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

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.