 1 3at  sin ( k 1√ 3 at sin θ ) k sin θ  − 2√ 3  sin ( k √ 3 2 at sin θ ) sin ( k √ 3 6 at sin θ ) k2 sin2 θ  , (21) 474 vol. 64 no. 5/2024 Acoustic radiation beam patterns of different polygonal . . . Y X c A D B E at 60o (a). Surface analysis. c A θ Q p R r Z (b). Radiation analysis. Figure 5. The geometry for the derivation of the directional factor of a triangular acoustic piston source. = i A R 2ei(ωt−kR)  8 3 √ 3 at 2  sin ( k 2√ 3 at sin θ ) k 2√ 3 at sin θ + 1 3 √ 3 at 2  sin ( k 1√ 3 at sin θ ) k 1√ 3 at sin θ  − 1 2 √ 3 at 2  sin ( k √ 3 2 at sin θ ) k √ 3 2 at sin θ · sin ( k √ 3 6 at sin θ ) k √ 3 6 at sin θ  , (22) pt = i A R 16 3 √ 3 at 2ei(ωt−kR)  sin ( k 2√ 3 at sin θ ) k 2√ 3 at sin θ + 1 8  sin ( k 1√ 3 at sin θ ) k 1√ 3 at sin θ  − 3 16  sin ( k √ 3 2 at sin θ ) k √ 3 2 at sin θ · sin ( k √ 3 6 at sin θ ) k √ 3 6 at sin θ  , (23) Ht = ∣∣∣∣∣∣ sin ( k 2√ 3 at sin θ ) k 2√ 3 at sin θ + 1 8  sin ( k 1√ 3 at sin θ ) k 1√ 3 at sin θ − 3 16  sin ( k √ 3 2 at sin θ ) k √ 3 2 at sin θ · sin ( k √ 3 6 at sin θ ) k √ 3 6 at sin θ  ∣∣∣∣∣∣. (24) 3.3. Rectangular and square pistons The rectangle and the square have identical geometries except that the latter is regular by having equal sides. Consequently, the directional factor when successfully derived for one can be used to predict the other without the repetition of the procedures. The acoustic element lying parallel to the Y -axis on the surface of the rectangular surface is shown in Figure 6. The surface is symmetrical since the centre C is located at the exact midpoint. Since AB = ac and dS = 2 ac dx, the complex pressure component pr can then be expressed as Equation (25). This equation is simplified from Equations (26)–(29) and the directional factor of the rectangular surface Hr is then isolated as given in Equation (30). Therefore, the directional factor of the square piston Hs takes the form of Equation (31). pr = i A r ei(ωt−kr) ∫ ar −ar cos (kx sin θ)2acdx, (25) = i A r ei(ωt−kr)2ac ∣∣∣∣ sin (kx sin θ) k sin θ ∣∣∣∣ ar −ar , (26) = i A r ei(ωt−kr)2ac ( sin (kar sin θ) + sin (kar sin θ) k sin θ ) , (27) = i A r ei(ωt−kr)4ac sin (kar sin θ) k sin θ , (28) 475 T. A. Yusuf, S. A. Aodu, A. O. Alabi Acta Polytechnica Y X c A B ac 45o (a). Surface analysis. c A θ Q p R r Z θ X (b). Radiation analysis. Figure 6. The geometry for the derivation of the directional factor of a rectangular acoustic piston source. pr = i A r ei(ωt−kr)4acar sin (kar sin θ) kar sin θ , (29) Hr = ∣∣∣∣ sin (kar sin θ) kar sin θ ∣∣∣∣ = ∣∣∣∣ sin (kar tan 45 sin θ) kar tan 45 sin θ ∣∣∣∣, (30) Hs = ∣∣∣∣ sin (kas sin θ) kas sin θ ∣∣∣∣ = ∣∣∣∣ sin (kas tan 45 sin θ) kas tan 45 sin θ ∣∣∣∣. (31) 3.4. Pentagonal piston Figure 7 shows the location of the centre C with respect to the top and bottom of the radiation surface of a regular pentagon. Just like the triangle, the surface is non-symmetrical since the point C is not located at the exact midpoint of the surface. The height of the extruded triangle as given in the figure allows us to find the HSL, AB of the acoustic element on that surface by the principle of similar triangles as given in Equation (32). Consequently, the integral for calculating the complex component of the pressure pp is given in Equation (33) and simplified through Equations (34)–(36). When further manipulated mathematically, Equation (36) can be expressed as given in Equation (37). From this equation, the directional factor for the pentagon surface Hp can be isolated as