Directional radiation of sound waves by a subwavelength source
Xu-Dong Fan, Yi-Fan Zhu, Bin Liang, Jian-chun Cheng, Likun Zhang
FFan et al.
Directional radiation of sound waves by a subwavelength source
Xu-Dong Fan, Yi-Fan Zhu, Bin Liang, ∗ Jian-chun Cheng, † and Likun Zhang ‡ Collaborative Innovation Center of AdvancedMicrostructures and Key Laboratory of Modern Acoustics,MOE, Institute of Acoustics, Department of Physics,Nanjing University, Nanjing 210093, P. R. China National Center for Physical Acoustics and Department of Physics and Astronomy,University of Mississippi, University, Mississippi 38677, USA (Dated: February 27, 2017)
Abstract
We propose and experimentally achieve a directional dipole field radiated by an omnidirectionalmonopole source enclosed in a subwavelength structure of acoustically hybrid resonances. Thewhole structure has its every dimension at an order smaller than the sound wavelength. Thesignificance is that the radiation efficiency is up to 2.3 of the radiation by traditional dipoleconsisting of two out-of-phase monopoles in the same space. This study eventually takes an essentialstep towards solving the long-existing barrier of inefficient radiation of directional sound waves inlow frequencies, and consequently inspires the ultimate radiation of arbitrary multipoles or even ahighly directional beam by a monopole in limited spaces.
PACS numbers: XX a r X i v : . [ phy s i c s . c l a ss - ph ] F e b irectional radiation of sound waves is of fundamental interest to the field of acoustics,with obvious potential to revolutionize various applications, such as focused sound wavesfor imaging, directional sound beams for underwater communication, among others. Yetin practice, from a classical perspective, any sound source with dimensions much smallerthan the wavelength will act as a monopole and always radiate sound energy equally inall directions while the multipole radiation is inefficient relevant to the monopole[1, 2]. Tocompensate the source size limitation, directional radiated fields at low frequency have to beproduced via arrays of sources[3] or sources that must move by long distance to synthesizea large-scale aperture in order for a source dimension comparable to the wavelength[4–8].This poses a fundamental barrier on radiation of directional sound waves at low frequenciesby a finite-size source, therefore preventing the effective downscaling of acoustic devices. Wepursue radiation of low-frequency directional sound waves by a subwavelength source.Many mechanisms have already been developed for controlling low-frequency soundradiation, which usually depend on coiling up space structures that force the sound totravel through a zigzag path to delay its propagation phase[9–12], or some hybrid resonantstructures that use arrays of resonators to accumulate a sufficient phase lag[3, 8]. Theexisting metastructures based on these mechanisms, despite their vanishing thickness, stillneed to have a large transverse dimension in terms of wavelength to form a directional beam.On the other hand, metamaterials with anisotropic parameters designed by coordinatetransformation technique can rotate the omni-directivity of a sound field but is still notable to reshape the directivity in subwavelength dimension[13]. Instead, we seek to developa structure enclosing a monopole that as a whole has a subwavelength dimension yet canradiate directional sound waves at low frequency.This letter takes the first step for breaking through the barrier to enable a high efficientproduction of a dipolar field from a deep-subwavelength structure wrapping around a simplepoint source. To elucidate our mechanism underlying such unusual phenomenon, we beginwith a revisit of the sound field radiated from a finite-size source q of a volume V , Φ( r ) = (cid:82) V q ( r ) G ( r , r ) d r , where G ( r , r )=e ik | r − r | / π | r − r | is the Green’s function of free spacewith k being the wave number[1]. By taking the multipole expansion around the sourcecenter r = r , Φ( r ) = (cid:90) V q ( r ) G ( r , r )[1 + p d ( r ) + ... ] d r , (1)where the first term is the monopole term, while the second term represents the dipole2adiation with the factor p d in the far field form as p d ( r ) = ( r − r ) · ∇ G ( r , r ) /G ( r , r ) ≈ − ik | r − r | cos θ (2)For a low-frequency airborne sound radiated in free space, one has k | r − r | (cid:28) V with V / (cid:28) λ ( λ is the wavelength), and hencethe dipole part would be trivial as compared to the monopole part, in accordance with theconventional notion that any source with subwavelength dimension always behaves like amonopole with omni-directional radiation. A directional radiation of low-frequency sound,as a consequence, has to depend on a sufficiently large | r − r | in terms of wavelength,which is realized by arranging small sources into an array with dimension comparable withwavelength[3] or by synthesizing a large-scale aperture with moving sources, as done intraditional methods.[4–8]Here we achieve the radiation of a dipole field from a monopole source enclosed by aproposed structure with deep-subwavelength dimension, based on an essentially differentmechanism that manipulates the effective wave number k = k eff instead of enlarging thedimension of source for achieving a large | r − r | . We substantially expand the effectivewave number k eff to make k | r − r | ∼
1, thereby satisfying the basic requirement that thewavelength and the source dimension must be comparable. We demonstrate this mechanismboth numerically and experimentally by converting a monopole into a dipole with near-unityefficiency with a deep-subwavelength structure.
