We established a method for generating an ultrasound beam that propagates along a narrow, isolated curved path and is accompanied by an arc-shaped airflow, and experimentally confirmed the actual generation of such a beam. The method employs a two-dimensional orthogonal phased array of ultrasound transducers, whose individual columns correspond to a line segment in a given beam trajectory. Each column of transducers yields a “ring” in which acoustic energy is concentrated. A cluster of these ultrasound rings arranged at sufficiently small spatial intervals interfere with adjacent rings, consequently forming a fine curved path of propagating ultrasound accompanied by an ultrasound-driven air flow. The positions of these rings can be electronically controlled and so can the position of the resulting beam path. We obtained isolated sharp arc-shaped airflows propagating over nearly 1 m in open space. Such airflows have never been reported and are extremely difficult to generate by the superposition of ordinary jet-driven airflows. Our achievement will lead to the ability to generate airflows along an arbitrarily designed three-dimensional arc-shaped path. This technique will be utilized in such unprecedented applications as midair transportation of gaseous substance or control of heat in the air while circumventing obstacles, which are intuitive, yet hard to achieve by other methods.

## I. INTRODUCTION

Acoustic streaming is the mass flow of a medium accompanied with intense sound propagation and has been a well-known nonlinear acoustic phenomenon for more than a century.^{1–5} Acoustic streaming is classified into three types of flows based on its spatial scale compared with the acoustic field wavelength: Rayleigh, Schlichting, and Eckart flows.^{6} The generation of Eckart flows has been chiefly reported with a packed fluid. By deftly controlling such flows in air, applications such as midair transportation of gaseous substances or localized control of the spatial heat distribution in living spaces can be conceived. Nevertheless, Eckart air flows have not currently been utilized in any practical form. One primary reason for this could be the need for a spatially localized acoustic intensity strong enough for generating a concentrated driving force distribution for acoustic streaming, which is not easy to achieve in practice with conventional sound sources.

The use of an ultrasound phased array, which is a cluster of transducers whose individual emission waveforms can be controlled, is a promising method for generating such a localized wavefield in an electronically controllable fashion.^{7} This area has recently been the focus of vigorous research for practical uses such as acoustic levitation^{8,9} and noncontact tactile stimulation,^{10–13} based on a localized acoustic radiation force, which is another primary nonlinear acoustic phenomenon.^{14} For effectively generating a radiation force, an acoustic field should be highly concentrated in a small region. In contrast, for acoustic streaming in a space of considerable spatial extent, a suitably designed stretching acoustic beam is required.

In our previous study, we demonstrated the generation of a narrow and straight beam stretching over a distance equal to tens of wavelengths, with a cross-sectional diameter comparable to the wavelength,^{15} based on the concept of a Bessel beam.^{16,17} The position and orientation of the beam were electronically steerable, along with an accompanying straight ultrasound-driven airflow. We also demonstrated that these straight flows could be applied to a midair fragrance transportation system^{18} or a remote cooling system.^{19}

The intrinsic degree of freedom (DoF) of an ultrasound phased array driven at a single frequency is given by the number of transducers, when the output intensity of all transducers is identical. This means that ultrasound propagation that is more spatially flexible than a straight beam can be expected.

The goal of the work described in this paper was to construct a single arc-shaped path of acoustic streaming out of a large number of locally accelerating regions in midair with a spatially designed acoustic power flux field in the form of arc-shaped acoustic beams emitted from an ultrasound phased array. We devised a method for generating such acoustic beams composed of many line segments in which there are intense acoustic fields whose phase delays are determined by the arc length of the beam path. Such an approach for generating a narrow arc-shaped path of flow acceleration in the air has never been reported in the literature.

Here, we refer to a related study in which a bottle-shaped acoustic beam was created out of sound emission from an annular transducer array.^{20} In that study, wavefronts that would yield a sound beam of a given shape were calculated based on the idea that the beam can be formed by a consecutive series of localized points of sound convergence, which are called caustics in optics. Although this method is technically sound, it is applicable only to axisymmetric volumetric beams and requires somewhat cumbersome calculations, such as solving nonlinear differential equations. In the process of creating a bottle-shaped beam, the generation of a two-dimensional arc-shaped beam was also demonstrated. However, our target is three-dimensional beam convergence, which cannot be accomplished by directly applying the method described in the literature, which is only valid for the two-dimensional problem. The two-dimensional approach, as in the original form, does not guarantee energy convergence of the beam in a direction perpendicular to the plane in which the beam is confined. This issue must be solved because ultrasound-driven flows that are sufficiently strong and localized can be generated only when three-dimensional sound convergence is achieved. Note that the caustic-based approach might be applicable to arbitrary shapes, but only with proper modifications to handle the three-dimensional convergence. Instead, we propose a simpler strategy to overcome this issue.