given in Equation (38), Applying trigonometry identity, this equation is further worked down through Equations (39)–(40). ap tan 72 ap tan 72 + (ap tan 54 − x) = ap AB ∴ AB = (1 + tan 54 tan 18) ap − x tan 18, (32) Pp = i A r ei(ωt−kR) ∫ 1 cos 54 a p −ap tan 54 cos (kx sin θ) · {(1 + tan 54 tan 18) ap − x tan 18} dx (33) = i A R 2ei(ωt−kR) { (1 + tan 54 tan 18) ap (∣∣∣∣ sin (kx sin θ) k sin θ ∣∣∣∣ ap sec 54 −ap tan 54 ) − tan 18 (∣∣∣∣x sin (kx sin θ) k sin θ ∣∣∣∣ ap sec 54 −ap tan 54 ) − tan 18 (∣∣∣∣cos (kx sin θ) k2 sin2 θ ∣∣∣∣ ap sec 54 −ap tan 54 )} (34) = i A R 2ei(ωt−kR) [ (1 + tan 54 tan 18) ap { sin (kap sec 54 sin θ) + sin (kap tan 54 sin θ) k sin θ } − { (tan 18 sec 54) ap sin (kap sec 54 sin θ) − (tan 18 tan 54) ap sin (kap tan 54 sin θ) k sin θ } − tan 18 { cos (kap sec 54 sin θ) − cos (kap tan 54 sin θ) k2 sin2 θ }] (35) 476 vol. 64 no. 5/2024 Acoustic radiation beam patterns of different polygonal . . . = i A R 2ei(ωt−kR) [ (1 + tan 54 tan 18 − tan 18 sec 54) ap { sin (kap sec 54 sin θ) k sin θ } + (1 + 2 tan 54 tan 18) ap { sin (kap tan 54 sin θ) k sin θ } + 2 tan 18 k2 sin2 θ [ sin { k ( sec 54 + tan 54 2 ) ap sin θ } sin { k ( sec 54 − tan 54 2 ) a p sin θ }]] , (36) Pp = i A R 2ap 2ei(ωt−kR) sec 54 {1 + tan 18 (tan 54 − sec 54)} { sin (kap sec 54 sin θ) kap sec 54 sin θ } + tan 54 (1 + 2 tan 54 tan 18) { sin (kap tan 54 sin θ) kap tan 54 sin θ } +tan 18 2  sin { k ( sec 54+tan 54 2 ) a p sin θ } k ( sec 54+tan 54 2 ) a p sin θ · sin { k ( sec 54−tan 54 2 ) a p sin θ } k ( sec 54−tan 54 2 ) a p sin θ , (37) Hp = ∣∣∣∣∣∣ sec 54 {1 + tan 18 (tan 54 − sec 54)} { sin (kap sec 54 sin θ) kap sec 54 sin θ } + tan 54 (1 + 2 tan 54 tan 18) { sin (kap tan 54 sin θ) kap tan 54 sin θ } +tan 18 2  sin { k ( sec 54+tan 54 2 ) a p sin θ } k ( sec 54+tan 54 2 ) a p sin θ · sin { k ( sec 54−tan 54 2 ) a p sin θ } k ( sec 54−tan 54 2 ) a p sin θ ∣∣∣∣∣∣ (38) = ∣∣∣∣∣∣sec 54 { 1 + ( sec 54 − tan 54 tan 108 )}{ sin (kap sec 54 sin θ) kap sec 54 sin θ } + tan 54 ( 1 − 2 tan 54 tan 108 ){ sin (kap tan 54 sin θ) kap tan 54 sin θ } − 1 2 tan 108  sin { k ( sec 54+tan 54 2 ) a p sin θ } k ( sec 54+tan 54 2 ) a p sin θ · sin { k ( sec 54−tan 54 2 ) a p sin θ } k ( sec 54−tan 54 2 ) a p sin θ ∣∣∣∣∣∣, (39) Hp = ∣∣∣∣∣∣sec 54 { 1 + ( sec 54 − tan 54 tan 108 )}{ sin (kap sec 54 sin θ) kap sec 54 sin θ } + tan3 54 { sin (kap tan 54 sin θ) kap tan 54 sin θ } − 1 2 tan 108  sin { k ( sec 54+tan 54 2 ) a p sin θ } k ( sec 54+tan 54 2 ) a p sin θ · sin { k ( sec 54−tan 54 2 ) a p sin θ } k ( sec 54−tan 54 2 ) a p sin θ ∣∣∣∣∣∣. (40) 3.5. Hexagonal piston The position of the centre C of a regular hexagon surface geometry, which is clearly symmetrical, is shown in Figure 8. The HSL, AB of the acoustic element on the surface can be found using the similar triangles approach as shown in Equation (41). The complex component of the pressure integral px is then given in Equation (42) and evaluated in Equation (43). On further simplification, the middle and the last terms in this equation vanishes and the final expression for the acoustic pressure then becomes Equation (44). Then, the directional factor is isolated as Equation (45). √ 3ax ax = AX AB = √ 3ax + √ 3ax − x AB ∴ AB = 2ax − 1√ 3 x, (41) 477 T. A. Yusuf, S. A. Aodu, A. O. Alabi Acta Polytechnica Y X c A B ap 54o 72o (a). Surface analysis. X c A θ Q p R r Z θ (b). Radiation analysis. Figure 7. The geometry for the derivation of the directional factor of a pentagonal acoustic piston source. Y