A comprehensive experimental, analytical and numerical study of the true focal region drift relative to the geometrical focus (focal shift effect) in acoustic focused beams and its nonlinear evolution is presented. For this aim, the concept of Fresnel number, proportional to the linear gain, is introduced as a convenient parameter for characterizing focused sources. It is shown that the magnitude of the shift is strongly dependent on the Fresnel number of the source, being larger for weakly focused systems where a large initial shift occurs. Analytical expressions for axial pressure distributions in linear regime are presented for the general case of truncated Gaussian beams. The main new contribution of this work is the examination of the connection between the linear and nonlinear stages of the focal shift effect, and its use for the estimation of the more complicated nonlinear stage. Experiments were carried out using a continuous-wave ultrasonic beam in water, radiated by a focused source with nominal frequency f=1 MHz, aperture radius a=1.5 cm and geometrical focal distance R=11.7 cm, corresponding to a Fresnel number N(F)=1.28. The maximum measured shifts for peak pressure and intensity were 4.4 and 1.1cm, respectively. The evolution of the different maxima with the source amplitude, and the disparity in their axial positions, is interpreted in terms of the dynamics of the nonlinear distortion process. Analytical results for the particular case of a sound beam with initial Gaussian distribution are also presented, demonstrating that the motion of peak pressure and peak intensity may occur in opposite directions.

A further refinement of the Howard-Kochar-Jain theorem is given which allows the estimation of the range of complex wave velocity for growing perturbations in a stratified shear flow. According to the results obtained, the boundary of this region depends both on the minimum Richardson number and on the wavenumber of the perturbations. The effect of external boundaries on the stability of parallel flows is defined. An estimate of the maximum rate of growth versus dimensionless wave-number is found. The theoretical results are compared with numerical computations and laboratory experiments of other authors.

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