Sound emission is inefficient at low frequencies as limited by source size. This letter presents enhancing emission of monochromatic monopole and multipole sources by enclosing the source with a subwavelength circular enclosure filled of an anisotropic material of a low radial sound speed. The anisotropy is associated with an infinite tangential density along the azimuth. Numerical simulations show that emission gain is produced at frequencies surrounding degenerate Mie resonant frequencies of the enclosure, and meanwhile the radiation directivity pattern is well preserved. The degeneracy is theoretically analyzed. A realization of the material is suggested by using a space-coiling structure.

Sound emission rate at low frequencies is extremely low because of the smallness of the source size relative to the large wavelength of low-frequency sound (D<λ). The sound power radiated to far field by a monopole source is proportional to (kD)2 (k=2π/λ is wavelength), and the power decreases proportionally for higher multipoles.1 This present study proposes to enclose monochromatic multipole sources of different orders (including monopole) in a deep subwavelength structure (e.g., Dλ) that can eventually enhance the source emission and meanwhile preserve the directivity pattern of the emitted sound waves.

Existing methods enhancing sound source emission somehow alters the directivity pattern of the emitted sound. A loudspeaker mouth adapted in a horn shape improves the emission but confines the radiation in limited space.2,3 A woofer diaphragm is mounted in a speaker cabinet to boost the low-frequency emission, but the emission is not omnidirectional, and additionally, there are requirements on damping sounds that get into walls of the cabinet.4 Recently, acoustic slab metamaterials were used to enhance emission of a monopole source in a subwavelength space, but the employed Fabry-Pérot resonances between two planar structures inevitably alter the omnidirectional radiation directivity.5 

This present work enhances emission of monopole and multipole sources without altering the directivity by enclosing the source in a subwavelength circular structure of an anisotropic material of degenerate Mie resonances. The anisotropy is associated with an infinite tangential density along the azimuthal direction. Numerical simulations of both monopole and multipole sources enclosed by the material show emission gain at frequencies surrounding Mie resonant frequencies of the material (Sec. 2). Enhancing emission of higher multipoles would otherwise be impossible by a subwavelength isotropic material of common Mie resonances whose resonant frequencies increase for higher multipoles. The anisotropic material presented here has the ability to degenerate the resonant frequencies of higher multipoles to the same frequency as dipole, as analyzed in Sec. 3, consequently enabling emission gain of higher multipoles by the subwavelength enclosure. A space-coiling structure is implemented to realize the anisotropic material for emission enhancement in practice (Sec. 4).

This section presents simulations of the emission enhancement by an conceptual enclosure of a subwavelength diameter D<λ, illustrated in Fig. 1(a) as a circular annulus (yellow). The enclosure has a constant radial sound speed cr much smaller than the sound speed in air, cair, and an extremely high density ρθ along the azimuthal direction. The low cr enables the enhancement to occur at low frequencies, while the infinite ρθ forces the enhancement of higher multipoles to degenerate at a same low frequency, as we will explain in Sec. 3.

Fig. 1.

(Color online) Emission enhancement of a monochromatic monopole source (wavelength λ and diameter 0.01D) by a circular enclosure of an outer and inner diameter D and 0.1D (not scaled in ratio for clear illustration). (a) A contrast of emitted sound fields (λ=5.26D) simulated with and without an enclosure (left half vs right half) of a radial density ρr=14.13ρair (ρair=1.21 kg/m3). (b) Same as (a) but for λ=7.14D and ρr=25πρairr/D. (c) The ratio of the radiated power calculated from simulations with and without the enclosure in (b) as a function of D/λ, illustrating emission enhancement surrounding three Mie resonances; the resonances are shown by the phase distribution of sound fields (insets). The simulated enclosures have a small radial sound speed cr=0.266cair (cair=343 m/s) to enable resonances at low frequencies (D/λ<1).

Fig. 1.

(Color online) Emission enhancement of a monochromatic monopole source (wavelength λ and diameter 0.01D) by a circular enclosure of an outer and inner diameter D and 0.1D (not scaled in ratio for clear illustration). (a) A contrast of emitted sound fields (λ=5.26D) simulated with and without an enclosure (left half vs right half) of a radial density ρr=14.13ρair (ρair=1.21 kg/m3). (b) Same as (a) but for λ=7.14D and ρr=25πρairr/D. (c) The ratio of the radiated power calculated from simulations with and without the enclosure in (b) as a function of D/λ, illustrating emission enhancement surrounding three Mie resonances; the resonances are shown by the phase distribution of sound fields (insets). The simulated enclosures have a small radial sound speed cr=0.266cair (cair=343 m/s) to enable resonances at low frequencies (D/λ<1).

