Far-field directivity of parametric loudspeaker arrays set on curved surfaces

Parametric loudspeaker arrays comprise arrangements of piezoelectric ultrasonic transducers that emit highly directive audible sound, thanks to the nonlinear parametric array phenomenon. Most parametric loudspeakers consist of planar arrays of transducers though, lately, devices have been developed which involve their distribution on curved surfaces. In this work, we present an extension of the recently proposed convolution model for planar arrays to predict the far field directivity of curved parametric loudspeakers. An expression is first given to compute the audible secondary pressure field generated by a single ultrasonic transducer placed at any point on a general curved surface, and pointing in its normal direction. Assuming weak non-linearity, the total audible pressure produced by all transducers on the surface is then recovered from the superposition principle. As an application, we predict the far-field pressure generated by an omnidirectional parametric loudspeaker consisting of hundreds of ultrasonic transducers set on a sphere. A critical aspect for the performance of the omnidirectional source is that of finding a proper distribution for them on the spherical surface. Two options are analyzed: getting an optimal solution for a Fekete-like problem, and resorting to an equal-area partitioning scheme, which is more feasible for a practical construction of the source. Numerical simulations are carried out for both alternatives.