Origami acoustics: using principles of folding structural acoustics for simple and large focusing of sound energy

A lossless, structural-acoustic model is formulated to gain essential insights by predicting the acoustic pressure at far field point $p(R,\beta ,\phi ,t)-a$ distance $R,$ elevation angle $\beta ,$ and azimuthal angle $\phi$ away from the origin—due to the single-frequency, normal surface acceleration of each parallelogram assembled into an array of the Miura-ori cells numbering ${M}_{x}$ by ${M}_{y}$ in the $x$ and $y$ axes, respectively, illustrated in figure 1(a). In other words, each parallelogram facet is assumed to vibrate like an ideal, rigid, baffled piston such that pressure waves are radiated into the acoustic fluid above the plane $(x,y,0).$ Because of the folding-induced rotation and translation of the parallelogram-shaped and piston-like radiators, the phase differences of pressure waves arriving at the field point from each facet superimpose uniquely based on the folding angle $\theta$ and field point location $(R,\beta ,\phi ).$ Thus, as provided by this new model formulation, a direct link is created between the kinematically-defined formation of the tessellated array and the acoustic spectral and spatial sensitivities for pressure radiation to the field point. In fact, this technical approach may be compared to certain shape-changing, adaptive radio frequency (RF) antennae [45, 46] including recent advancements to create frequency selective surfaces by folding RF antennae in patterns inspired by origami tessellations [47].

When the mechanical impedance of a baffled, vibrating structural surface is much greater than the acoustic impedance (such as for a stiff structural panel interfaced with air, hereafter assumed), the relation between the spatially distributed harmonic oscillations of the surface and the resulting pressure change at a point in space away from the surface is examined by evaluating Rayleigh's integral [43]:

$$\begin{eqnarray}&&p(R,\beta ,\phi ,t)={\rm{j}}\displaystyle \frac{{\rho }_{0}\omega {u}_{0}}{2\pi }{{\rm{e}}}^{{\rm{j}}\omega t}\displaystyle {\int }_{A}\displaystyle \frac{{{\rm{e}}}^{-{\rm{j}}kr}}{r}{\rm{d}}A.\end{eqnarray} \tag{ 1 }$$

Figure 1(b) details the structural acoustic unit cell for a folded configuration $\theta \gt 0$ while figure 1(c) presents the unfolded topology and corresponding notations. From equation (1), one has that ${\rho }_{0}$ = 1.104 kg m⁻³ is the air density; $\omega$ is the angular frequency of oscillation of the surface; ${u}_{0}$ is the spatially-uniform, normal, velocity amplitude of the structure-fluid interfacing surface; $A$ is the surface area of the array; $k=\omega /c$ is the acoustic wavenumber, with $c$ = 340 m s⁻¹ as the sound speed; while $r$ is the distance from an oscillating differential area element ${\rm{d}}A$ to the far field pressure point $p(R,\beta ,\phi ,t)$ that is a distance $R$ from the reference origin and angularly positioned according to angles $\beta$ in elevation and $\phi$ in azimuth. The Miura-ori cell is repeated in the $x-y$ plane by ${M}_{x}$ and ${M}_{y}$ times, respectively, to construct the tessellated array, figure 1(a). Focusing first on the unfolded topology, figure 1(c), it is assumed that each parallelogram-shaped, piston-like facet contributes to the total integration of equation (1) while the array is itself enclosed in a rigid baffle where the normal velocity is zero. It is also assumed that the array remains attached to the $x-y$ plane and does not exhibit modal oscillations associated with the whole structural dimensions. These three assumptions are easily met in the experiments as described in the following section.