Close modal

We simulate the emission in two dimensions using COMSOL (acoustic pressure module).6 A monochromatic monopole source of diameter 0.01D is centered in the enclosure and emits sound waves in air. The monopole source is simulated with a constant source strength, i.e., a constant volume flow rate of the source, which is a universal model in reality.7,8 The anisotropy of the enclosure is simulated with a large number of laminated isotropic layers along the azimuthal direction.

The enhancement is illustrated in Figs. 1(a) and 1(b) by the contrast of sound fields emitted by a monopole source in free space and enclosed by the enclosure. The omnidirectional radiation pattern is clearly unchanged. We characterize the enhancement by the ratio of radiated sound power P1 in the presence of the enclosure to the power P0 radiated in free space. The powers are calculated from the far field. The P1/P0 values are 76 and 294 for the cases in Figs. 1(a) and 1(b), where the enclosure has a subwavelength dimension of D=0.19λ and 0.14λ, respectively. The results Figs. 1(a) and 1(b) are for two enclosures of the same cr but different radial density ρr (see Fig. 1).

The P1/P0 value is further shown in Fig. 1(c) as a function of D/λ. The result is for the enclosure in Fig. 1(b), revealing that the emission gain P1/P01 is produced at the range of D/λ = [0.01, 0.22], [0.37, 0.49], and [0.68, 0.77]. The gain surrounds three maxima of Mie resonant frequencies: P1/P0 = 294, 60, 33 at D/λ = 0.14, 0.43, 0.73, respectively. The phase distribution of sound fields in the enclosure [insets in Fig. 1(c)] shows a pattern of the monopole Mie resonance.

Numerical results for a dipole source centered in the same enclosure in Fig. 1(b) also show three peak values P1/P0 = 397, 135, 86 in the subwavelength regime of D/λ = 0.29, 0.58, 0.87, respectively. The results are shown in Fig. 2(a) where the insets show the pattern of the lowest three dipole Mie resonances corresponding to the three P1/P0 peaks. The emisssion gain is produced at the whole range of D/λ = [0.17, 0.97]. Here the dipole source is simulated in COMSOL by a constant dipole source strength of a diameter 0.01D.

Fig. 2.

(Color online) Emission enhancement of monochromatic multipole sources using the enclosure in Fig. 1(b) that has an infinite azimuthal density ρθ (i.e., an extreme anisotropy) to degenerate resonant frequencies of higher multipoles. (a) Results as in Fig. 1(c) but for a dipole source also show three resonant peaks at low frequencies (D/λ<1). (b) Theoretical calculations of the three D/λ values for higher multipoles show a degeneracy for different multipole orders m1 (black dots), in contrast to an increase with m for an isotropic enclosure of ρθ=ρr (red dots). The green circles are simulated results for a realistic structure presented in Fig. 3(a).

Fig. 2.

(Color online) Emission enhancement of monochromatic multipole sources using the enclosure in Fig. 1(b) that has an infinite azimuthal density ρθ (i.e., an extreme anisotropy) to degenerate resonant frequencies of higher multipoles. (a) Results as in Fig. 1(c) but for a dipole source also show three resonant peaks at low frequencies (D/λ<1). (b) Theoretical calculations of the three D/λ values for higher multipoles show a degeneracy for different multipole orders m1 (black dots), in contrast to an increase with m for an isotropic enclosure of ρθ=ρr (red dots). The green circles are simulated results for a realistic structure presented in Fig. 3(a).

Close modal

The enhancement can be understood as the improvement of radiation efficiency, following from the interaction of the enclosure with the near field. Without the enclosure, the majority of emitted low-frequency energy is stored as near-field oscillation rather than radiated to the ambient medium. The storage of sound energy in the near field is a result of the large radiation reactance, which is irrelevant to sound dissipation. The enclosure modifies the source's environment to convert sound energy stored in near fields to that radiated to far fields.

This section analyzes how the extreme anisotropy (ρθ) of the material enables the degeneracy of the Mie resonant frequencies of multipole sources of different orders m=integer (with an azimuthal dependence eimθ).

