Robust acoustic pulling using chiral surface waves
RRobust acoustic pulling using chiral surface waves
Neng Wang , Ruo-Yang Zhang , and C. T. Chan Institute of Microscale Optoelectronics, Shenzhen University, Shenzhen 518060, China Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay,Hong Kong, China*Corresponding to: [email protected]
Abstract :We show that long-range and robust acoustic pulling can be achieved by using a pair of one-waychiral surface waves supported on the interface between two phononic crystals composed ofspinning cylinders with equal but opposite spinning velocities embedded in water. When the chiralsurface mode with a relative small Bloch wave vector is excited, the particle located in theinterface waveguide will scatter the excited surface mode to another chiral surface mode with agreater Bloch wave vector, resulting in an acoustic pulling force, irrespective of the size andmaterial of the particle. Thanks to the backscattering immunity of the chiral surface waves againstlocal disorders, the particle can be pulled following a flexible trajectory as determined by theshape of the interface. As such, this new acoustic pulling scheme overcomes some of thelimitations of the traditional acoustic pulling using structured beams, such as short pullingdistances, straight-line type pulling and strong dependence on the scattering properties of theparticle. Our work may also inspire the application of topological acoustics to acousticmanipulations. . Introduction
The ability of acoustic waves to exert radiation forces and torques on matter enables contactlessand noninvasive acoustic manipulations, which have found fruitful applications in various areasranging from physics [1-4], to chemistry[5, 6] and biology [7-10]. Among these manipulations, theacoustic pulling [11-18], which refers to pulling particles using the acoustic waves towards thesource, is perhaps most amazing. In addition to being a counter-intuitive phenomenon, theacoustic pulling also provides a new mechanism for the acoustic manipulations apart from thelevitation [19-22], trapping [23-26] and binding [27, 28].The acoustic pulling force stems from the backward momentum gained by the particle from theacoustic wave during the scattering process. Due to the momentum conservation law, this requiresthat the scattered wave has a greater forward momentum than the incident one. To achieve pullingin free space, structured beams such as a Bessel beam with a very large cone angle [11-15] ormultiple beams with giant obliquely incident angles [16, 17], are required to minimize the forwardmomentum of the incident wave. The technical difficulty on realizing long-range Bessel beamsand the finite beam waists of the obliquely incident beams limits the application scenarios ofacoustic pulling in reality. The size, shape and material of the particle also need to be carefullycustomized to enhance the forward scattering and reduce the backward scattering simultaneouslyso that the forward momentum of the scattered wave can be large enough. Therefore, the freespace acoustic pulling is not only limited to short-range in general but also dependent strongly onthe scattering properties of the manipulated object.In this paper, we propose a way to achieve long-range and robust acoustic pulling for arbitraryparticles located in an interface waveguide sandwiched between two phononic crystals composedof oppositely spinning cylinders embedded in water. On the interface of the two phononic crystals,there are two topologically protected chiral surface states propagating in the same direction, theone-way propagation nature of the two surface states is independent of the shape of the interfaceas well as the local disorder. The channel sandwiched by the two phononic crystals can have sharporners which do not induce the state switching of the two states, inside the channel the particlecan be pulled towards the source irrespective of its size and material, following a trajectoryconfined by the channel. Although a similar idea of optical pulling using the photonic chiralsurface waves [29] has been proposed recently, the acoustic pulling using the acoustic chiralsurface waves has its unique advantage that it can be used to pull much larger particles with muchstronger forces. Here, by rigorously calculating the acoustic forces using the closed-surfaceintegral of the time-averaged radiation stress tensor [11-13], and quantitatively analyzing themwith the response theory [30], we show that once the surface mode with the relatively small Blochwave vector is excited, the longitudinal (along wave propagating direction) acoustic force actingon the particle located inside the channel is mostly negative. Moreover, the averaged longitudinalforce for the particle moving over a lattice constant along the wave propagating direction isalways negative. The negative nature of the averaged longitudinal force is guaranteed by the Blochwave vector difference between the two surface modes. Therefore, for a particle made of anymaterial, possessing any size, it can always be pulled backward along the interface waveguide bythe surface waves due to the net pulling force.
