Ultrasonic Wave Propagation in Heterogeneous Media

Piezoelectric ultrasonic transducers typically employ composite structures to improve their transmission and reception sensitivities. The geometry of the composite is regular with one dominant length scale and, since these are resonant devices, this dictates the central operating frequency of the device. In order to construct a wide bandwidth device it would seem natural therefore to utilise resonators that span a range of length scales. In this paper we consider such a device and build a theoretical model to predict its performance. A fractal medium is used as this contains a wide range of length scales and yields to a renormalisation approach. The propagation of an ultrasonic wave in this heterogeneous medium is then analysed and used to construct expressions for the electrical impedance, and the transmission and reception sensitivities of this device as a function of the driving frequency.