Absorbing boundary conditions for the Westervelt equation

The focus of this work is on the construction of a family of nonlinear absorbing boundary conditions for the Westervelt equation in one and two space dimensions. The principal ingredient used in the design of such conditions is pseudo-differential calculus. This approach enables to develop high order boundary conditions in a consistent way which are typically more accurate than their low order analogs. Under the hypothesis of small initial data, we establish local well-posedness for the Westervelt equation with the absorbing boundary conditions. The performed numerical experiments illustrate the efficiency of the proposed boundary conditions for different regimes of wave propagation

Customization of the acoustic field produced by a piezoelectric array through interelement delays

A method for producing a prescribed acoustic pressure field from a piezoelectric array was investigated. The array consisted of 170 elements placed on the inner surface of a 15 cm radius spherical cap. Each element was independently driven by using individual pulsers each capable of generating 1.2 kV. Acoustic field customization was achieved by independently controlling the time when each element was excited. The set of time delays necessary to produce a particular acoustic field was determined by using an optimization scheme. The acoustic field at the focal plane was simulated by using the angular spectrum method, and the optimization searched for the time delays that minimized the least squared difference between the magnitudes of the simulated and desired pressure fields. The acoustic field was shaped in two different ways: the −6 dB focal width was increased to different desired widths and the ring-shaped pressure distributions of various prescribed diameters were produced. For both cases, the set of delays resulting from the respective optimization schemes were confirmed to yield the desired pressure distributions by using simulations and measurements. The simulations, however, predicted peak positive pressures roughly half those obtained from the measurements, which was attributed to the exclusion of nonlinearity in the simulations