Simulations of the nonlinear acoustic pressure field without using the parabolic approximation

The popular KZK equation has been widely used to model the nonlinear acoustic wave field. However, since it is based on a parabolic approximation, it has various limitations, both at close distances from the transducer and at large radial extents from the beam axis or, in case of phased arrays, at steering angles larger than, say, 20°. The purpose of this work is to model the nonlinear acoustic wave field without using the parabolic approximation. The model is based on the full nonlinear wave equation known as the Westervelt equation. This equation is solved numerically using an explicit finite difference scheme with fourth order centred space differences and second order centered or second order backward time differences. To check the validity of our model, numerical results for the lossless linear case were compared to analytical results. Moreover, the nonlinear beam profiles were compared to the results of a numerical solution of the KZK equation and to hydrophone measurements performed with a single element, unfocused transducer with a radius of 6.35 mm and a centre frequency of 1 MHz. We used a pulsed excitation of 6 cycles with pressure levels of 40, 220 and 500 kPa. The axial beam profiles and the lateral beam profiles at several distances were measured. For a low drive level, our numerical predictions show a good agreement with the analytic solution as well as with the measured data for all fundamental frequency beam profiles. At higher excitations, our simulations give a significant second harmonic component at 15 db below the fundamental level, which is in agreement with the measurements. The numerical data reproduces all the features at the fundamental and harmonic frequency beam profiles. For regions where the KZK equation is known to be less accurate, our model performs better. The sidelobes created at larger observation angles (above 20°) and the nearfield variations are accurately reproduced. The performance of our model is good since it predicts accurately the harmonic field at any position with respect to the transducer. It can be used for the whole scanning sector that is required for harmonic beam profile simulations for phased arrays.