Geometrically, the distance $r$ is related to the reference distance $R$ according to

$$\begin{eqnarray}&&r={[{R}^{2}+{r}_{0}^{2}-2R{r}_{0}\sin \beta \cos (\phi -\alpha )]}^{1/2}.\end{eqnarray} \tag{ 2 }$$

This expression can be reduced assuming that the location of the pressure point is in the far field. Subject to far field approximation [48], the $r$ becomes $r\approx R$ in the denominator of equation (1), while the complex exponential (oscillatory) argument must retain the fundamental phase information according to

$$\begin{eqnarray}&&r\approx R-r\,\sin \,\beta \,\cos (\phi -\alpha ).\end{eqnarray} \tag{ 3 }$$

Substituting these results along with geometric relations—$x=r\,\cos \,\alpha ,$ $y=r\,\sin \,\alpha ,$ and ${\rm{d}}A={\rm{d}}x{\rm{d}}y-into$ equation (1), one obtains for the unit cell

$$\begin{eqnarray}&&{p}_{{\rm{cell}}}(R,\beta ,\phi ,t)={\rm{j}}\displaystyle \frac{{\rho }_{0}\omega {u}_{0}}{2\pi }{{\rm{e}}}^{{\rm{j}}\omega t}\displaystyle \frac{{{\rm{e}}}^{-{\rm{j}}kR}}{R}\\ &&\quad \quad \quad \quad \quad \quad \quad \times \displaystyle {\int }_{A}{{\rm{e}}}^{{\rm{j}}({{\rm{\Psi }}}_{1}x+{{\rm{\Psi }}}_{2}y)}{\rm{d}}x{\rm{d}}y\end{eqnarray} \tag{ 4 }$$

given that ${{\rm{\Psi }}}_{1}=k\,\sin \,\beta \,\cos \,\phi$ and ${{\rm{\Psi }}}_{2}=k\,\sin \,\beta \,\sin \,\phi$.

Due to the folding of the Miura-ori array, the unfolded spatial extents ${S}_{0},$ ${L}_{0},$ and ${V}_{0}$ , figure 1(c), are modified to [28]:

$$\begin{eqnarray}&&S=b\displaystyle \frac{\cos \,\theta \,\tan \,\gamma }{\sqrt{1+{\cos }^{2}\theta {\tan }^{2}\gamma }},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&L=a\sqrt{1-{\sin }^{2}\theta {\sin }^{2}\gamma },\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&V=b\displaystyle \frac{1}{\sqrt{1+{\cos }^{2}\theta {\tan }^{2}\gamma }},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&H=a\,\sin \,\theta \,\sin \,\gamma ,\end{eqnarray} \tag{ 8 }$$

where $H$ is the height of the Miura cell induced by fold angles $\theta \gt 0,$ figure 1(b). During folding, the geometric center of each parallelogram facet increases in height from the reference plane by the same amount $H/2.$ This quantity is negligible in comparison to $R$ when employing far field approximations, and thus is safely neglected from the following computations of the Rayleigh's integral since it contributes to a minor change in the term that influences all angular coordinates uniformly according to $\tfrac{{{\rm{e}}}^{-{\rm{j}}kR}}{R}\to \tfrac{{{\rm{e}}}^{-{\rm{j}}k(R-H/2)}}{R-H/2}.$ On the other hand, the folding causes each facet to translate and rotate with respect to the reference $(x,y,z)$ system. These motions modify the appropriate area integration limits for each facet in equation (4) and alter the far field elevation angle $\beta$ region over which a given facet contributes a directly radiated pressure wave to the far field point. To account for these influences of folding, equation (4) is written for the unit cell as

