Nonlinear Wave Equation for Ultrasound Beam in Nonuniform Bubbly Liquids

Abstract

In our previous paper (Kanagawa et al., J. Fluid Sci. Tech., 5, 2010), we have proposed a systematic method for derivation of various types of nonlinear wave equations for plane waves in bubbly liquids. The method makes use of an asymptotic expansion with multiple scales in terms of a small wave amplitude as an expansion parameter and a set of scaling relations of physical parameters, based on basic equations of two-fluid model of bubbly flows. In this paper, we extend the method so as to handle a weakly diffracted ultrasound beam in a quiescent liquid containing a number of spherical gas bubbles distributed with a weak nonuniformity. Because of the high expandability of the original method, the extension can be accomplished by adding a scaling relation of the diameter of the beam to the original set of scaling relations. As a result, we derive a generalized Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation [or a generalized Kadomtsev-Petviashvili (KP) equation] for a long wave and low frequency case.

Two types of nonlinear wave equations for diffractive beams in bubbly liquids with nonuniform bubble number density

“…We proceed without explicitly showing these values to ensure that the contribution of each term on the right-hand side of Eq. (11) is visible in the final result; b 3 does not appear in the final results for any of the problems studied by our group (Yano et al, 2006;Kanagawa et al, 2010Kanagawa et al, , 2011aYano et al, 2013) including the present analysis.…”

“…The wave motion is extended in two ways (see also Kanagawa et al, 2011a) from the plane wave problem considered by Kanagawa et al (2010). The first extension is based on the condition that the typical wavelength L Ã is sufficiently small relative to D Ã ,…”

“…Although the Davey-Stewartson equation (i.e., the twodimensional NLS equation) and the KP equation were derived Akhatov and Khismatullin, 2001) for uniform bubbly liquids from the mixture model (e.g., Nigmatulin, 1991), dissipation and nonuniform effects were neglected and the two-fluid model was not utilized. As in our previous derivation (Kanagawa et al, 2010(Kanagawa et al, , 2011a, we employ the basic equations in the two-fluid model (Egashira et al, 2004;Yano et al, 2006); the broad applicability of this model to wave propagation analysis was revealed by Kanagawa et al (2011b). To demonstrate the wider applicability of our method, this paper tackles the derivation of two nonlinear wave equations.…”

“…In this paper, we extend our method to encompass a sound beam in a nonuniform bubbly liquid without these restrictions. The method is extended by the following two steps (Kanagawa et al, 2011a;Yano et al, 2013):…”

“…In Sec. V, we review the derivation of the generalized KZK equation [the detailed procedure is found in Kanagawa et al (2011a)]. In Sec.…”