Distortion and harmonic generation in the nearfield of a finite amplitude sound beam

Abstract

Distortion and harmonic generation in the nearfield of a finite amplitude sound beam are considered, assuming time-periodic but otherwise arbitrary on-source conditions. The basic equations of motion for a lossy fluid are simplified by utilizing the parabolic approximation, and the solution is derived by seeking a Fourier series expansion for the sound pressure. The harmonics are governed by an infinite set of coupled differential equations in the amplitudes, which are truncated and solved numerically. Amplitude and phase of the fundamental and the first few harmonics are calculated along the beam axis, and across the beam at various ranges from the source. Two cases for the source are considered and compared: one with a uniformly excited circular piston, and one with a Gaussian distribution. Various source levels are used, and the calculations are carried out into the shock region. The on-axis results for the fundamental amplitude are compared with results derived using the linearized solution modified with various taper functions. Apart from a nonlinear tapering of the amplitude along and near the axis, the results are found to be very close to the linearized solution for the fundamental, and for the second harmonic close to what is obtained from a quasilinear theory. The wave profile is calculated at various ranges. An energy equation for each harmonic is obtained, and shown to be equivalent within our approximation to the three-dimensional version of Westervelt’s energy equation. Recent works on one-dimensional propagation are reviewed and compared.

Numerical method for calculating nonlinear sound propagation in full acoustic field

“…(5) represents the diffraction, the third term is the dissipation, and the fourth term is the nonlinearity in sound propagation. The equation can be solved numerically after converting it into Fourier series equations by the implicit finite-difference method [3].…”

“…Nonlinear sound propagation in the far-field acoustics has been studied by numerically solving the KhokhlovZaboloskya-Kuznetsov (KZK) equation [1][2][3], which is derived from the fluid dynamic equation under the assumption of parabolic approximation [4]. It is well known that the approach based on the KZK equation is efficient enough in the far-field acoustics, whereas it is not accurate enough in the near sound source owing to the parabolic nature of the equation [5,6].…”

Simulation of nonlinear propagation of biomedical ultrasound using pzflex and the Khokhlov-Zabolotskaya-Kuznetsov Texas code

“…Aanonsen and Tjøtta 9 employed a Fourier series expansion and solved the equation in the frequency domain. Mixed time frequency domains approaches have also been employed including capturing discontinuities.…”

Acoustic microscopy and nonlinear effects in pressurized superfluid helium

“…(1) Tjotta and Tj0tta [24] and Aanonsen, Barkve, Tjctta. and Tjotta [25] [28] made detailed evaluations of the numerical algorithms needed to accomplish a spectral solution of the KZK equation. The problem we address, unlike those already studied, is that the following conditions apply simultaneously: (1) There is large focusing gain, (2) the aperture angle is relatively large (so that the so-called Fresnel approximation is invalid), and (3) there is significant pump depletion of the first harmonic of the input acoustic power.…”

“…[25] and [26]). This is justified by the ability of the KZK equation to take into account moderate nonlinearities, diffraction, and absorption in sound beams of significant amplitude, as in our case, where the usual quasi-linear approaches are rendered inapplicable.…”