$$\begin{eqnarray}&&{p}_{{\rm{cell}}}(R,\beta ,\phi ,t)={\rm{j}}\displaystyle \frac{{\rho }_{0}\omega {u}_{0}}{2\pi }{{\rm{e}}}^{{\rm{j}}\omega t}\displaystyle \frac{{{\rm{e}}}^{-{\rm{j}}kR}}{R}\displaystyle \sum _{i=1}^{4}{v}_{i},\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{v}_{1}=\displaystyle {\int }_{0}^{{S}_{0}}\displaystyle {\int }_{{y}_{1}(x)}^{{y}_{2}(x)}{{\rm{e}}}^{{\rm{j}}{{\rm{\Psi }}}_{11}y}{\rm{d}}y\cdot {{\rm{e}}}^{{\rm{j}}{{\rm{\Psi }}}_{12}x}{\rm{d}}x\,{\rm{for}}\,{\rm{parallelogram}}\,1,\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{v}_{2}=\displaystyle {\int }_{S}^{S+{S}_{0}}\displaystyle {\int }_{{y}_{3}(x)}^{{y}_{4}(x)}{{\rm{e}}}^{{\rm{j}}{{\rm{\Psi }}}_{21}y}{\rm{d}}y\cdot {{\rm{e}}}^{{\rm{j}}{{\rm{\Psi }}}_{22}x}{\rm{d}}x\,{\rm{for}}\,{\rm{parallelogram}}\,2,\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{v}_{3}=\displaystyle {\int }_{0}^{{S}_{0}}\displaystyle {\int }_{{y}_{5}(x)}^{{y}_{6}(x)}{{\rm{e}}}^{{\rm{j}}{{\rm{\Psi }}}_{31}y}{\rm{d}}y\cdot {{\rm{e}}}^{{\rm{j}}{{\rm{\Psi }}}_{32}x}{\rm{d}}x\,{\rm{for}}\,{\rm{parallelogram}}\,3,\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{v}_{4}=\displaystyle {\int }_{S}^{S+{S}_{0}}\displaystyle {\int }_{{y}_{7}(x)}^{{y}_{8}(x)}{{\rm{e}}}^{{\rm{j}}{{\rm{\Psi }}}_{41}y}{\rm{d}}y\cdot {{\rm{e}}}^{{\rm{j}}{{\rm{\Psi }}}_{42}x}{\rm{d}}x\,{\rm{for}}\,{\rm{parallelogram}}\,4,\end{eqnarray} \tag{ 13 }$$

where, for the Miura-ori cell in the ith row of an array with ${M}_{y}$ rows, the integration limits are

$$\begin{eqnarray}&&{y}_{1}(x)=\displaystyle \frac{{V}_{0}}{{S}_{0}}x;{y}_{2}(x)={L}_{0}+\displaystyle \frac{{V}_{0}}{{S}_{0}}x,\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{y}_{3}(x)=-\displaystyle \frac{{V}_{0}}{{S}_{0}}x+{V}_{0}+\displaystyle \frac{{V}_{0}}{{S}_{0}}S;\,{y}_{4}(x)=-\displaystyle \frac{{V}_{0}}{{S}_{0}}x+{L}_{0}\\ &&\quad \quad \quad \,\,+{V}_{0}+\displaystyle \frac{{V}_{0}}{{S}_{0}}S,\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{y}_{5}(x)=\displaystyle \frac{{V}_{0}}{{S}_{0}}x+L;\,{y}_{6}(x)=L+{L}_{0}+\displaystyle \frac{{V}_{0}}{{S}_{0}}x,\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{y}_{7}(x)=-\displaystyle \frac{{V}_{0}}{{S}_{0}}x+L+{V}_{0}+\displaystyle \frac{{V}_{0}}{{S}_{0}}S;{y}_{8}(x)=-\displaystyle \frac{{V}_{0}}{{S}_{0}}x\\ &&\quad \quad \quad \,\,\,\,+L+{L}_{0}+{V}_{0}+\displaystyle \frac{{V}_{0}}{{S}_{0}}S.\end{eqnarray} \tag{ 17 }$$

The complex exponential arguments utilize the expressions

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{11}=k\,\sin (\beta +\theta )\cos \,{\phi }_{1};{{\rm{\Psi }}}_{12}=k\,\sin (\beta +\theta )\sin \,{\phi }_{1}\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{21}=k\,\sin (\beta +\theta )\cos \,{\phi }_{2};{{\rm{\Psi }}}_{22}=k\,\sin (\beta +\theta )\sin \,{\phi }_{2}\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{31}=k\,\sin (\beta -\theta )\cos \,{\phi }_{1};{{\rm{\Psi }}}_{32}=k\,\sin (\beta -\theta )\sin \,{\phi }_{1}\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{41}=k\,\sin (\beta -\theta )\cos \,{\phi }_{2};\,{{\rm{\Psi }}}_{42}=k\,\sin (\beta -\theta )\sin \,{\phi }_{2}\end{eqnarray} \tag{ 21 }$$