We created an arc-shaped beam whose shape and location can be electronically determined with simple calculations. The beam has sufficiently high localized acoustic intensity for generating an Eckart flow on a curved path. We conducted numerical simulations on the generated acoustic intensity field to demonstrate that the field around the beam path was parallel to the tangential directions on line segments of the beam path. These flows could serve as a midair conveyor of gas or heat that circumvents obstacles from behind, which is quite difficult to realize with the superposition of jets or straight ultrasound-driven flows.

## II. METHODS

Figure 1 shows the phased array of 40 kHz ultrasonic transducers utilized in the experimental setup, which contained 1992 transducers arranged in a two-dimensional lattice of 28 rows and 72 columns. The diameter of each transducer was 10 mm, and the total array size was $768mm\xd7302.8mm$. There was one row and three separate columns where no transducers were embedded, as seen in Fig. 1. These empty row and columns have little effect on the ultrasound focusing since the transducers are compactly mounted on the rest of the radiation plane. The array contained field programmable gate arrays (FPGAs) and CPUs to control each transducer using a standard PC communicating via EtherCAT (Beckhoff Automation GmbH, Germany). Each transducer was supplied with driving currents via individual amplifier ICs implemented on the substrates of the phased arrays.

Suppose that the sound field is time-harmonic where all sound sources are sinusoidally driven with a common output angular frequency. Let $P(r)$ (Pa) be the complex sound amplitude and $V(r)$ (m/s) be the complex particle velocity at a location $r:=[xyz]$ (m). Note that these quantities correspond to the RMS value of the instantaneous sound pressure $p(t)$ and the instantaneous particle velocity $v(t)$ as follows:

where *t* (s) denotes time, $\omega (Hz)$ is the driving frequency, and $j=\u22121$. It is known that the acoustic intensity vector $I(r)$ ($W/m2$), which indicates the time-averaged energy flux per unit area, is given by

where $Re[\u22c5]$ and $\u22c5\u2217$ denote the real part and the complex conjugate of $\u22c5$, respectively.^{21} In a sound field with sinusoidal sources, the driving force to the medium per unit mass $F(r)$ (N/kg) is known to be proportional to the acoustic intensity vector $I(r)$ ($W/m2$)

where $\rho (kg/m3$) is the medium density, $c(m/s)$ is the sound velocity in the medium, and $\alpha (m\u22121$) is the sound absorption coefficient.^{22} This driving force is understood as a time-averaged nonlinear effect of intense sound propagation. These equations indicate that the acceleration field of the medium can be designed by an appropriately generated sound intensity field.

As stated above, we intend to create a curved stream by combining adjacent local tangential acceleration vectors provided by the ultrasound field. Here, we give a geometric description of the procedure for calculating the phase shift at each transducer in relation to a given beam path. We begin with a coordinate system shown in Fig. 2. The beam path $L$ located in the $zx$-plane is parametrically written as $l(t)=[lx(t)0lz(t)]$, with an arbitrary parameter $t>0$. We assume that $l(0)$ gives the starting point of the arc located on the array plane. Next, for every column of the phased array $xi,i=1,\u2026,N,$ where $N$ is the number of columns, we obtain a tangent that passes through the center of the column $[xi00]$ along with the tangential point $li$ on $L$. $li$ corresponds to a specific value of the parameter $t=ti$, and thus, $li:=l(ti)=[lx(ti)0lz(ti)]$. After obtaining the value of $ti$ for $li$, we calculate the arc length from $l0$ to $li$ on $L$ by

We drive all transducers on column $xi$ so that their emissions are in phase at $li$. As a result, an annular region including $li$ is in phase, where intense convergence of acoustic energy is observed. With a sufficient number of these “rings” so that the phase delay of the consequent wavefield on the beam path is proportional to the arc length of the path, the overlapping of the rings yields one arc-shaped path within which intense propagating ultrasound is confined. Here the proper phase delay at $li$ is assumed to be given by $ksi$, with $k$ denoting the wavenumber. Hence, the proper phase shift $\theta ij$, where $j$ denotes the row index on the $y$ axis of the array, should be given so that it cancels the phase delay through propagation $k|rij\u2212li|$, where the transducer position is denoted by $rij=[xiyj0]$. Finally, the following equation gives the phase shift to be added at the transducers