X c A B ax60o 60o (a). Surface analysis. X c A θ Q p R r Z θ (b). Radiation analysis. Figure 8. The geometry for the derivation of the directional factor of hexagonal acoustic piston source. Px = i A R ei(ωt−kR) ∫ √ 3ax − √ 3ax cos (kx sin θ) · ( 2ax − 1√ 3 x ) dx (42) = i A R 2ei(ωt−kR) { 2ax (∣∣∣∣ sin (kx sin θ) k sin θ ∣∣∣∣ √ 3ax − √ 3ax ) − 1√ 3 (∣∣∣∣x sin (kx sin θ) k sin θ ∣∣∣∣ √ 3ax − √ 3ax ) − 1√ 3 (∣∣∣∣cos (kx sin θ) k2 sin2 θ ∣∣∣∣ √ 3ax − √ 3ax )} , (43) Px = i A R ei(ωt−kR)8ax ( sin ( k √ 3ax sin θ ) k sin θ ) = i A R ei(ωt−kR)8 √ 3ax 2 sin ( k √ 3ax sin θ ) k √ 3ax sin θ , (44) Hx = ∣∣∣∣∣ sin ( k √ 3ax sin θ ) k √ 3ax sin θ ∣∣∣∣∣ = ∣∣∣∣ sin (kax tan 60 sin θ) kax tan 60 sin θ ∣∣∣∣. (45) 3.6. Heptagonal piston The centre C of the regular heptagon with respect to the top and bottom of the non-symmetrical surface is as shown in Figure 9. By the similar triangle principle, the HSL, AB can be determined as given in Equation (46) 478 vol. 64 no. 5/2024 Acoustic radiation beam patterns of different polygonal . . . Y X c A B ah 64.3o 51.43o (a). Surface analysis. X c A θ Q p R r Z θ (b). Radiation analysis. Figure 9. The geometry for the derivation of the directional factor of heptagonal acoustic piston source. and the complex part of the pressure integral can, therefore, be expressed as in Equation (47). As can be clearly observed from these equations, the heptagon indicates almost an identical geometry with the pentagon analysed in Section 3.4. Thus, without any repetition of the procedure, it is easy to determine that Equation (47) takes the final form of Equation (48). Consequently, the directional factor follows a similar pattern as previously done in Section 3.4, and therefore expressed as Equation (49). ah tan 51.43 ah = ah tan 51.43 + (ah tan 64.3 − x) AB ∴ AB = (1 + tan 64.3 tan 38.57) ah − x tan 38.57, (46) Ph = i A r ei(ωt−kR) ∫ 1 cos 64.3 ah −ah tan 64.3 cos (kx sin θ) · 2 {(1 + tan 64.3 tan 38.57) ah − x tan 38.57} dx, (47) Ph = i A R 2ah 2ei(ωt−kR) [ sec 64.3 {1 + tan 38.57 (tan 64.3 − sec 64.3)} { sin (kah sec 64.3 sin θ) kah sec 64.3 sin θ } + tan 64.3 (1 + 2 tan 64.3 tan 38.57) { sin (kah tan 64.3 sin θ) kah tan 64.3 sin θ } +tan 38.57 2 [ sin { k ( sec 64.3+tan 64.3 2 ) ah sin θ } k ( sec 64.3+tan 64.3 2 ) ah sin θ · sin { k ( sec 64.3−tan 64.3 2 ) ah sin θ } k ( sec 64.3−tan 64.3 2 ) ah sin θ ]] , (48) Hh = ∣∣∣∣∣sec 64.3 { 1 + ( sec 64.3 − tan 64.3 tan 128.6 )}{ sin (kah sec 64.3 sin θ) kah sec 64.3 sin θ } + tan3 64.3 { sin (kah tan 64.3 sin θ) kah tan 64.3 sin θ } − 1 2 tan 128.6 [ sin { k ( sec 64.3+tan 64.3 2 ) ah sin θ } k ( sec 64.3+tan 64.3 2 ) ah sin θ · sin { k ( sec 64.3−tan 64.3 2 ) ah sin θ } k ( sec 64.3−tan 64.3 2 ) ah sin θ ]∣∣∣∣∣. (49) 3.7. Octagonal piston Figure 10 shows the centre C of the regular octagon piston surface with a nearly identical symmetrical geometry to the hexagon. Following the same analytical procedures as previously undertaken in Section 3.5, the HSL, AB, the complex pressure component, po, and the directional factor are derived as expressed in Equations (50)–(52), respectively. a0 a0 = a0 + a0 ( 1 + √ 2 ) − x AB ∴ AB = ( 2 + √ 2 ) ao − x, (50) 479 T. A. Yusuf, S. A. Aodu, A. O. Alabi Acta Polytechnica Y X c A B ao 67.5o 45o (a). Surface analysis. X c A θ Q p R r Z θ (b). Radiation analysis. Figure 10. The geometry for the derivation of the directional factor of octagonal piston source. po = i A r ei(ωt−kR) ∫ (1+ √ 2)a0 −(1+ √ 2)a0 cos (kx sin θ) · 2 {( 