having defined

$$\begin{eqnarray*}&&{\phi }_{1}=\phi +\alpha -{\alpha }^{* },\end{eqnarray*}$$

$$\begin{eqnarray*}&&{\phi }_{2}=\phi -\alpha -{\alpha }^{* },\end{eqnarray*}$$

$$\begin{eqnarray*}&&{\alpha }^{* }=(\pi /2)-\gamma ,\end{eqnarray*}$$

$$\begin{eqnarray*}&&\alpha (\theta )={\cos }^{-1}[\cos \,\theta \,\tan \,\gamma {(1+{\cos }^{2}\theta {\tan }^{2}\gamma )}^{-\mathrm{1/2}}].\end{eqnarray*}$$

The computation of the Rayleigh's integral equation (9) is subject to the following constraints

$$\begin{eqnarray}&&{\rm{if}}\,\beta +\theta \lt \pi /2\,{\rm{and}}\,\beta -\theta \gt -\pi /2,\,{\rm{then}}\,{p}_{{\rm{cell}}}(R,\beta ,\phi ,t)\\ &&\quad ={\rm{j}}\displaystyle \frac{{\rho }_{0}\omega {u}_{0}}{2\pi }{{\rm{e}}}^{{\rm{j}}\omega t}\displaystyle \frac{{{\rm{e}}}^{-{\rm{j}}kR}}{R}\displaystyle \sum _{i=1}^{4}{v}_{i}.\end{eqnarray} \tag{ 22 }$$

For elevation and folding angle combinations outside of the range indicated in equation (22), one of the facet types, numbered 1–4 in figure 1(c), exhibits an acoustic shadow with respect to the point in the far field, as illustrated in the inset of figure 1(b). As a result, predictions may be compromised by the fact that other acoustic phenomena like reflection and diffraction would non-trivially contribute to the radiated acoustic pressure at the field point, although only the direct, line-of-sight radiation is accounted for in Rayleigh's integral. Thus, model results determined for angular locations that fall outside of the constraints of equation (22) are omitted. We may also anticipate that model predictions approaching these limits exhibit progressively poorer accuracy with respect to the more nuanced acoustic wave front that truly reaches the field point.

Equation (22) therefore predicts the sound pressure directional sensitivities of the Miura-ori acoustic transducer unit cell. Array design principles are employed to account for the assembly of the cells into an array that spans the $x-y$ plane. Namely, the directivity of arrays of directional transducers may be determined by the product of the transducer unit cell directivity function and the array directivity function [14]. Here, the unit cell directivity is given in equation (22) by the summation. To account for ${M}_{x}$ number of these columns and ${M}_{y}$ number of these rows in the two-dimensional planar array, the array directivity function is [49]:

$$\begin{eqnarray}&&D(\beta ,\phi )=\displaystyle \frac{\sin [{M}_{x}kS\,\sin \,\beta \,\cos \,\phi ]}{{M}_{x}\,\sin [kS\,\sin \,\beta \,\cos \,\phi ]}\\ &&\quad \quad \quad \quad \quad \times \displaystyle \frac{\sin [{M}_{y}kL\,\sin \,\beta \,\sin \,\phi ]}{{M}_{y}\,\sin [kL\,\sin \,\beta \,\sin \,\phi ]}.\end{eqnarray} \tag{ 23 }$$

The product of this directivity, equation (23), with the directivity determined from the summation in equation (22) yields the total Miura-ori acoustic array directivity function. Multiplying the total directivity by the leading terms gives the harmonic pressure change (above and below static pressure) at the field point

$$\begin{eqnarray}&&p(R,\beta ,\phi ,t)={\rm{j}}\displaystyle \frac{{\rho }_{0}\omega {u}_{0}}{2\pi }{M}_{x}{M}_{y}{{\rm{e}}}^{{\rm{j}}\omega t}\displaystyle \frac{{{\rm{e}}}^{-{\rm{j}}kR}}{R}D(\beta ,\phi )\\ &&\quad \quad \quad \quad \quad \quad \times \displaystyle \sum _{i=1}^{4}{v}_{i}.\end{eqnarray} \tag{ 24 }$$

Hereafter, the pressure equation (24) is normalized with respect to the total number of transducer facets given by the product $4{M}_{x}{M}_{y}$ so that arrays of different numbers of facets and cells can be more effectively compared.

