Numerical modeling of finite-amplitude sound beams: Shock formation in the near field of a cw plane piston source

Abstract

Two theoretical models and the corresponding numerical codes for the description of nonlinear acoustic beams radiated from intense cw sources in water are presented. In the first model, diffraction effects are included using the Rayleigh integral, whereas nonlinearity and thermoviscous absorption are accounted for in a quasi-plane approximation. The simulations are performed in the time domain using the code previously developed for single-pulse propagation in medium having arbitrary frequency-dependent absorption. The second model is based on the Khokhlov–Zabolotskaya–Kuznetsov equation, which, contrary to the first model, accounts for diffraction in the parabolic approximation. The simulations are performed in the frequency domain using a novel algorithm that has been developed. A variable number of harmonics, which follows the nonlinear broadening of the wave spectrum are employed in the algorithm to speed up calculations. In order to prove the validity and the accuracy of the two codes developed, the simulation of diffraction and nonlinear effects in the near field of an intense ultrasound circular piston source in water is performed. The results of modeling obtained by both codes are compared with each other and with known experimental data, and are found to be in a good agreement. Frequency-domain code is then used for detailed study of the strongly nonlinear regime of propagation, when shocks are developed in the waveform close to the source. It is demonstrated that diffraction plays a major role in shock formation. Development of two shocks in each cycle and their further collision is predicted. It is also shown that nonlinear propagation and shock formation result at some distance in the two times excess of peak positive pressure in comparison with the maximum value obtained in the case of linear propagation. The beam total power decay due to formation of shocks as a function of the propagation distance is compared with the intensity in a plane wave propagation without diffraction. It is shown that nonlinear energy decay starts earlier for the beam, but decreases slower over longer distances.