The resulting acoustic intensity vectors at the tangential points are parallel to the tangential directions on the beam path since their $y$ components cancel in a summation process of all emissions from the transducers in the column, which is line-symmetrical about the $zx$-plane. The two essential quantities here are the tangential points $li$ and the arc length $si$, both of which can be calculated with an algebraic or numerical procedure without much difficulty. A physically intrinsic constraint on the feasibility of the beam path is that the radius of curvature at any point on the path cannot be lower than the wavelength.

## III. NUMERICAL AND PHYSICAL EXPERIMENTS

We numerically simulated the spatial distribution of the acoustic intensity vector field from the determined phase shifts at the transducers. We set the ultrasound frequency to 40 kHz and the sound velocity to 340 m/s. The arrangement of transducers corresponds to the actual devices. In the experiment, the beam path was parametrically written as $x(t)=1.4t,z(t)=\u22121.176t2+1.176t\u22120.39$. The primary reason for choosing this set of parameter values is that the bending part of the beam should be within the frontal region of the finite-sized array. Another important factor is that the beam should adequately extend in both $x$ and $z$ directions for successful observation of the flow. With the transducer location $rij$ and the individual phase shift $\theta ij$ and the constant $A(m3/s$), the complex pressure $Pij(r)$ and complex particle velocity $Vij(r)$ that the corresponding transducer generates at $r$ were calculated as follows:

where $r=|rij\u2212r|$. The acoustic intensity vector $I(r)$ could be obtained as the product of the sum of those quantities with respect to all transducers

We calculated the phase shift of each transducer with the following procedure. We divided the beam path into segments and obtained their individual tangents along with their colliding points with the array, $[xc00]$. Consequently, a set of tangents were allocated to a region occupied by each transducer. We performed Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) interpolation, which gave the phase shift as a piecewise cubic function of the $x$ coordinates of the colliding points, $\theta =\theta (x)$. Substituting the center location of the corresponding transducer in the piecewise cubic function gave the final phase shift. For transducers whose $y$ coordinate is not zero, the obtained tangents were tilted toward the $y$ axis so that the colliding $y$ coordinate matched that of the transducer, prior to the PCHIP interpolation.

Next, We measured the spatial distribution of the airflow velocity and acoustic pressure of the actually generated ultrasound beam, where the phase shifts of the transducers were calculated with the aforementioned numerical simulations. For the measurement of the velocity, we used a hot-wire anemometer (KANOMAX CLIMOMASTER 6501-C0) with a probe (KANOMAX 6533-21) attached to a robot hand (FANUC M-710iC 20L). The probe had a spherical sensing tip of 2.5 mm in diameter, which was azimuthally non-directive. For capturing the sound pressure, we used a standard microphone (Type 4138), with a pre-amplifier (Type 2670) and an amplifier (NEXUS Type 2690), all of which are products of Brüel & Kjær, Denmark. In the measurement of the flow velocity, we performed temporal averaging of the sensor output voltage for 10 s because of the aerodynamic turbulence. It should be noted that some temporal fluctuation still remained after this procedure, though it was much reduced.

Figures 3 and 4 show the result of simulations and measurements. Figure 5 depicts the calculated phase shifts for the beam path. The calculated pressure distribution indicates that the beam has a shape that is defined by the given parametric curve. Correspondingly, the simulated acoustic intensity vectors are observed to be parallel to the beam with their intensified magnitude on the beam. These results indicate that the acceleration of the air flow near the beam is expected to be parallel to the tangential direction of the beam propagation, whereas little acceleration is seen at the regions away from the beam. As for the measurements, it is confirmed that the captured pressure distribution is well-predicted by the numerical simulation. Note that in the acoustic measurement, the output level of the phased array is intentionally lowered compared to the possible maximum so that clipping of the captured signal could be avoided. In the wind velocity measurement, the output level of the phased array was set to the maximum, resulting in around 4 kPa (166 db SPL) at the maximum RMS pressure in the ultrasound field. The intensity of the velocity is observed to be localized around the beam. At the same time, it is seen that the flow spread at the farther end of the beam, which is presumably because of the inertial behavior of the air in the absence of the flow-accelerating wavefield.