2 + √ 2 ) ao − x } dx = i A R ei(ωt−kR)4ao 2 ( 4 + 3 √ 2 ) sin ( k ( 1 + √ 2 ) a0 sin θ ) k ( 1 + √ 2 ) a0 sin θ , (51) H0 = ∣∣∣∣∣ sin ( k ( 1 + √ 2 ) a0 sin θ ) k ( 1 + √ 2 ) a0 sin θ ∣∣∣∣∣ = ∣∣∣∣ sin (ka0 tan 67.5 sin θ) ka0 tan 67.5 sin θ ∣∣∣∣. (52) 3.8. Nonagonal piston Like the heptagon, Figure 11 clearly shows that a non-symmetrical surface geometry of a regular nonagon, which is closely similar to that of the pentagon studied in Section 3.4. Then, AB and the complex pressure component, pn are given as Equations (53) and (54), respectively. Hence, it is easy to identify that the directional factor is in the form given in Equation (55). AB = (1 + tan 70 tan 50) an − x tan 50, (53) pn = i A R 2an 2ei(ωt−kR) sec 70 {1 + tan 50 (tan 70 − sec 70)} { sin (kan sec 70 sin θ) kan sec 70 sin θ } + tan 70 (1 + 2 tan 70 tan 50) { sin (kan tan 70 sin θ) kan tan 70 sin θ } +tan 50 2  sin { k (sec 70+tan 70) 2 a n sin θ } k (sec 70+tan 70) 2 a n sin θ · sin { k (sec 70−tan 70) 2 a n sin θ } k (sec 70−tan 70) 2 a n sin θ , (54) Hn = sec 70 { 1 + ( sec 70 − tan 70 tan 140 )}{ sin (kan sec 70 sin θ) kan sec 70 sin θ } + tan3 70 { sin (kan tan 70 sin θ) kan tan 70 sin θ } − 1 2 tan 140 [ sin { k ( sec 70+tan 70 2 ) a n sin θ } k ( sec 70+tan 70 2 ) a n sin θ · sin { k ( sec 70−tan 70 2 ) a n sin θ } k ( sec 70−tan 70 2 ) a n sin θ ] . (55) 3.9. Decagonal piston Like the octagon in Section 3.7, Figure 12 indicates a clear symmetrical similarity with the hexagon studied in Section 3.5. Consequently, following similar analytical procedures while avoiding repetitions, the directional factor of the octagonal piston can be expressed as: Hd = ∣∣∣∣ sin (kad tan 72 sin θ) kad tan 72 sin θ ∣∣∣∣. (56) 480 vol. 64 no. 5/2024 Acoustic radiation beam patterns of different polygonal . . . Y X c A B an 70o 40o (a). Surface analysis. X c A θ Q p R r Z θ (b). Radiation analysis. Figure 11. The geometry for the derivation of the directional factor of nonagonal acoustic piston source. Y X c A B ad 72o 36o (a). Surface analysis. X c A θ Q p R r Z θ (b). Radiation analysis. Figure 12. The surface geometry for the derivation of the directional factor of decagonal acoustic piston source. 4. The analysis and characterisation of the beam patterns For each piston surface, the beam pattern is determined using Equation (57) where H is the directional factor and Hmax is its maximum value. The comparative plots of the beam patterns for all the pistons calculated at different diameters of the standard circular surface using the respective directional factors derived in Section 3 is shown in Figure 13. The circular diameters range from 0.28–3.0λ since the directivity of a circular transducer is high enough when the diameter is greater than 2λ [26]. The quantitative comparison of the results indicating the half-power (−3 dB) beam width (BW) of the main lobe as well as the side lobe level (SLL) of the beam patterns is presented in Table 1. beam pattern = 20 log ∣∣∣∣ H Hmax ∣∣∣∣. (57) These results show that all the pistons of the same surface area exhibit a similar performance with a circular piston having a diameter not greater than 0.28λ or being as large as 0.43λ when the triangle is excluded. Meanwhile, the rectangular and circular pistons of equal surface area still maintain similar radiation patterns as long as the circular diameter is still within a half wavelength. Above this, the odd-sided non-symmetrical polygon surfaces appear to have a relatively lower