II. Chiral surface states
The chiral surface states in classical systems have been predicted [31-33] and realized [34,35] in2-dimensional magneto-optical photonic crystals by breaking time-reversal symmetry in analogyto the quantum Hall effect in electronic systems. The acoustic counterparts were proposed [36-38]in phononic crystals using circulating fluid coatings, where the circulating flows break thetime-reversal symmetry. This acoustic chiral edge transport was observed in experiment veryrecently [39].Following the idea of Ref. [36], we consider the phononic crystal consisting of spinning cylinderswith the spinning angular frequency Ω arranged into a square lattice (lattice constant a ) in xy plane embedded in water with mass density and sound speed c . When the cylinders aretatic, the mass density and sound speed inside are and c , respectively. A thin, soundpermeable and unspinning shell is coated on each cylinder to avoid the direct contact between thecylinder and water, so that the spinning will not drive the water to move. Taking both the Dopplereffect and Coriolis force into account, the n th order Mie coefficient of the spinning cylinder for atime-harmonic ( i t e ) incident acoustic wave is given by [40, 41] ( ) ( ) '( ) ,( ) ( ) '( ) n n n c n n c n cn n n n c n n c n c R J k r k J r J k rD R H k r k J r H k r (1)where n J and (1) n H are the Bessel function and Hankel function of the first kind, c r is the radius ofthe cylinder, / k c is the wavenumber in water, ( 4 ) / n M c with ( )
M i n and c being the sound speed inside a static cylinder, and the auxiliary functionis expressed as (2 ) '( ) 3 ( ) / ( ) .(4 )( ) n n c n n c n cn M J r inM J r rR M M (2)When , Eq. (1) is reduced to the Mie coefficient of a static cylinder. The band dispersionfor the lattice structure containing the spinning cylinders can be obtained by using the multiplescattering method and applying the periodic boundary conditions [42, 43]. When the cylinders arestatic, a two-fold degeneracy at M point is enforced by the C v and time-reversal symmetry, asshown in Fig. 1(a). As the cylinders are spinning, the time-reversal symmetry is broken which liftsthe degeneracy and opens a topologically nontrivial band gap with gap Chern number 1 (-1) foranticlockwise (clockwise) spinning cylinders. For example, for radius c r a and spinningfrequency ca , there is a complete band gap ranging from ca to ca , seeFig. 1(b). Let us consider the interface between the two phononic crystals composed of oppositelyspinning cylinders with the same spinning speed. Since the difference of the gap Chern numbersof the two sides is 2, there will exist two topologically protected chiral interface states propagatingalong the same direction at any frequency within the bulk band gap.he dispersion relations of surface states can be calculated using the multiple scattering method inconjunction with supercell calculations [44]. In calculations, the supercell contains one layer along x direction and 18 layers along y direction. In order to form a closed system with purely realeigen-frequencies, all the boundaries of the supercell are connected according to periodicboundary conditions. Cylinders in the upper 9 layers have spinning frequency , while cylindersin the lower 9 layers have spinning frequency . In addition, the upper 9 and lower 9 layers arefurther separated by a water channel of width a (therefore the dimension of the supercell is a a ), as shown in Fig. 2(a). The band structures of the bulk (black dotted lines) and surface(red and blue dotted lines) states are shown in Fig. 2(a). We can see that there are four surfacestates at each frequency ranging from ca to ca corresponding to the band gap in Fig.1(b). Two are propagating on the top/bottom edge (blue dotted lines) of the supercell and two arepropagating on the middle (red dotted lines) interface of the supercell. At frequency cf a ,the four eigen pressure field patterns corresponding to the four surface modes are shown in Fig. 2(b). It is clearly seen that modes A and D are even while modes B and C are odd under the mirrorreflection about the central