Active structural acoustic illusions

We present active manipulation of the structural vibrations of an elastic body to generate an acoustic illusion. The resultant illusion misrepresents the nature, size and number of objects in the exterior acoustic domain. We demonstrate our technique, herein termed active structural acoustic illusion, using an elastic cylindrical shell. The radial motion of the shell at its cavity resonance frequencies is actively modified using localised mechanical forces. Acoustic illusions are generated to resemble the scattered acoustic field by one or more rigid cylinders of different size and location.

active control techniques. Employing an adaptive controller allows for variation in frequency and physical system behaviour to be accommodated, thus yielding the ability to misrepresent the nature of an object being perceived in real time.

Methods
Active control configuration and theory. The active control system comprises a multichannel control configuration with W control forces and L error sensors. We minimise a quadratic cost function based on a feedforward adaptive least-mean-square algorithm 32 . The first stage is the removal of the acoustic field at the error sensor locations (sound cancellation). The second stage generates the desired acoustic illusional field (sound reproduction). Our control approach is applied to a two-dimensional elastic cylindrical shell. The physical system responses associated with the original and illusional objects must be established prior to implementing the control process. The mathematical formulation to describe the acoustic fields of an elastic cylindrical shell (original object) and illusion object (one or two rigid cylinders) is provided in the Supplementary Material 33,34 . The quadratic cost function corresponding to the sum of the squared acoustic pressures at each error sensor location is given by H where e is a vector denoting the acoustic field at the error sensors and H denotes the Hermitian transpose operator. The error vector can be expressed as structural illusion where p structural denotes the shell's structure-borne sound pressure arising from excitation due to a plane wave (Eq. (3) in the Supplementary Material) or a monopole source (Eq. (4) in the Supplementary Material). Z is an × L W matrix of complex transfer functions representing the radiated field due to W control forces (Eq. (7) in the Supplementary Material is the sound field for single force excitation of the shell). p illusion is a vector representing the acoustic pressure of the desired illusion at the L error sensor locations (Eq. (13) in the Supplementary Material represents the acoustic pressure at a single error sensor). q is the vector of W optimised complex control force amplitudes. Substituting Eq. (2) into Eq. (1), differentiating the resulting expression with respect to the real and imaginary components of q and equating to zero, that is, ∂ ∂ = J q / 0 , the optimised control force amplitudes can be obtained as The inversion of the matrix Z Z H is carried out using a Moore-Penrose pseudoinverse function. Since Z Z H is a Hermitian, positive-definite matrix, the inverse always exists and is unique 22 . The inversion is well-behaved provided the number of error sensors is greater than the number of control sources as this ensures the least-mean-squares problem is overdetermined. The aforementioned active control method utilises a priori knowledge of the incident field. There are limited strategies to overcome this issue, however the ability to measure and deduce key information about the incident field can be found in [35][36][37] .
We demonstrate our active structural acoustic illusion technique on a 2D elastic cylindrical shell of thickness h = 4 mm and radius a = 0.6 m measured to mid-plane thickness, as shown in Fig. 1. The shell material is polyvinyl chloride with properties corresponding to density 1300 kg/m 3 , Young's modulus 2.9 GPa and Poisson's ratio The primary acoustic field is due to an incident plane wave or monopole source located 4a upstream of the shell. 0.3. The interior and exterior fluid media correspond to air with density and sound speed of .
1 225 kg/m 3 and 343 m/s, respectively. Localised control forces were distributed equally around the shell. Similarly, error sensors were distributed circumferentially around the shell at a radial distance of 3a from the shell centre.

