To ensure the safe operation of many safety critical structures such as nuclear plants, aircraft and oil pipelines, non-destructive imaging is employed using piezoelectric ultrasonic transducers. These sensors typically operate at a single frequency due to the restrictions imposed on their resonant behaviour by the use of a single length scale in the design. To allow these transducers to transmit and receive more complex signals it would seem logical to use a range of length scales in the design so that a wide range of resonating frequencies will result. In this thesis we derive a mathematical model to predict the operating characteristics of an ultrasound transducer that achieves this range of length scales by adopting a fractal architecture; the fractal in this case being the Sierpinski gasket. Expressions for the electrical and mechanical fields that are contained within this structure are expressed in terms of a finite element basis. A renormalisation approach is then used to calculate the key components from the discrete matrices that arise. The propagation of an ultrasonic wave in this transducer is then analyzed and used to derive expressions for the non-dimensionalised electrical impedance and the transmission and reception sensitivities as a function of the driving frequency. Comparing these key performance measures of the fractal transducer to an equivalent standard (Euclidean) design shows that the fractal devices have a significantly higher reception sensitivity and a significantly wider bandwidth.