line of the supercell (shown by the black dashed lines in Fig 2(b)).From the dispersion relations, we also see that both modes B and D have positive group velocitiesindicating that the corresponding surface waves are propagating along the positive- x direction.To show that the surface states are backscattering immuned against local disorders, we did fullwave simulations of the surface waves propagating on the interface and being scattered by aparticle using multiple scattering technique. In simulations, 50 supercells in total are used, and anunspinning cylindrical particle is arbitrarily inside the water channel between the upper and lowerphononic crystals, see Fig 2(c). To prevent the particle from entering the phononic crystals, weassume there are two hard walls (extreme thin and permeable to acoustic wave) placed at y a , respectively. Elastic collision occurs when the particle hits the hard walls.Therefore, there is a a a water gap between the two phononic crystals. Two line sourceswith opposite initial phases are placed at y a , respectively, to excite the odd mode B, Wecan see that the surface wave can only propagate rightward indicating that the surface wave cannly transport unidirectionally. Before the wave being scattered by the particle, the pressure fielddistribution is odd about the middle line of the water gap since only mode B is excited. After thesurface wave is scattered by the particle, the odd mirror symmetry of the field distribution isbroken because a portion of mode B is scattered into the even mode D. However, the fielddistribution on the left hand side of the particle retains the odd symmetry indicating the surfacewaves are not backward scattered. III. Acoustic force calculation
For a time-harmonic incident acoustic wave, the time-averaged acoustic force acting on a particlesubmerged in water can be evaluated by integrating the stress tensor T over a surface enclosingthe particle [11-13], T , S d F S (3)where the stress tensor is expressed as * *20 020 0 | |T ( ) I ,4 4 2 pc u u uu (4)with p and u being the pressure and velocity field, and I being a 2 by 2 identity matrix. Using theidea of Lorenz-Mie theory [45-49], the integral can be greatly simplified when the closed surfaceis a circle. Then the x and y components of the acoustic force can be obtained as (see details aboutthe derivation in the appendix) Im (2 ),Re (2 ), x n n n n n nny n n n n n nn pF b b q b b qc kpF b b q b b qc k (5)where, p is the amplitude of the line source, n b is the scattering coefficient of the particle whichis obtained through the multiple scattering calculations, and / n n n q b D with n D being theMie coefficient of the particle.or a cylindrical particle with radius s r a , we calculated the acoustic forces acting on theparticle as a function of the location ( , ) s s x y of the particle confined in a rectangular regioncentered on the x axis (see the dashed rectangle in Fig. 2(c)), and the result is shown in Fig. 3(a).The rectangular region has a length a and a width 0.45 a . Because the system preserves thetranslational symmetry, the acoustic force acting on the particle will repeat when the particlemoves a lattice constant a along x direction. In Fig. 3(a), the arrows represent the acoustic forcevectors and their lengths denote the force magnitudes. It can be seen that most of the acousticforces point leftward. For each s y , the spatially averaged longitudinal forces over a latticeconstant
1( ) ( , ) ax s x s s s
F y F x y dxa are calculated and shown in Fig. 3(b) by the black squares.We can see that ( ) x s F y for every s y is negative. Thus although the particle may be subject topushing forces at some locations, as a whole it will be accelerated leftward when it moves freelyinside the gap. Even if the radius and the material component of the particle are changed, ( ) x s F y are still negative for all s y , as shown by the curves of red disks and blue triangles in Fig. 3(b). Inthe following, we will show that so long as the incident surface wave is purely composed of modeB, ( ) x s F y is negative irrespective of the particle’s size and material.
