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Title: Linear/Nonlinear Acoustic Wave Propagation Through Ideal Fluid With Inclusion
Authors: LIU GANG
Keywords: linear/nonlinear acoustic wave, westervelt equation, compressible potential flow theory, conformal mapping, perturbation
Issue Date: 15-Apr-2011
Citation: LIU GANG (2011-04-15). Linear/Nonlinear Acoustic Wave Propagation Through Ideal Fluid With Inclusion. ScholarBank@NUS Repository.
Abstract: This work proposed several analytical model for the linear/nonlinear acoustic wave propagating through the ideal fluid with inclusion embedded. The conformal mapping together with the complex variables method were applied to solve the linear acoustic wave scattering by irregular shaped inclusion. Subsequently, we use the perturbation method to analytically solve the nonlinear acoustic wave interact with the regular shaped inclusion by expand the nonlinear governing equation into linear homogeneous/non-homogeneous equations. In general, these two methods are versatile to obtain the analytical solutions for two classes of problems: the linear problems with complex boundary conditions and the nonlinear problem with more complex governing equations. For the linear model, we analytically obtained the two dimensional general solution of Helmholtz equation, shown as Bessel function with mapping function as the argument and fractional order Bessel function, to study the linear acoustic wave scattering by rigid inclusion with irregular cross section in an ideal fluid. Based on the conformal mapping method together with the complex variables method, we can map the initial geometry into a circular shape as well as transform the original physical vector into corresponding new expressions in the mapping plane. This study may provide the basis for further analyses of other conditions of acoustic wave scattering in fluids, e.g. irregular elastic inclusion within fluid with viscosity, etc. Our calculated results have shown that the angle and frequency of the incident waves have significant influence on the bistatic scattering pattern as well as the far field form factor for the pressure in the fluid. Moreover, we have shown that the sharper corners of the irregular inclusion may amplify the bistatic scattering pattern compared with the more rounded corners. For the part of nonlinear acoustic wave propagation, we adopted two nonlinear models to investigate the multiple incident acoustic waves focused on certain domain where the nonlinear effect is not negligible in the vicinity of the scatterer. The general solutions for the one dimensional Westervelt equation with different coordinates (plane, cylindrical and spherical) are analytically obtained based on the perturbation method with keeping only the second order nonlinear terms. Separately, introducing the small parameter (Mach number), we applied the compressible potential flow theory and proposed a dimensionless formulation and asymptotic perturbation expansion for the velocity potential and enthalpy which is different from the existing (and more traditional) fractional nonlinear acoustic models (eg. the Burgers equation, KZK equation and Westervelt equation). Our analytical solutions and numerical calculations have shown the general tendency of the velocity potential and pressure to decrease w.r.t. the increase of the distance away from the focused point. At least, within the region which is about 10 times the radius of the scatterer, the non-linear effect exerts a significant influence on the distribution of the pressure and velocity potential. It is also interesting that at high frequencies, lower Mach numbers appear to bring out even stronger nonlinear effects for the spherical wave. Our approach with small parameter for the cylindrical and spherical waves could serve as an effective analytical model to simulate the focused nonlinear acoustic near the scatterer in an ideal fluid and be applied to study bubble cavitation dynamic associated with HIFU in our future work.
Appears in Collections:Master's Theses (Open)

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