We write the acoustic pressure fields in the enclosure region as p=R(r)(Aeimθ+Beimθ)eiωt where DI/2rD/2 (DI is the inner diameter of the enclosure). The radial function R(r) subjects to

ρrrddr(rρrdRdr)+(ω2cr2ρrm2ρθr2)R=0,
(1)

which is derived from the wave equation [(ρ¯¯)1p]+(ω2/B)p=0 of an anisotropic medium with ρ¯¯ being its mass density tensor and B bulk modulus. Equation (1) explicitly shows that an infinite ρθ decouples the multipole order m from R(r), and further indicates that ω is scaled by the radial sound speed cr of the enclosure. Our enclosure has a small cr, enabling the emission enhancement to systematically occur at low frequencies for both monopole (m = 0) and multipole (m0) sources.

The solutions of Eq. (1) for ρr=constant are analytically obtained as

p={Jv(ωr/cr)Hv(1)(ωr/cr)}{eimθeimθ}eiωt,v=mρrρθ,
(2)

where {} means linear combination, and Jv(·) and Hv(1)(·) are the Bessel and first-kind Hankel functions of order v. The v is forced to be zero by the extreme anisotropy ρθ, and thus the radial functions are J0(·) and H0(1)(·) for multipoles of arbitrary order m.

The resonant frequencies for an enclosure of an inhomogeneous density ρr(r) in general, like the one in Fig. 1(b) with ρr=25πρairr/D, are determined by discretizing the enclosure into N homogeneous rings of different radial density ρr(i)=ρr(ri) (i=1,2,...,N) and applying the solution Eq. (2) to each layer. The radial components of sound fields follow as

R0(r)=aJm(ωr/cair)forrr0DI/2,
(3a)
Ri(r)=biH01(ωr/cr)+ciJ0(ωr/cr)forri1rri(1iN),
(3b)
RN+1(r)=dHm1(ωr/cair)+Jm(ωr/cair)forrrND/2,
(3c)

where the 2(N+1) unknown coefficients (a,bi,ci,d) and unknown resonant frequencies form an eigen-value problem. It is solved by scattering matrices formed from matching acoustic pressure and velocity at interfaces: Ri(r)=Ri+1(r) and [1/ρr(i)][dRi(r)/dr]=[1/ρr(i+1)][dRi+1(r)/dr] at r = ri (i=0,1,2,...,N; ρr(0)=ρr(N+1)ρair). Calculations with N = 1 can give results of an enclosure of a constant density, like the one in Fig. 1(a). For the enclosure in Fig. 1(b) of an inhomogeneous density, the resonant frequencies are calculated with N = 20 that gives convergent solutions.

The D/λ values calculated for the enclosure in Fig. 1(b) are shown in Fig. 2(b) by black dots for the lowest three resonances for multipole sources of different orders. The resonances for higher multipoles all occurs in the subwavelength region D/λ<1 as in the cases of a monopole (m = 0) and dipole (m = 1). The frequencies for multipoles of different orders (m0) are degenerate at the same frequency as predicted. There is a frequency difference between the cases of a monopole m = 0 and multipoles m0. The difference is due to a contrast of acoustic fields in the inner region r<DI/2 where Jm=0(kr1)1 for a monopole source but Jm0(kr1)0 for a multipole source.

The degeneracy of the Mie resonant frequencies presented here differs from the common Mie resonances in an isotropic enclosure, whose resonant frequencies monotonically increase with multipole order m [red dots in Fig. 2(b)]. The degenerate Mie resonances enables simultaneous enhancement of multipole modes of a monochromatic source.

This section presents the implementation of a practical structure to realize the conceptual enclosure in Fig. 1(b). The structure is shown in Fig. 3(a) which has outer and inner diameters (D and DI=0.1D) that are same as the conceptual enclosure. The structure can be made of materials, e.g., brass (gray part), which are rigid enough for a stark contrast of acoustic impedance to the filled air (white part). The air-filled channels of the structure elongate the “acoustic path” (red line) to support the low value of cr, while the rigid walls that separate the channels support the anisotropy of an extremely high ρθ. Such a structure of space-coiling configuration by some modifications has other applications such as for reduced sound transmission9 and directivity sensing,10 where the anisotropic feature was not modeled though.

Fig. 3.

(Color online) A realistic structure for emission enhancement. (a) The structure is made of rigid materials (gray part), where the coiled air-filled channels elongate the sound path (red line) to reduce the equivalent sound speed cr along the radial direction and the rigid walls between the channels support the azimuthal anisotropy ρθ. (b) Phase distribution of sound fields radiated from a monopole source as simulated at three resonant frequencies [cf. Fig. 2(c)]. (c) Same as (b) but for a dipole source. (d,e) Comparison of far-field directivity with and without the enclosure simulated for the lowest resonance in (b) and (c), respectively.