IV. Interpretation using the response theory
The response theory was proposed previously to explain the optical force in multi-port photonicsystems [30]. We apply this response theory to analyze the acoustic force in our system, as opticaland acoustic forces can be described by the same mathematics.According to the response theory adapted to our system, the longitudinal acoustic force acting onthe particle fulfills , ) ,
B Dx s s B Ds s
F x y I Ix x (6)where ( )
B D
I I and ( )
B D are the intensity and phase of mode B (D) component in the scatteredwave, and ( , ) s s x y is the location of the particle. Taking mode B as an example, the pressure fieldcan be expressed as ,0 ( , ) B ik x u x y e , where ( , ) ( , ) u x y u x a y is the periodic part of the Blochfunction and ,0 B k is the Bloch vector in the first Brillouin zone ( ,0 B ka a ). According toperiodicity of the Bloch functions, mode B actually consists of all surface wave componentswhich are plane waves in the x direction with discrete longitudinal wave vectors , ,0 B n B k k n n Za . Here we express the pressure field of n th surface wave component ofmode B as , , ( , ) B n ik xB n g k y e . The amplitude | ( , ) | g k y and phase arg[ ( , )] g k y of the wavecomponent with wave vector k at vertical coordinate y can be evaluated using the Fouriertransform of the pressure field ( , ) p x y ,
1( , ) ( , ) , x ikxx g k y p x y e dxx x (7)where x and x are the starting and ending points for the Fourier transform. The intensity ofthe surface wave component is proportional to | ( , ) | g k y at any given y .In Fig. 4, we plotted | ( , ) | g k y at different vertical coordinates y for both modes B and D. Asexpected, the discrete peaks of | ( , ) | g k y are spaced by the reciprocal lattice vector a , as shownin the insets in Fig. 4. For mode B, | ( ,0) | g k are equal to zero due to the odd parity of the fielddistribution. It is worth noticing that both modes B and D are dominated by their zeroth planewave components, although some contributions from higher order components arise for large y due to the appearance of evanescent waves close to the spinning cylinders. Therefore, we canapproximate modes B and D reasonably well by their zeroth components and use the two zerothwaves to study the acoustic force according to the response theory. As the intensities of the zerothaves of modes B and D are proportional to | ( , ) | ( ), g k y B D , respectively, thelongitudinal force at position ( , ) s x y has the following expression under the zero orderapproximation | ( , ) | | ( , ) |( , ) arg[ ( , )] (1 ) arg[ ( , )],| ( , ) | | ( , ) | s B s Bx s s B s Di B s i B s g k y g k yF x y g k y g k yg k y x g k y x (8)where i g and s g are the Fourier transforms of the incident and scattered fields respectively.The phase and amplitude of the incident field acting on the particle will determine the amplitudeand phase of the scattered field. If mode B is excited, the incident field intensity obeys evensymmetry about the central vertical line of the supercell, see the inset of Fig. 5(a). Thus theintensities of modes B and D of the scattered wave are also of even symmetry about the line s x ,as shown in Fig. 5(a). When the particle moves a lattice constant along x direction, the phase ofthe incident wave will increase by ,0 B k a . The particle is the source of the scattered waves. Whenthe particle moves a lattice constant, the scattered waves decrease a phase of ,0 B k a and ,0 D k a relative to the incident wave, respectively. Therefore, the total phases of modes B and D ofthe scattered wave increase by and ,0 ,0 ( ) B D k k a , respectively. As a result, B is oscillatingwhile D decreases monotonically when ,0 ,0 ( ) 0 B D k k a as the particle moves along x direction, as shown in Fig. 5(b). According to Eq. (8), the oscillating of ~ B s x will contribute nonet force over a lattice constant, while the monotonous decreasing of ~ D s x will lead to anegative force component. Therefore, the net optical force along x direction is always negative.In Fig. 5 (c) and (d), we compared the acoustic forces calculated by the rigorous expressions, Eq.