New approaches to nonlinear diffractive field propagation

Abstract

In many domains of acoustic field propagation, such as medical ultrasound imaging, lithotripsy shock treatment, and underwater sonar, a realistic calculation of beam patterns requires treatment of the effects of diffraction from finite sources. Also, the mechanisms of loss and nonlinear effects within the medium are typically nonnegligible. The combination of diffraction, attenuation, and nonlinear effects has been treated by a number of formulations and numerical techniques. A novel model that incrementally propagates the field of baffled planar sources with substeps that account for the physics of diffraction, attenuation, and nonlinearity is presented. The model accounts for the effect of refraction and reflection (but not multiple reflections) in the case of propagation through multiple, parallel layers of fluid medium. An implementation of the model for axis symmetric sources has been developed. In one substep of the implementation, a new discrete Hankel transform is used with spatial transform techniques to propagate the field over a short distance with diffraction and attenuation. In the other substep, the temporal frequency domain solution to Burgers' equation is implemented to account for the nonlinear accretion and depletion of harmonics. This approach yields a computationally efficient procedure for calculating beam patterns from a baffled planar, axially symmetric source under conditions ranging from quasilinear through shock. The model is not restricted by the usual parabolic wave approximation and the field's directionality is explicitly accounted for at each point. Useage of a harmonic-limiting scheme allows the model to propagate some previously intractable high-intensity nonlinear fields. Results of the model are shown to be in excellent agreement with measurements performed on the nonlinear field of an unfocused 2.25-MHz piston source, even in the near field where the established parabolic wave approximation model fails. Next, the model is used to compare the water path and in situ fields of a medical ultrasound device. Finally, the model is used to calculate the spatial heating rate associated with a nonlinear field and to simulate the phenomenon of saturation-induced beam broadening.

Fast prediction of pulsed nonlinear acoustic fields from clinically relevant sources using time-averaged wave envelope approach: Comparison of numerical simulations and experimental results

“…The comparison of the experimental results and numerical simulations has shown that the TAWE approach is well suited to predict (to within ± 1 dB) both the spatialtemporal and spatial-spectral pressure variations in the pulsed nonlinear acoustic beams. The obtained results indicated that implementation of the TAWE approach enabled shortening of computation time in comparison with the time needed for prediction of the full 4D pulsed nonlinear acoustic fields using a conventional (Fourier-series) approach [2]. The reduction in computation time depends on several parameters, including the source geometry, dimensions, fundamental resonance frequency, excitation level as well as the strength of the medium nonlinearity.…”

On the use of Gegenbauer reconstructions for shock wave propagation modeling

“…Since this is essentially an excess-absorption scheme, while it maintains the accuracy of low frequency, it reduces the accuracy of the high frequencies and inevitably distorts the shock fronts. 1,9 Fortunately, recent studies on Gegenbauer reconstructions [11][12][13][14] show that if the function under test is piecewise analytic/smooth, high frequency components can be recovered by essentially using only the low frequency components and the Gibbs effect can be completely removed. This is realized by reconstructing a rapidly converging series based on the expansions in Gegenbauer polynomials.…”

“…It is well known that this type of wave profile can be extremely hard to model by frequency-domain methods, 1 since the Fourier expansion of a shock wave has a slow convergence rate. Therefore, to model a shock wave using the frequencydomain approach, the spectral components typically have to include hundreds or even thousands of harmonics, resulting in intolerable computation time.…”

Simulations of nonlinear continuous wave pressure fields in FOCUS

“…Nonlinearity is taken into consideration in the Angular Spectrum Approach by adding a nonlinear substep after each linear diffractive substep over a distance z according to [4] p n (z, r) = p n (z, r) + j β π f 0 z …”

“…Another method for simulating nonlinear wave propagation is the angular spectrum approach (ASA) [4]. These two models have been integrated into FOCUS, the 'Fast Object-Oriented C++ Ultrasound Simulator' (http://www.egr.msu.edu/∼fultras-web) to facilitate convenient nonlinear ultrasound simulations.…”