Fig. 3.

(Color online) A realistic structure for emission enhancement. (a) The structure is made of rigid materials (gray part), where the coiled air-filled channels elongate the sound path (red line) to reduce the equivalent sound speed cr along the radial direction and the rigid walls between the channels support the azimuthal anisotropy ρθ. (b) Phase distribution of sound fields radiated from a monopole source as simulated at three resonant frequencies [cf. Fig. 2(c)]. (c) Same as (b) but for a dipole source. (d,e) Comparison of far-field directivity with and without the enclosure simulated for the lowest resonance in (b) and (c), respectively.

Close modal

The effective parameters (cr, ρr, ρθ) of the structure [cf. Fig. 3(a)] depend on geometrical parameters DI, D, channel number M, channel width w, and sound path length L of an individual channel. We derive this dependence as

cr=(DDI)cair2L,ρr=4πLρairrMw(DDI),ρθ,DI/2rD/2.
(4)

Briefly, cr is derived from the conservation of the sound traveling time along the radial direction, (DDI)/2cr=L/cair; ρr is derived from the impedance matching condition, i.e., crρr equals cairρair scaled by the structure's air-to-brass filling ratio that varies along r. We use 10 channels (M = 10) to maintain reasonable resolutions along the azimuth (as we will illustrate), though the choice is not unique. Then, to resemble the expected cr=0.266cair and ρr=25πρairr/D (cf. Fig. 1), we determine L=1.69D and w=0.03D. The resultant structure is the one illustrated in Fig. 3(a).

We conduct numerical simulations to validate the omnidirectional enhancement of emission. We use a centered source of various order m, and sweep D/λ values by sweeping frequencies. The D/λ values corresponding to the resonant peaks simulated in lossless cases are shown by green circles in Fig. 2(b), which agree well with the results of the conceptual enclosure [black dots in Fig. 2(b)].

The losses would reduce the enhancement near resonances. The amount of reduction depends on the structure size. As an example, for a structure of D = 10 cm, numerical simulations with losses included11 give three peak values of enhancement as P1/P0 = (81, 14, and 6) at D/λ = (0.13, 0.43, and 0.72), in comparison with the lossless case of (267, 52, and 28) at (0.14, 0.44, and 0.73), respectively. The phase distributions of sound fields in the structure at resonances are shown in Figs. 3(b) and 3(c) for monopole and dipole sources, respectively. Even with the losses included, the directivity pattern is still well preserved, as shown in Figs. 3(d) and 3(e) for the cases of monopole and dipole sources, respectively.

We have presented an extremely anisotropic enclosure in a subwavelength regime for omnidirectionally enhancing emission of monochromatic multipole sources of different orders. Degenerate Mie resonances induced by the extreme anisotropy lead to the emission enhancement at the same low frequencies, which guarantees the subwavelength feature of the enclosure. A structure whose anisotropy and degeneracy were not modeled in prior studies is herein used for achieving the emission enhancement. Our simulations validate that the emission is enhanced at frequencies surrounding the resonant frequencies, while the original directivity is preserved. The results are examined in two-dimensional sound fields, though it is possible to exend to three-dimensional designs.

The resonance-based emission enhancement has apparent disadvantages like the fact that (i) in the region between resonances, the amplitude of the emission relative to that of the source in free space becomes less than one [cf. Fig. 1(c)], and (ii) the amplitude of the resonant peaks decrease by increasing order of the resonances [cf. Figs. 1(c) and 2(a)]. With the degenerate resonances concerned here that do not depend on the multipole order, the emission of a real source composed by different multipoles will be primarily enhanced around the resonances. It would be of interest to extend the enhancement also to broadband sound sources. For that case maybe non-degenerate Mie resonances could be useful.

The concept of enclosing a sound source to enhance the emission could be an analogue of the quantum Purcell effect that modifies the spontaneous emission rate of a quantum source by changing the surrounding environment.12–16 The enhancement presented here remains to be experimentally verified, and if achieved, would be of broad impacts in various areas where minimized transducers are essential for efficient emission of either omnidirectional or directional sound waves.

This work is supported by King Abdullah University of Science and Technology, and by the University of Mississippi through the start-up fund and the 2017 College of Liberal Arts Summer Research Award.

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