(5), (lines) with the results calculated by the response theory, Eq. (8), (circles) for the particlemoving along x direction at two different vertical coordinates. The results obtained by the twoalculation methods show a very good agreement, showing that the response theory works well.The slight discrepancy between the two approaches is due to the neglect of contributions from thehigher order plane wave components of modes B and D. VI. Conclusion
In summary, using a pair of chiral surface waves supported on the interface between two phononiccrystals with broken time-reversal symmetry, we can achieve a new type of acoustic pulling. Thisacoustic pulling mechanism has advantages that are absent in traditional optical pulling schemes,such as a long pulling distance, flexible pulling trajectory and is independent of the particle’s sizeand material. The band structures, transporting and scattering properties of the chiral surfacewaves have been studied using the multiple scattering technique. The acoustic forces acting on aparticle changing with the particle’s location inside the interface waveguide have been calculatedrigorously using the Lorenz-Mie formula and analyzed according to the response theory inconjunction with the symmetry analysis. When the incident chiral surface mode has a smallerBloch wave vector, the excitation of the other chiral surface mode due to the scattering by theparticle will result in an averaged pulling force acting on the particle. The particle can be pulledtowards the source irrespective of its size and material. Owing to the backscattering immunity ofthe chiral surface waves against local disorders, the channel between the two phononic crystalsdoes not need to be straight line. If the defect on the interface does not break the C symmetry[30], such as a corner with 90 o bending angle, mode transform cannot occur before the wave isscattered by the particle. So the particle can be pulled continuously in a channel that possesses anarbitrary number of right-angled bends. As there is no restriction on the length of the channel, along pulling distance can be easily achieved. Our work shows that topological sound waves can beused to control particles more proficiently than ordinary acoustic waves. Acknowledgments : We thank Prof. Zhao-Qing Zhang for constructive suggestions. This work issupported by National Natural Science Foundation of China (NSFC) through No. 11904237 andong Kong Research Grants Council through grant No. AoE/P-02/12 and 16303119.
Appendix : Lorenz-Mie theory for acoustic force
Choosing the center of the cylindrical particle as the origin, the pressure fieldat ( , ) r r according to the Mie theory [49] can be expressed as (1)0 0 0 ( ) [ ( ) ( )] , inn n n nn p p q J k r b H k r e r (A1)where n b is the scattering coefficient of the particle, and / n n n q b D with n D being the Miecoefficient of the particle. The velocity field is calculated according to ˆ ˆ{[ '( ) '( )] [ ( ) ( )] } . inn n n n r n n n nn i pip inq k J k r b k H k r e q J k r b H k r e er u (A2)The acoustic force is obtained by substituting Eqs. (A1) and (A2) into Eq. (3). Due to the law ofmomentum conservation, the closed surface in the integral can be arbitrary. For the sake ofsimplicity, we choose the closed surface as a circle with infinite radius centered at the origin. Atthe infinity r , using the asymptotical formulae of Bessel and Hankel functions, ( )2 4( )2 4 ni xn n ni xn n nJ x x H x ex xnJ x x H x i ex x (A3)and ignoring the high order terms of r , the pressure and velocity fields are reduced to