MODEL
The artificial structure to implement our distinctive idea, shown in
Fig.1(a) , is formed bycoupling a straight cylindrical tube of constant cross-sectional area S shown in Fig.1(b) withtwo facing spiral tubes shown in
Fig.1(d) and (e) . The two facing tubes, spirally wrappedaround the straight tube with their openings being attached at two opposite openings in themiddle of that tube, maximally compress the dimension of the resulting device. And the twoparts shown in
Fig.1(c) and (f ) are connected to the straight cylindrical tube on the bothends to optimize the radiation by improving the impedance matching. While guaranteeingthe deep subwavelength scale of the overall size of the whole structure, the combination ofstraight and spiral tubes supports hybrid internal resonances that substantially slow down3
IG. 1. Diagram of the artificial structure. The structure in (a) is composed of the several partsin (b-f). the transmission wave and change the effective wave number k = k eff to make k | r − r | ∼ Fig.1(b) . With a cross-sectional dimension D ( S = πD / th mode to propagate. Without consideringthe near field effect caused by the approximation of a point source to a plane wave in thesubwavelength waveguide, the sound pressure and volume velocity at the left end of thetube (i.e., the location of the point source) give: p = A exp( ik x ) + B exp( − ik x ), and U = Sρc [ A exp( ik x ) − B exp( − ik x )], where k is wave number in the air, A and B arethe complex amplitudes of incident wave and reflected wave respectively. Correspondingly,the sound pressure and volume velocity at the other end of the tube gives: p = A exp( ik x ),and U = Sρc A exp( ik x ). At the junction, the continuity of pressure and velocity requires: A + B = A = p , and Sρc ( A − B ) = U + Sρc A , where p and U are the sound pressureand volume velocity at the opening of the branch, respectively. It follows that the reflectedcoefficient and transmitted coefficient are r p ≡ B A = − Z Z + 2 Z b , t p ≡ A A = − Z b Z + 2 Z b (3)4here Z = ρc /S is the acoustic impedance of the central tube, and Z b = Z Z / ( Z + Z )is the combined acoustic impedance of the two facing spiral tubes. ρ and c is the mediumdensity and sound speed respectively, and Z , Z are the acoustic impedances of the twobranches, which are obtained by the impedance transfer equation to be: Z = ρ c S Z ∗ − i ρ c S tan( k L ) ρ c S − iZ ∗ tan( k L ) (4a) Z ∗ = i ρ c S h cot( k l ) (4b)where S = πd / S h = πd h / Fig.1(b) and the spiral tubes shown in
Fig.1(d) and (e) , and L and l are the lengthsof the neck and the spiral tubes respectively. From the analytical formulae derived above,it can be anticipated that at the right end of the straight tube, the wave field would be ofnearly equal strength and opposite phase as compared to the source located at the otherend of that tube, if the structural parameters of straight and spiral tubes could be adjustedappropriately. This corresponds to a dramatic increase of the effective wave vector thatensures k | r − r | ∼ SIMULATIONS AND MEASUREMENTS
Now we simulate the directional radiation of low-frequency sound with the commercialCOMSOL MULTIPHYSICS software based on the finite element method. The density andsound speed of artificial structure are set as ρ = 1250 kg/m and c = 2700 m/s, respectively,and the background material applied are air. The standard parameters used for air underan ambient pressure of 1 atm at 20 ◦ C are mass density ρ = 1 .
21 kg/m and sound speed c = 343 m/s. Perfectly matched layers (PMLs) are imposed on the outer boundaries ofsimulated domain to eliminate the interference from reflected wave.Experiments to verify and demonstrate our conversion of monopole source into a dipoleare conducted in anechoic chamber. Figure 2(a) shows the sample of the artificial structureand the experiment setup is shown in
Fig.2(b) . The solid sample is fabricated withpolylactic acid (PLA) plastic via integrated 3D printing technology (Stratasys Dimension5
IG. 2. Experiment setup. (a)Sample photo (b)Diagram of measurement environment.
Elite, 0.08 mm in precision) to meet the requirement of the theoretical design. The materialof sample is treated acoustically rigid due to the huge impedance mismatch between the solidand air. A loudspeaker (18 mm in length and 10mm in width) emitting a monochromaticwave as the acoustic source is placed inside the sample. For each measurement, two 1/4-inmicrophones (B&K type 4961) are placed at designated positions to sense the local pressure:one is mounted at a fixed position to detect the pressure as signal 1 and the other is moveableto scan the pressure field as signal 2. By using the software PULSE LABSHOP, the crossspectrum of the two signals gathered by the two microphones is obtained, for which signal1 works as a reference and signal 2 as an input signal. The pressure field is retrieved byanalyzing the cross spectrum and recording the magnitude and phase at different spatialpositions within the measured region.
Figure 3 shows the simulated results and measured data of the amplitude and phasedifferences on the two openings of the artificial structure with the sound pressure distributioninside the cylindrical tube at the resonant frequency being inserted. From the result, we canobtain that the amplitude difference is zero and the phase difference is π at around 352.9Hz,which is the frequency to realize a dipole-like directivity as implied by the black arrow. Andthe corresponding L structure /λ = 0 .