A conventional metric is the sound pressure level (SPL), computed from the pressure change $p(R,\beta ,\phi ,t),$ equation (24), with respect to a reference pressure, here ${p}_{{\rm{ref}}}$ = 20 μPa. The SPL is employed here to quantify the means by which topological reconfiguration of the tessellated array results in transmission of the acoustic energy to the field point at the angular coordinates $(\beta ,\phi ).$ By the usage here, the SPL is evaluated in decibels, according to convention

$$\begin{eqnarray}&&{\rm{SPL}}=20\,{\mathrm{log}}_{10}\left[\displaystyle \frac{{p}_{\mathrm{rms}}(R,\beta ,\phi ,t)}{{p}_{{\rm{ref}}}}\right],\end{eqnarray} \tag{ 25 }$$

where the subscript rms indicates the root-mean-square value.

The far field pressure spans azimuthal angles $\phi \in [0,\pi ]rad$ and elevation angles $\beta \in (-\pi /2,\pi /2)rad$ where $\beta =0$ is termed 'broadside' radiation [43]. In the following studies, we use the results of equation (25) to characterize the potential for this new concept of integrating origami-based engineering design and structural acoustics to cultivate large and simple acoustic energy-focusing capabilities via the folding of a Miura-ori tessellated transducer array.

Several steps are undertaken to fabricate the experimental specimens whose measurements are reported here. Additional experimental details are given in the supplementary information. Specimens are created using polypropylene sheet (McMaster-Carr 1451T21) of thickness 0.762 mm for the underlying structural surface of the array. The shape of a Miura-ori pattern is cut and scored on the sheet using a laser cutter (Full Spectrum Laser H-20x12). The edge lengths and edge angle of the specimens are, respectively, a = 33 mm, b = 30 mm, and γ = 50°. Once scored, the Miura-ori pattern on the sheet is appropriately folded.

Piezoelectric PVDF, of 28 μm thickness, with sputtered Cu–Ni electrodes (Measurement Specialties, 1-1003702-7) is cut into Miura-ori unit cell shapes with edge lengths $a$ and $b$ scaled to approximately 80% of the size used to prepare the polypropylene sheet. The size reductions are intended to prevent shorting between adjacent cells of PVDF. The laser cutter is used for cutting the PVDF. Copper tape with electrically conductive adhesive is bonded to adjacent PVDF cells and one final lead is used for connection to the amplified excitation; all upper electrodes are connected together and likewise for all lower electrodes. Thin, double-sided tape is used to bond the PVDF shapes to the polypropylene sheet at a time when it is folded to approximately an angle $\theta \ =\ 30^\circ ,$ which readily enables unfolding $\theta \ =\ 0^\circ$ and permits greater folding to approximately $\theta \ =\ 60^\circ$ at which extent the adherence of the PVDF to the folded cell is moderately reduced due to stretching. Two external leads are used to connect all upper or lower electrodes of the tessellated array; these two leads are then connected to the amplified excitation signal. The specimen whose measurements are reported here is shown in figure 2(a).