2( ) ( cos ) ,2( ) ( sin ) , n n iy inn n nn iy inn n nn p p q y b e exip q y ib e ex ru r (A4)where , / 2 / 4 n x kr y x n . Substituting Eq. (A4) into Eq. (3), n nn nn x x S iy iyn n n n n niy iyn n n n n n niyn n n n n n f e d p r dq y ib e q y ib epc k q y ib e q y ib eq y b e q y b e S u * *1 1 1 ) }.( cos )( cos )] nn n iyiy iyn n n n n n n q y b e q y b e (A5)Note that sin sin( ) cos ,cos sin ,2sin sin( ) cos ,cos sin ,2 , , n n n n n n n n n n nn n n n ni iiy iy iy iy iy iy y y y y yy y y y ye e e ie e e e ie (A6)acoustic force along x direction becomes | | { 2 2 }2| | Im (2 ). x n n n n n n n n n n n nn n n n n n nn pf ib q ib b ib q ib b iq b iq bc kp b b b q q bc k (A7)Similarly, the force along y direction is | | Re (2 ). y n n n n n nn pf b b p b b pc k (A8) References:
1. A. Eller, J. Acoust. Soc. Am. 43, 170 (1968).2. J. Wu, J. Acoust. Soc. Am. 89, 2140 (1991).3. D. Baresch, J.-L. Thomas, and R. Marchiano, Phys. Rev. Lett. 024301 (2016).4. A. Anhauser, R. Wunenburger, and E. Barsselet, Phys. Rev. Lett. 109, 034301 (2012).5. S. Santesson and S. Bilsson, Anal. Bioanal. Chem. 378, 1704 (2004).66. R. Tuckermann, L. Puskar, M. Zavabeti, R. Sekine, and D. McNaughton, Anal. Bioanal. Chem.94, 1433 (2009).7. J. Shi, D. Ahmed, X. Mao, S.-C. S. Lin, A. Lawit, and T. J. Huang, Lab Chip 9, 2890 (2009).8. X. Ding et al., PNAS 109, 11105 (2012).9. F. Guo et al., PNAS 113, 1522 (2016).10. W. J. Xie, C. D. Cao, Y. J. Lu, Z. Y. Hong, and B. Wei, Appl. Phys. Lett. 89, 214102 (2006).11. P. L. Marston, J. Acoust. Soc. Am. 120, 3518 (2006).12. P. L. Marston, J. Acoust. Soc. Am. 122, 3162 (2007).13. P. L. Marston, J. Acoust. Soc. Am. 125, 3539 (2009).14. L. Zhang et al., Phys. Rev. E 84, 035601(R) (2011).15. L. Zhang et al., J. Acoust. Soc. Am. 131, EL329 (2012).16. S. Xu, C. Qiu, and Z. Liu, Europhys. Lett. 99, 44003 (2012).17. C. E. M. Demore et al., Phys. Rev. Lett. 112, 174302 (2014).18. F. G. Mitri, J. Appl. Phys. 117, 094903 (2015).19. E. H. Trinh, Rev. Sci. Instrum. 56, 2059 (1985).20. A. L. Yarin, M. Pfaffenlehner, and C. Tropea, J. Fluid Mech. 356, 65 (1998).21. E. H. Brandt, Nature 413, 474 (2001).22. A. Marzo, S. A. Seah, B. W. Drinkwater, D. R. Sahoo, B. Long, and S. Subramanian, Nat.Coummun. 6, 8661 (2015).23. H. M. Hertz, J. Appl. Phys. 78, 4845 (1995).24. J. Lee, S.-Y. Teh, A. Lee, H. H. Kim, C. Lee, and K. K. Shung, Appl. Phys. Lett. 95, 073701(2009).25. D. J. Collins, C. Devendran, Z, Ma, J. W. Ng, A. Neild, and Y. Ai, Sci. Adv. 2, e1600089(2016).6. S. C. Takatori, R. D. Dier, J. Vermant, and J. F. Brady, Nat. Commun. 7, 10694 (2016).27. F. Guo, P. Li, J. B. French, Z. Mao, H. Zhao, S. Li, N. Nama, J. R. Fick, S. J. Benkovic, and T.J. Huang, Proc. Natl. Acad. Sci. 112, 43 (2015).28. A. Marzo and B. W. Drinkwater, Proc. Natl. Acad. Sci. 116, 84 (2019).29. D. Wang, C.-W. Qiu, P. T. Rakich, and Z. Wang, CLEO (2015).30. P. T. Rakich, M. A. Popovic, and Z. Wang, Opt. Express , 18116 (2009).31. F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).32. S. Raghu and F. D. M. Haldane, Phys. Rev. A 78, 033834 (2008).33. Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. Lett. 100, 013905(2008).34. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, Nature , 772 (2009).35. Y. Poo, R.