14 (i.e., kL structure = 0 .
9) and L tube /λ = 0 .
08 (i.e., kL tube = 0 . Figures 4(a) and display the simulated spatial distribution of pressure fields forsound radiated from low-frequency monopole sources with and without the designed artificialstructure respectively.
Figure 4(a) shows that in the absence of any artificial structure, themonopole source radiates sound equally in all directions as expected. In the presence of our6
IG. 3. Results of the amplitude difference and phase difference between the two openings of theartificial structure. Blue and red solid lines are simulated results of amplitude difference and phasedifference versus frequency with star and triangle mark being the corresponding measured data.Black arrow points at around 352.9 Hz. (Insert: Sound pressure distribution inside the cylindricaltube at the resonant frequency) designed structure, the original radiation field produced by the monopole source is convertedinto a dipole with near-unity efficiency, as shown in
Fig.4(b) , revealing the effectiveness ofour scheme of generating directional radiation for low-frequency sound.
Figure 4(c) shows the simulated results and the measured data of the directivity.Measured results agree well with the simulated ones, with both proving that our artificialstructure can convert a monopole source to radiate sound waves like a dipole source insubwavelength dimension.For a better comparison, we also plot the radiation of a simple dipole in free space in
Fig.4(d) . The artificial dipole realized by the combination of a single monopole source andthe subwavelength structure physically avoids the destructive interference of the two sourceswith opposite phases in a subwavelength interval. The radiation efficiency is 2.3 of thetraditional dipole consisting of two out-of-phase monopole sources separated at a distanceequal to the dimension of the structure L structure = 0 . λ . This amplification is beneficialfrom our structure that supports hybrid resonances in the tubes.7 IG. 4. Sound pressure fields and directivity patterns of (a) a monopole source without structure,(b) a monopole source with structure. (c) Simulated and measured directivity patterns of theartificial dipole. (d) Sound pressure fields and directivity patterns of a dipole source. The workingfrequency is around 352.9Hz.
DISCUSSION
We have demonstrated the conversion of acoustic monopole source to dipole in asubwavelength source space via numerical simulations and experimental measurements.The whole structure has its every dimension at an order smaller than the sound wavelength,however, the radiation efficiency is up to 2.3 of the direct radiation of a dipole consisting oftwo out-of-phase monopoles seperated at a subwavelength distance equal to the dimensionof our structure.The design is beneficial from the coupling resonances of sound waves in our structure.While it is expected from the known properties of acoustic resonances to provide phaseshift in limited spaces [1], our structure has used a configuration of spiral tubes that enablethe resonances to be effectively excited in a way that the coupling of sound energy in thestructure leads to the desired phase delay for the efficient radiation of directional soundwaves from an omnidirectional monopole enclosed by our structure that has a subwavelenghdimension as a whole.The scheme proposed here only works within a narrow frequency band due to themechanism of resonances, but the narrow-band sound source is also an important category8f acoustic speakers/transducers that traditionally was associated with the resonances ofthe crystal of the transducers, though it is of interest to have a directional radiation ofbroad-band sources by a subwavelength source that we pursue here. The results inspirethe further study of converting monopole to arbitrary multipoles or even to a desireddirectional beam in subwavelength dimension with much higher radiation efficiency. Besides,the directional radiation of low frequency sound is also important to help understandthe phenomena in nature such as for the dolphin to use directional sound for long-rangecommunication even with its subwavelength vocal source. Our scheme takes the advantagesof simpleness and compactness compared with existing traditional methods, which may openan avenue to solve the long-existing problem of omnidirectional radiation in low frequencyin subwavelength dimension.This work was supported by a start-up fund of the University of Mississippi, the NationalNatural Science Foundation of China (Grant Nos. 11634006 and 81127901), and A ProjectFunded by the Priority Academic Program Development of Jiangsu Higher EducationInstitutions. ∗ [email protected] † [email protected] ‡ [email protected][1] Morse P. M. C. and Ingard K. U., “Theoretical acoustics,” Princeton university (1968).[2] Daniel A. Russell, Joseph P. Titlow, and Ya-Juan Bemmen, “Acoustic monopoles, dipoles,and quadrupoles: An experiment revisited,” American Journal of Physics , 660 (1999).[3] Yong Li, Xue Jiang, Bin Liang, Cheng Jianchun, and Likun Zhang, “Metascreen-basedacoustic passive phased array,” Phys. Rev. Appl. , 024003 (2015).[4] A. J. Berkhout, D. de Vries, and Vogel P., “Acoustic control by wave field synthesis,” TheJournal of the Acoustical Society of America , 2764 (2016).[5] L. Azar, SHi Y., and S. C. Wooh, “Beam focusing behavior of linear phased arrays,” NDTand E International , 189 (2000).[6] H.S.C. Wang, “Performance of phased-array antennas with mechanical errors,” IEEETransactions on Aerospace and Electronic Systems , 535 (1992).
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