Figure 6 shows the water vapor redirected by the generated ultrasound-driven flow. The vapor was guided in different directions depending on its location along the beam path. However, those directions were not always observed to coincide with the tangent of the beam path at those points. This is because what is directly controlled by the ultrasound is not the direction of the flow but the acceleration of the medium, which means that the streamlines of the flow are not guaranteed to be the same as the beam path. It is also observed that a slight deviation from the beam path resulted in a change in guided direction because of the viscosity of the air [see Fig. 6(d) in comparison with (b)]. Nevertheless, the ultimate goal of this research, that is, guiding an air stream along an arbitrary curved path, was partially achieved in the context of localized control of the acceleration of the medium.

## IV. DISCUSSIONS AND CONCLUSIONS

The method we proposed here associates one column of the array with one line segment on the beam path. However, for some shapes of beams, this one-to-one correspondence does not hold. For example, some array columns might be allocated to two or more tangential points on the curved path. For those cases, a complex output amplitude of the transducers can be obtained as the sum of individual output amplitudes that correspond to the tangential points, based on the principle of superposition. In this case, the amplitude is not uniform among all transducers.

In relation to this, one might desire to obtain an arbitrary beam path that is no longer confined to a specific plane, as in the above-stated formulation. Although no numerical or experimental verification is given in this paper, we briefly describe a possible modification of our method to achieve this. First, the tangents at every point on the beam path and their crossing points at the array are obtained. Next, a line segment on the array that is perpendicular to the horizontal (parallel to the array surface) component of the incident tangent is obtained. Then, the rest of the calculation can be done in the same fashion as in the xz-plane case by replacing an array column with a set of transducers on the line segment. When two or more tangents are allocated to one transducer, superposition of the corresponding output amplitudes would be required, as in the case discussed above.

Our proposed method is theoretically valid regardless of the driving frequency as long as the number of array columns is so large that the cluster of generated acoustic rings overlap in sufficiently large portions to construct a continuous beam. In our setup, the mean distance between each ring, which is equivalent to the mean length of the line segments in the beam path allocated to each array column, was approximately two to three wavelengths. Although the largest intervals of these rings that yield proper beam convergence have not been investigated in the current study, this result is reasonable because each ring has its radial margin spread over more than several wavelengths. When applied to a higher frequency, such as the MHz range, the number of array columns must be accordingly larger to guarantee sufficiently fine segmentation of the beam path and the transducers must be in an adequately dense arrangement. In addition, sound attenuation during propagation in those frequency ranges is less negligible than at lower frequencies. For underwater cases, compared to midair, these issues are less influential thanks to the longer wavelength and smaller attenuation.

Rigorously, what we achieved in this paper is a sequential cluster of locally controlled accelerations. That is, we did not directly create a curved streamline. Although we suppose that a streamline might emerge as an aggregated path of those local accelerations, it is obvious that this is not perfectly correct because of the inertial behavior of the air. Our measurements just indicated that the magnitude of the streaming velocity has a shape similar to that of the beam. At this time, it has not been verified if the streamlines are parallel to the beam propagation direction. The only criteria for creating a curved beam in this paper is just that the acceleration direction on the path should coincide with the tangential direction. Nevertheless, an arc-shaped acoustic beam with three-dimensional convergence and accompanying flows with location-dependent acceleration was experimentally demonstrated as a first step in the investigation of ultrasound-driven flow fields.

We will continue the research for ultrasound-driven flows that take on some additional advantageous properties according to the applications. With adequate modifications, the proposed method can be applied to three-dimensional parametric curves that are not confined in a planar space. These modifications could be useful in practice for circumventing obstacles in the environment. In optics, several arc-shaped beams have been discussed and observed.^{23,24} There is a possibility that those techniques can be partially incorporated in our method to achieve these goals.

## ACKNOWLEDGMENTS

This work was supported by JSPS Kakenhi (Nos. 18H01458 and 16H06303).

## REFERENCES

*Nonlinear Acoustics*, edited by M. F. Hamilton and D. T. Blackstock (Academic Press, San Diego, 1998).

*Proceedings of Eurohaptics 2008*(2008), pp. 504–513.

*Proceedings of IEEE Haptics Symposium*(2018), pp. 340–343.

*Springer Handbook of Acoustics*, edited by T. Rossing (Springer, 2014).