-X. Wu, Z. Lin, Y. Yang, and C. T. Chan, Phys. Rev. Lett. , 093903 (2011).36. Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, B. Zhang, Phys. Rev. Lett. 114, 114301(2015).37. A. B. Khanikaev, R. Fleury, H. Mousavi, and A. Alu, Nat. Commun. 6, 8260 (2015)38. X. Ni, C. He, X.-C. Sun, X. Liu, M.-H. Lu, L. Feng, and Y.-F. Chen, New. J. Phys. 17, 053016(2015).39. Y. Ding, Y. Peng, Y. Zhu, X. Fan, J. Yang, B. Liang, X. Zhu, X. Wan, and J. Cheng, Phys. Rev.Lett. 122, 014302 (2019).40. D. Censor and J. Aboudi, Journal of Sound Vibration 19, 437 (1971).41. D. Zhao, Y.-T. Wang, K.-H. Fung, Z.-Q. Zhang, and C. T. Chan, Phys. Rev. B 101, 054107(2020).42. J. S. Faulkner, Phys. Rev. B 19, 6186 (1979).43. S. K. Chin, N. A. Nicorovici, and R. C. McPhedran, Phys. Rev. E 49, 4590 (1994).4. X. Zhang, L.-M. Li, Z.-Q. Zhang, and C. T. Chan, Phys. Rev. B 63, 125114 (2000).45. G. Gouesbet, J. Quant. Spectrosc. Radiat. Transf. 110, 1223 (2009).46. G. Gouesbet and G. Grehan, Generalized Lorenz-Mie Theories (Springer, Berlin, 2011).47. G. Gouesbet and J. A. Lock, Appl. Opt. 52, 897 (2013).48. N. Wang, J. Chen, S. Liu, and Z. Lin, Phys. Rev. A 87, 063812 (2013).49. C. F. Bohren and D. R. Huffman, absorption and Scattering of Light by Small particles (JohnWiley and Sons, New York, 1983).ig. 1. Band structures of phononic crystals composed of (a) static cylinders and (b) spinningcylinders arranged into a square lattice embedded in water. The parameters of the water and staticcylinder are
10 / , 1489 / , 1.3 10 / , 400 / kg m c m s kg m c m s , the radius of thecylinder is c r a , and the spinning circular frequency is a c .ig. 2. (a) Bulk (black) and surface (red and blue) state dispersion relations calculated usingmultiple scattering method in conjunction with a supercell calculation. (b) Pressure field patternsfor the four surface modes. (c) Full wave simulations for the one-way surface wave beingscattered by a cylindrical particle. The spinning cylinders are shown by the black disks in (b) and(c). In (c), the scatter is represented by the white disk, and the two line sources (red and blue dots) (1)0 0 0 1 ( ) p H k r and (1)0 0 0 2 ( ) p H k r are located at ( 10,0.2) r and ( 10, 0.2) r ,espectively.Fig. 3. (a) The acoustic forces acting on the particle as a function of the location ( , ) s s x y of theparticle within a rectangular region as marked in Fig. 2(c). The parameters of the cylinder are s s s r a kg m c m s . (b) The spatially averaged longitudinal acousticforce x F as functions of s y for different particles. For the red black symbol lines, the massdensity and sound speed of the particle are s s kg m c m s , while for the bluesymbol lines, the mass density and sound speed of the particle are ' 1.3 / , ' 340 / s s kg m c m s .Fig. 4. (a) , | ( , ) | B n g k y and (b) , | ( , ) | D n g k y as functions of the vertical coordinate y . Theinsets show | ( , ) | g k y as functions of plane wave vector k at y a .ig. 5. (a) and (b) the intensities , and relative phases , B D of modes B and D of zeroorder in the scattered wave which are normalized by the intensity of the incident wave as functionsof the particle location s x . The vertical coordinate of the particle is s y a . The insect in (a)shows the intensity distributions of eigenmodes B and D. (c) and (d) the longitudinal acousticforces x F acting on the particle as functions of the particle location s x which are calculated usingthe Lorenz-Mie formula Eq. (5) (red lines) and response theory Eq. (8) (blue circles). The verticalcoordinates of the particle in (c) and (d) are s y a and s y a respectively. Theparameters of the particle are s r a , s s kg m c m s