IMA Journal of Applied Mathematics

December 11, 2014

IMA Journal of Applied

  1. Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK
  1. *Corresponding author: ebrahem.algehyne{at}, anthony.mulholland{at}
  • Received April 22, 2014.
  • Revision received March 26, 2015.
  • Accepted April 21, 2015.


Piezoelectric ultrasonic transducers usually 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 utilize resonators that span a range of length scales. In this article, we derive a mathematical model to predict the dynamics of a fractal ultrasound transducer; 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. The propagation of an ultrasonic wave in this transducer is then analysed and used to derive expressions for the non-dimensionalized electrical impedance and the transmission and reception sensitivities as a function of the driving frequency. Comparing these key performance measures to an equivalent standard (Euclidean) design shows some benefits of these fractal designs.

  • © The Authors 2015. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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