Ultrasound Beam Radiated From Circular Piston in Bubbly Liquids: Nonlinear Analysis for Envelope of Short Waves With Diffraction Effect Available to Purchase

A weakly diffracted ultrasound beam radiated from a circular piston placed in a bubbly liquid is formulated in terms of a wave equation based on scaling relations of physical parameters [1]: typical propagation speed, period, wavelength, and diameter of beam. We derive a nonlinear evolution equation for the modulation of quasi-monochromatic waves for the case of a short wavelength with a moderately high frequency from a set of basic equations for bubbly flows: conservation equations of mass and momentum for gas and liquid in a two-fluid model, Keller’s equation for bubble wall motion, state equations for gas and liquid, and so on. The compressibility of liquid is taken into account, and thus the waves are attenuated due to bubble oscillations. The viscosity of gas, heat conduction in gas and liquid, and phase change across bubble wall are ignored. As a result, the nonlinear Schro¨dinger equation for the envelope of the beam with diffraction effect is derived from the basic equations. For plane waves the diffraction term does not appear, and hence our equation is reduced to the original nonlinear Schro¨dinger equation [1].