Results
All results were generated using MATLAB. The number of control forces and error sensors implemented in the control configuration for each illusion case study were determined using a similar procedure outlined in ref. 38 based on a percentage error function given by illusion controlled illusion where p controlled is a vector of the actively modified acoustic pressure at the error sensors. The number of control forces and error sensors for each illusion were obtained for the minimum value of ∆.
Single body illusions. We first demonstrate results for single body illusions misrepresenting the nature, size and location of the cylindrical shell. Figure 2(a) presents the acoustic field arising from a monopole source located at a 4 upstream of the cylindrical shell at an excitation frequency of 283.4 Hz. This frequency corresponds to an internal (0,2)-resonance within the shell cavity as indicated by the two radial nodal lines 39 . Significant scattering can be observed downstream of the shell. Figure 2(b) shows the desired acoustic illusional fields associated with larger and smaller rigid cylinders that are shifted from the centre of the original cylindrical shell. The top panel corresponds to the scattered acoustic field for a rigid cylinder of radius 2a centred at a radial distance of a and an angle of 120° to the horizontal axis from the centre of the shell. The bottom panel corresponds to the scattered acoustic field for a rigid cylinder of radius . a 0 5 centred at a radial distance of . a 2 5 and an angle of 120° to the horizontal axis from the centre of the shell. Greater asymmetric acoustic scattering is observed for the larger cylinder. The active structural acoustic illusions are presented in Fig. 2(c), in which the radial motion of the shell in Fig. 2(a) is actively modified such that the structure-borne pressure yields the acoustic field in Fig. 2(b). Beyond the radial location of the error sensors (indicated by a white dashed line), the actively generated acoustic field matches the desired illusional field. The control configuration utilised 19 forces and 35 error sensors for Fig. 2  www.nature.com/scientificreports www.nature.com/scientificreports/ Multiple body illusions. We now generate active illusions which are capable of misrepresenting the size, location and number of objects. Figure 3(a) shows the acoustic field due to an incident plane wave impinging on the elastic cylindrical shell at an excitation frequency of 387.6 Hz. This frequency corresponds to an internal (0,3)-resonance within the shell cavity as indicated by the three radial nodal lines 39 . The desired acoustic field corresponds to acoustic scattering by two rigid cylinders. The first case comprises two cylinders each of radius a and located 2a directly above and below the centre of the shell, resulting in symmetric scattering ( Fig. 3(b), top). The second case comprises a rigid cylinder of radius . a 0 5 located 2a directly above the centre of the shell in conjunction with a rigid cylinder of radius . a 1 5 located . a 1 5 directly below the centre of the shell, resulting in asymmetric scattering (Fig. 3(b), bottom). Figure 3(c) presents the acoustic illusions whereby the actively modified shell displacement has effectively reproduced the scattered acoustic field arising from multiple objects. The control configuration utilised 21 forces and 61 error sensors for Fig. 3(c) (top), and 21 forces and 50 error sensors for Fig. 3(c) (bottom).

Limitation of active structural acoustic illusions. The performance of the proposed technique to
actively generate an acoustic illusion is examined. Figure 4 presents the deviation (top row) and acoustic directivity (bottom row) of the actively controlled field with respect to the desired illusional field. The deviation corresponds to the absolute difference between the actively controlled acoustic field (in dB) and the desired acoustic illusion (in dB). The directivity corresponds to the absolute acoustic pressure (in Pascal) evaluated for all field points along the circumference at a specific radial distance. The acoustic field of the original system corresponds to that for a cylindrical shell of radius a under monopole excitation at 283.4 Hz, as shown in Fig. 2(a). For the deviation in Fig. 4(a), the desired acoustic illusion corresponds to that for a rigid cylinder of radius . a 0 5 concentrically located with the original cylindrical shell. Similarly, for the deviation in Fig. 4(b,c), the desired acoustic illusion corresponds to that for a rigid cylinder of radius . a 0 5 at a radial distance of . a 1 5 and . a 2 5 in the forward-scatter region, respectively. It is important to note that the error sensors must always fully enclose the boundary of the illusion object and are located at a radial distance of 3a. In all cases considered, the deviation between the actively controlled field and desired illusional field beyond the perimeter of the error sensors is less than 1 dB. When the illusion cylinder is completely enclosed by the original cylindrical shell (Fig. 4(a)), the illusional field takes effect beyond the surface of the shell. However, as the illusion cylinder is moved away from the original cylindrical shell (Fig. 4(b,c)), the illusional field takes effect beyond the error sensor perimeter. Furthermore, as the distance between the original and illusion objects increases, the performance of the acoustic illusion decreases, with a corresponding increase in control effort. The control configurations to generate the results in Fig. 4(a-c)   www.nature.com/scientificreports www.nature.com/scientificreports/ sensors, respectively. The acoustic directivity for each case shown in the bottom row of Fig. 4 highlights that there is no noticeable deviation between the actively controlled pressure and desired illusion pressure.

Discussion
In summary, we present a method whereby the radial motion of an elastic cylindrical shell is actively modified to generate an acoustic illusion. Illusions were generated to resemble the acoustic field due to scattering by a rigid object of different size and location, as well as an illusion associated with acoustic scattering by multiple bodies. A correlation in superior performance of the active structural acoustic illusion with greater proximity between the original and desired objects was observed. Our approach presents a foundation into practical implementation of acoustic illusions and can be employed in applications where masking of unwanted sound or misrepresentation of targets during stealth operations is required.

Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.