beam width compared to the symmetrical ones. This finding is contrary to what is reported by [19] who adopted the square as a rectangular piston and reported that octagonal, hexagonal, rectangular, and circular pistons exhibit completely omnidirectional beam patterns when the surface area is equivalent to the circular diameter of 0.5λ. 481 T. A. Yusuf, S. A. Aodu, A. O. Alabi Acta Polytechnica Circular Triangular Rectangular Square Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal -90 -60 -30 0 30 60 90-40 -30 -20 -10 0 B ea m P at te rn ( dB ) (a). 0.28λ. -90 -60 -30 0 30 60 90-40 -30 -20 -10 0 B ea m P at te rn ( dB ) (b). 0.43λ. -90 -60 -30 0 30 60 90-40 -30 -20 -10 0 B ea m P at te rn ( dB ) (c). 0.5λ. -90 -60 -30 0 30 60 90-40 -30 -20 -10 0 B ea m P at te rn ( dB ) (d). 1.0λ. -90 -60 -30 0 30 60 90-40 -30 -20 -10 0 B ea m P at te rn ( dB ) (e). 2.0λ. -90 -60 -30 0 30 60 90-40 -30 -20 -10 0 B ea m P at te rn ( dB ) (f). 3.0λ. Figure 13. The beam pattern of polygonal piston surfaces of equal surface area with a circular piston of different diameters. Piston sources The diameter of the circular piston 0.28λ 0.43λ 0.5λ 1.0λ 2.0λ 3.0λ BW [°] BW [°] BW [°] BW [°] SLL [dB] BW [°] SLL [dB] BW [°] SLL [dB] Circular 180 180 180 61.9 - 29.8 −17.6 19.8 −17.5 Triangular 180 84.3 70.5 33.6 −15.1 16.7 −15.1 11.1 −15.1 Rectangular 180 180 180 68.6 - 32.7 −13.3 21.7 −13.2 Square 180 180 173.1 59.9 - 28.9 −13.3 19.2 −13.2 Pentagonal 180 180 121.2 51.7 - 25.2 −14.2 16.8 −14.2 Hexagonal 180 180 136.5 55.4 - 26.9 −13.3 17.9 −13.2 Heptagonal 180 180 127.6 53.3 - 26.0 −13.4 17.3 −13.4 Octagonal 180 180 130.6 54.1 - 26.3 −13.3 17.5 −13.2 Nonagonal 180 180 127.5 53.3 - 26.0 −13.3 17.3 −13.3 Decagonal 180 180 128.2 53.5 - 26.1 −13.3 17.3 −13.2 Table 1. The −3 dB beam pattern of the polygonal pistons of the same surface area with a circular piston of different diameters. Meanwhile, the circular piston is better than all the polygons in terms of its side lobe level. However, it is worse in terms of its beam width compared to the rectangular piston and not much significantly different from the square. Coincidentally, the closeness observed in the radiation beam patterns between the square and circular pistons had equally been reported for their radiation impedances [27]. The side lobe levels remain the same at less than −13 dB for all other polygons but lower than −14 dB for the triangular and pentagonal pistons. Notably, the radiation pattern of the triangular piston is completely unique from other polygon surfaces. It has 482 vol. 64 no. 5/2024 Acoustic radiation beam patterns of different polygonal . . . a narrower beam width, one side lobe higher than the others, and the lowest side lobe level next to the circular piston. Perhaps, this explains why it does not appear to have any practical application, nor is it well known in the literature [28]. For the number of sides higher than five, the radiation beam characteristics are almost identical for all the polygons. Nevertheless, the hexagonal piston seems to be the best while heptagonal, nonagonal, and decagonal ones are generally the same. One would expect that the performance characteristics of the acoustic polygonal pistons become closer to the circular piston as the number of sides increases. However, this result has shown that neither the surface area nor the number of sides of the polygon influences the radiation pattern. Rather, it is the combination of the perimeter and the