Tetsuya Kanagawa

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

This paper theoretically treats the weakly nonlinear propagation of diffracted sound beams in nonuniform bubbly liquids. The spatial distribution of the number density of the bubbles, initially in a quiescent state, is assumed to be a slowly varying function of the spatial coordinates; the amplitude of variation is assumed to be small compared to the mean number density. A previous derivation method of nonlinear wave equations for plane progressive waves in uniform bubbly liquids [Kanagawa, Yano, Watanabe, and Fujikawa (2010). J. Fluid Sci. Technol. 5(3), 351-369] is extended to handle quasi-plane beams in weakly nonuniform bubbly liquids. The diffraction effect is incorporated by adding a relation that scales the circular sound source diameter to the wavelength into the original set of scaling relations composed of nondimensional physical parameters. A set of basic equations for bubbly flows is composed of the averaged equations of mass and momentum, the Keller equation for bubble wall, and supplementary equations. As a result, two types of evolution equations, a nonlinear Schrödinger equation including dissipation, diffraction, and nonuniform effects for high-frequency short-wavelength case, and a Khokhlov-Zabolotskaya-Kuznetsov equation including dispersion and nonuniform effects for low-frequency long-wavelength case, are derived from the basic set.

Nonlinear Wave Propagation in Bubbly Liquids

Weakly nonlinear wave equations for pressure waves in bubbly liquids are derived in a general and systematic way based on the asymptotic expansion method of multiple scales. The derivation procedure is explained in detail with a special attention to scaling relations between physical parameters characterizing the wave motions concerned. In the framework of the present theory, one can systematically deal with various weakly nonlinear wave motions for various systems of governing equations of bubbly liquids, thereby deriving such as the Korteweg–de Vries–Burgers equation, the nonlinear Schrodinger equation, and the Khokhlov–Zabolotskaya–Kuznetsov equation. In this sense, the method may be called a unified theory of weakly nonlinear waves in bubbly liquids.

Two types of propagations of nonlinear sound beams in nonuniform bubbly liquids

Weakly nonlinear propagation of diffracted sound beams in nonuniform bubbly liquids is theoretically examined. The spatial distribution of the number density of the bubbles, initially in a quiescent state, is assumed to be a slowly varying function of the spatial coordinates; the amplitude of variation is assumed to be small compared to the mean number density. Two types of nonlinear wave equations for progressive quasi-plane beams in weakly nonuniform bubbly liquids are then systematically derived via the method of multiple scales. The diffraction effect is incorporated by adding a relation that scales the circular sound source diameter to the wavelength into the original set of scaling relations composed of nondimensional physical parameters. A set of basic equations for bubbly liquids is composed of the averaged equations of mass and momentum in a two-fluid model, the Keller equation for bubble wall, the equation of state for gas and liquid, the mass conservation equation inside the bubble, and the bal...

Effects of Liquid Compressibility on Mathematical Well-Posedness of Two-Fluid Model Equation for Bubbly Flows

It is well-known that the conventional two-fluid model equation system is mathematically ill-posed as initial-value problems. We recently demonstrated that the model of a virtual mass force has a significant effect on the mathematical well-posedness and pointed out that the model should satisfy the invariance under the Galilean transformation so that the system can be well-posed. In our previous study, we clarified that the liquid compressibility generates a “fast mode” in addition to a well-known “slow mode” for the wave propagation in bubbly flows based on two-fluid model. In this study, we investigate effects of the liquid compressibility on the mathematical well-posedness of the two-fluid model equation.

Algebraic reconstruction technique considering curved ray for sound-speed tomography with ring-array transducer

Our objective is to develop an ultrasound treatment and diagnosis integrated system for breast cancer. Ultrasound computed tomography (UCT) in imaging and high intensity focused ultrasound (HIFU) in therapy was integrated to achieve ideal treatment system. Profiles of sound speed and attenuation obtained by UCT has informative parameters to correct deformation of HIFU beam. We try to develop an imaging system using ring-array transducer with 1024-elements, multiplexer connecting 1024 to 256 and Verasonics programmable imaging system with 256 channels. First, an iterative Simultaneous Algebraic Reconstruction Technique (SART) reconstruction methods with an assumption of straight path was employed. SART was applied to projection data calculated by a FEM simulator treating actual curved ray caused by tissue inhomogeneity. In these results, estimated error in sound speed difference was 53%. Then we introduced Linear Travel-time Interpolation (LTI) to SART to implement effects of curved ray. LTI is a ray traci...

Numerical study on acoustics in bubbly liquids based on weakly nonlinear wave equations

This paper numerically examines weakly nonlinear propagation of plane progressive waves in liquids uniformly containing many gas bubbles. Solving two types of nonlinear wave equations via a finite difference method clarifies and depicts the following results: (i) the dissipation and dispersion effects appear, (ii) nonlinearity generates higher harmonics, and (iii) dispersion causes the separation of each wavenumber component with its own phase velocity.

Derivation of Effective Wave Equation for Very-High-Frequency Short Waves in Bubbly Liquids

The present study theoretically investigates the propagation of plane progressive waves of very high frequency in a quiescent liquid containing many spherical gas bubbles. We focus on a compressibility of liquid phase that has been neglected in many previous studies, and treat wave propagation in a mode, i.e., “Fast mode”, induced by the liquid compressibility. Waves in Fast mode propagate with large phase velocity exceeding the sound speed in liquid. We derive an effective wave equation for high frequency short wave in Fast mode. By using the derivation method for effective wave equations in bubbly liquids recently proposed by our group, a linear equation with dissipation and dispersion effects can be derived from a set of basic equations in a two-fluid model.

Derivation of nonlinear wave equations for ultrasound beam in nonuniform bubbly liquids

Weakly nonlinear propagation of diffracted ultrasound beams in a nonuniform bubbly liquid is theoretically studied based on the method of multiple scales with the set of scaling relations of some physical parameters. It is assumed that the spatial distribution of the number density of bubbles in an initial state at rest is a slowly varying function of space coordinates and the amplitude of its variation is small compared with a mean number density. As a result, a Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation with dispersion and nonuniform effects for a low frequency case and a nonlinear Schrodinger (NLS) equation with dissipation, diffraction, and nonuniform effects for a high frequency case, are derived from the basic equations of bubbly flows.