A linear system model is developed for a composite piezoelectric transducer with multiple matching layers. The large number of degrees of freedom in this model and the sensitivity of the device performance to these parameters make the inverse problem, of choosing these parameters to optimize the device's performance, an extremely difficult task. However, by accepting a small number of assumptions on the nature of the propagating waves in the device a Chebyshev polynomial approach is used to reduce the degrees of freedom and arrive at an algorithm for a device design. A typical device is investigated using this approach and it is shown that a threefold improvement in the device performance can be achieved over the standard single-matching-layer design. The modelling also indicated that great care has to be taken in bonding the matching layers together to ensure the homogeneity of the bonds. A transducer with four matching layers was subsequently manufactured and the experimental results show a significant improvement in the device performance, in agreement with the model predictions.