symmetry of the surface. At equal surface area, all the polygons are higher in perimeter than the circle while their perimeter decreases as the number of sides increases in order of their list in Table 1. The triangle, which has the highest perimeter, exhibits the narrowest beamwidth due to its non-symmetry while the rectangle and square ones have the first and second overall dominance. Consequently, the surface symmetry factor takes precedence over the perimeter such as in the case of the hexagon and octagon relative to the pentagon and in the case of the last four polygons relative to one another. 5. Validation by finite element method To prove the correctness of the directional factors of each acoustic polygonal piston source proposed in this study, the finite element method (FEM) was employed for the validation of the radiation patterns using the Pzflex® commercial software. The first eight pistons were selected while the remaining two were left out since their directional factors are essentially the same with the polygons having either the odd or even number of sides and their performance is not dissimilar with the heptagonal piston. Each of the piston surfaces was modelled as the head masses of a Tonpilz transducer made of alu- minium using the FEM as shown for the standard circular source in Figure 14a. The tail mass at the rear end is a brass material serving as a rigid baffle while the piezo-ceramic drive section is sandwiched in- between. The selected piezoelectric ceramic material was KP14 (Kyoungwon Ferrite Co. Ltd). The head mass was designed at the equivalent circular diameter of 1.0λ for all the radiating surfaces corresponding to the equivalent surface area derived for each piston source in the theoretical analysis. Since the trans- ducer vibrates in the longitudinal mode, the thickness of the drive section and all other dimensions were determined by the resonance frequency of operation. For the finite element analysis (FEA), water was mod- elled at the front of each surface as shown for the standard circular head mass in Figure 14b. The final mesh of the structure was simulated at an element size of 1 32 factor of the wavelength. The absorption boundary condition was applied all around the water to avoid a loss of acoustic signals being transmitted through the application of a pressure pulse in the form of a sine wave function and 1 volt of electrical energy to drive the ceramics. Figure 15 shows the models of the selected polygon surfaces analysed fol- lowing similar procedures. The result of the comparison between the theoretical beam patterns and that obtained from the FEA for each of the surfaces are shown in Figure 16. Except with a little difference of about 2 dB in the triangular case, the excellent agreements observed in Figure 16 for all the selected cases is a strong attestation to the validity of the directional factors invented for the radiating polygonal pistons. 6. Summary The directional factors of acoustic polygonal piston sources with surfaces ranging from three to ten sides have been derived and used to characterise their radiation beam patterns in comparison with the standard circular piston. With the exception of the triangular surface, which has a unique characteristics, the study indicates that (a). Without water. (b). With water. Figure 14. Finite element model of Tonpilz with a circular head mass diameter of 1.0λ. 483 T. A. Yusuf, S. A. Aodu, A. O. Alabi Acta Polytechnica (a). Triangular. (b). Rectangular. (c). Square. (d). Pentagonal. (e). Hexagonal. (f). Heptagonal. (g). Octagonal. Figure 15. The finite element model of Tonpilz with head mass of different radiating polygon surfaces of equal surface areas with the circular head mass diameter of 1.0λ. (a). Circular. (b). Triangle. (c). Rectangular. (d). Square. (e). Pentagonal. (f). Hexagonal. (g). Heptagonal. (h). Octagonal. Figure 16. The comparison between the FEA and theoretical radiation beam patterns of different polygon surfaces of acoustic piston of equal surface areas with a circular piston diameter of 1.0λ. 484 vol. 64 no. 5/2024 Acoustic radiation beam patterns of different polygonal . . . even-sided (symmetrical) and odd-sided (non-symmetrical) polygons, respectively, has the general directional factor of the form: ∣∣∣∣∣∣ sin { ka tan ( ∅ 2 ) sin θ } ka tan ( ∅ 2 ) sin θ ∣∣∣∣∣∣, and: sec ( ∅ 2 )1 +  sec ( ∅ 2 ) − tan ( ∅ 2 ) tan ∅   sin { ka sec ( ∅ 2 ) sin θ } ka sec ( ∅ 2 ) sin θ  + tan3 ( ∅ 2 ) sin { ka tan ( ∅ 2 ) sin θ } ka tan ( ∅ 2 ) sin θ  − 1 2 tan ∅  sin [ ka { sec ( ∅ 2 )+tan ( ∅ 2 ) 2 } sin θ ] ka { sec ( ∅ 2 )+tan ( ∅ 2 ) 2 } sin θ · sin [ ka { sec ( ∅ 2 )−tan ( ∅ 2 ) 2 } sin θ ] ka { sec ( ∅ 2 )−tan ( ∅ 2 ) 2 } sin θ , where a is the half of the side length, ϕ is the interior angle. All the polygons show a similar performance to a circular piston of a diameter not more than 0.28λ while the same is true for all but the triangle when the circular diameter is up to 0.43λ. In terms of the side lobe levels, the radiating beam pattern of the circular piston is better than all the polygons while all except the triangle and pentagon are identical. However, the beam width of the circular piston is lower compared to the rectangular piston and almost identical with the square piston when the circular diameter is above one wavelength. The main beam widths for the non-symmetrical polygons are relatively narrower than for the symmetrical types. 7. Conclusion This study reveals that the perimeter and symmetry of the surface are the two factors that combine to influence the radiating beam pattern of an acoustic piston rather than the surface area or the number of sides. 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Master’s thesis, Texas Tech University, 1999. 486 https://doi.org/10.3390/s20072148 https://doi.org/10.1088/1742-6596/1237/4/042063 https://doi.org/10.1121/1.400543 https://doi.org/10.1590/S1678-58782011000400004 https://doi.org/10.1109/EMBC.2012.6347472 https://doi.org/10.18421/tem33-06 https://doi.org/10.1121/1.2108997 https://doi.org/10.1109/ICEE50728.2020.9777041 https://doi.org/10.1155/2021/6670277 https://doi.org/10.1007/s13198-020-01046-y https://doi.org/10.35940/ijitee.j9194.0881019 https://doi.org/10.1016/j.sna.2021.112843 https://doi.org/10.1109/LSP.2018.2876066 https://doi.org/10.3390/app11062702 https://doi.org/10.1143/JJAP.37.3166 https://doi.org/10.21105/joss.03740 https://doi.org/10.1121/1.4964632 Acta Polytechnica 64(5):470–486, 2024 1 Introduction 2 Equivalent surface area of the piston sources 3 Derivation of the directional factors of the acoustic piston surfaces 3.1 Circular piston 3.2 Triangular piston 3.3 Rectangular and square pistons 3.4 Pentagonal piston 3.5 Hexagonal piston 3.6 Heptagonal piston 3.7 Octagonal piston 3.8 Nonagonal piston 3.9 Decagonal piston 4 The analysis and characterisation of the beam patterns 5 Validation by finite element method 6 Summary 7 Conclusion References