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In figure 2(a), it is seen that a perimeter of non-activated facets surrounds the part of the tessellated array that is covered and activated by the PVDF. This approach is found to better baffle the array at the boundary between active and non-active facets, because the folded edges are immediately connected to other non-active surfaces (those with zero normal velocity). Practically, such an approach satisfies the baffle-related assumptions inherent in the model formulation. Beyond the effective baffle created by the non-active perimeter of facets, a more traditional baffle of medium density fiberboard (MDF) holds the transducer array in the plane during experimentation (supplementary figure 1). To entirely prevent curvature of the specimen due to the constraints imposed by the MDF baffle, small pins secure the bottom-most nodes of the tessellated array to a hard plane upon which the array rests. This setup is then positioned on a bed of acoustic foam housed in a chamber which is anechoic for frequencies greater than approximately 1.5 kHz. The array specimen is positioned at the center of a hemispherical track of radius 717.55 mm. This radial distance to the specimen plane is sufficiently well into the acoustic far field according to the standard requirements for the frequencies measured here [48]. The track is used as a guide for a traversing microphone (PCB Piezotronics 130E20) to move along elevation angle $\beta \in [0,\,5\pi /12]rad$ with $\phi \ =\ 0^\circ$ that measures the far field acoustic pressure radiated from the transducer at the center, figure 2(b). For the experiments, a function generator (Siglent SDG1025) creates a single frequency wave sent to an amplifier (AudioSource AMP100) and then through an audio output transformer (RadioShack 2731380) to increase the voltage delivered to the piezoelectric PVDF that drives the transducer array. A National Instruments data acquisition system (NI 9220) obtains the pressure measurements which are evaluated through the MATLAB Data Acquisition Toolbox for processing. Data is digitally filtered using a fourth-order Butterworth filter from 1 to 18 kHz prior to evaluating the far field SPL.

A comparison of the far field (SPL, dB ref. 20 μPa) predicted from our analytical model and that measured experimentally for two folding angles and three frequencies is presented in figure 3. The top row shows the experimental data across all elevation angles $\beta$ measured while the bottom row presents analytical results for elevation angles that do not induce acoustic shadows. The model predictions in the bottom row are truncated within the elevation angles around broadside ($\beta =0)$ that meet criteria according to equation (22). The measurements in the top row are shown with lighter shading if such elevation locations are not valid with respect to the model and thus do not permit comparison.

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At the low 3 kHz frequency, figure 3 column (a), the SPL is seen to gradually reduce as elevation angle $\beta$ increases for the slightly folded topology $\theta \ =\ 9^\circ .$ The increased folding angle θ = 32° leads to greater directional uniformity in energy propagation. These trends are observed experimentally (top row figure 3(a)) as well as predicted by the model (bottom row), and are consistent with low frequency trends for (non-foldable) acoustic arrays in general [43]. The experimental measurements at 3 kHz exhibit minor deviations in the smoothness of SPL change according to variation in elevation angle. This may be explained by the fact that only 6.33 acoustic wavelengths separate the specimen from the microphone measurement point; thus, although this is technically a far field location [48], it is possible that near field effects may distort the measurements at this low frequency to cause the less smooth change in SPL according to change in elevation angle.

At 5 kHz, figure 3 column (b), a pressure node of significant SPL decrease appears around elevation angle $\beta =\pi /4rad$ (analytical) and $\beta =\pi /3rad$ (experimental) that rotates in angular location as folding increases. Thus, an observer in the far field remains in a zone of relative silence by rotating along $\beta$ respecting the reference $x-y$ plane while the tessellated array is folded from θ = 9° to 32°. Due to the inherent losses in air and imperfections in fabrication, we experimentally measure that this pressure node at 5 kHz is approximately 20 dB less in amplitude than the broadside ($\beta \ =\ 0)$ SPL, which is a two order-of-magnitude difference in sound power. In other words, folding the array when driven at 5 kHz introduces a powerful sound-focusing functionality by modulating the breadth of the major lobe that is centered around broadside. At 9 kHz with the array folded to $\theta \ =\ 9^\circ ,$ solid curves in figure 3 column (c), two pressure nodes appear experimentally and analytically, effecting about 20 dB depth each (100 times power difference) in our measurements. Upon folding from $\theta \ =\ 9^\circ$ to 32°, dashed curves in figure 3 column (c), one node of pressure remains but at a different orientation in elevation $\beta ,$ likewise exemplifying a novel, folding-induced means to tune the concentration of energy focused within the major lobe.

The adaptive acoustic performance observed in figure 3 according to the array actuation frequency encourages an exploration of how the energy is variably focused across a wide bandwidth of frequencies at broadside, which is a common location at which acoustic waves are sought to be directed since it helps to characterize fundamental array capabilities [11]. Figure 4 presents exemplary model results for the influences of driving frequency and the edge angle $\gamma$ on broadside SPL for the 3 × 2 cell array. For clearer understanding of the influence of the edge angles $\gamma$ upon the system design, three 3 × 2 tessellated array configurations are shown in figure 4(b) each having a different edge angle.

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For the mostly unfolded array configurations $\theta \ =\ 9^\circ ,$ figure 4(a), an insignificant adaptation in SPL is effected for a given excitation frequency by adjustment of the edge angle $\gamma$ design variable. At higher frequencies and greater fold angles, figures 4(b) and (c), a substantially greater change in the normalized SPL may occur by tailoring the array edge angle. The white data points exemplify such change. At fold angle $\theta \ =\ 32^\circ$ and edge angle γ = 29.7°, figure 4(b), the acoustic energy around 3.3 kHz is over 25 dB greater than that at 5.1 kHz (>300 times difference in power). In contrast, figure 4(c) shows that the same frequency and edge angle comparisons for an array folded to θ = 60° reverses the energy focusing trend: now, the sound focused at broadside at 5.1 kHz is 20 dB greater than that achieved at 3.3 kHz. In other words, the spectral sensitivities feature significant and tunable acoustic power focusing at broadside according to the reversible array folding and facet design.

The results of figures 4(b) and (c) suggest that dynamic folding can facilitate significant, on-demand shaping of single-frequency sound power received by an observer in the far field at broadside. Thus, in contrast to active control of each individual transducer in a conventional two-dimensional array (i.e. beamforming), the straightforward folding of the tessellated architecture can give rise to large sound focusing at a location in space (i.e. beamfolding). In this way, the kinematic reconfiguration of the Miura-ori-based array, governed by the relations equations (5)–(8), results in a pre-determined degree of energy focusing, thus directly and simply relating the tessellated array configuration and design to acoustic performance characteristics.

As determined by the model and presented in figure 5, we characterize the influence of fold angle $\theta$ upon the far field SPL of a Miura-ori-based array spanning 3 × 2 and 10 × 9 cells' worth of the tessellation. Unshaded areas in figure 5 indicate that acoustic shadows may occur and the model composition is not suitable for prediction under such circumstances. For the 3 × 2 array at 5 kHz, figure 5(a), folding the array from about 10° to 40° can modulate the SPL by approximately 30 dB while further folding to 60° can raise up the SPL once again by 20 dB from the local minima near 40°. Similar influences are observed for the 10 × 9 array, figure 5(c), but the greater number of array elements dramatically intensifies the energy concentration to the major lobe centered on broadside. While several sidelobes of reduced amplitude appear, they are separated from the major lobe by deep pressure nodes that yield effective silence in the far field. Figures 5(b) and (d) show that similar trends occur for the 3 × 2 and 10 × 9-sized arrays, respectively, at 9 kHz, although now a local maxima of SPL occurs around a fold angle of 40° and the major lobes are considerably narrower than at 5 kHz, providing a more intense focusing of 9 kHz acoustic waves at broadside. Resulting from the topological reconfiguration, it is apparent that folding a tessellated acoustic array introduces a simple means to substantially tailor and morph the acoustic energy received in the far field. For example, at 9 kHz and with a 3 × 2 Miura-ori tessellated arrays, figure 5(b), dynamic folding from about 15° to 47° can promote about a 26 dB change in the SPL, which is over a 300 times difference in acoustic power. Indeed, our experimental demonstrations of the dramatic acoustic amplitude modulation driven by folding (figure 6, supplementary figure 3, and listen to supplementary audio files 1–4) exemplify the potency of this new bridge between acoustics and origami-based science and design.

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