Acoustic propagation in dispersions in the long wavelength limit

The problem of scattering of ultrasound by particles in the long wavelength limit has a well-established solution in terms of Rayleigh expansions of the scattered fields. However, this solution is ill-conditioned numerically, and recent work has attempted to identify an alternative method. The scattered fields have been expressed as a perturbation expansion in the parameter Ka (the wavenumber multiplied by the particle radius), which is small in the long wavelength region. In the work reported here the problem has been formulated so as to be valid for all values of the thermal wavelength, which varies in order of magnitude, from much smaller to much larger than the particle size in the long wavelength region. Thus the present solution overlaps the limiting solutions for very small thermal wavelength (geometric theory) and very large thermal wavelength (low frequency) previously reported. Close agreement is demonstrated with the established Rayleigh expansion solution

A perturbation solution for long wavelength thermoacoustic propagation in dispersions

AbstractIn thermoacoustic scattering the scattered field consists of a propagating acoustic wave together with a non-propagational “thermal” wave of much shorter wavelength. Although the scattered field may be obtained from a Rayleigh expansion, in cases where the particle radius is small compared with the acoustic wave length, these solutions are ill-conditioned. For this reason asymptotic or perturbation solutions are sought. In many situations the radius of the scatter is comparable to the length of the thermal wave. By exploiting the non-propagational character of the thermal field we obtain an asymptotic solution for long acoustic waves that is valid over a wide range of thermal wavelengths, on both sides of the thermal resonance condition. We show that this solution gives excellent agreement with both the full solution of the coupled Helmholtz equations and experimental measurements. This treatment provides a bridge between perturbation theory approximations in the long wavelength limit and high frequency solutions to the thermal field employing the geometric theory of diffraction

Acoustic scattering by a spherical obstacle: modification to the analytical long-wavelength solution for the zero-order coefficient

Classical long wavelength approximate solutions to the scattering of acoustic waves by a spherical liquid particle suspended in a liquid (an emulsion) show small but significant differences from full solutions at very low kca (typically kca 0.1, where kc is the compressional wavenumber and a the particle radius. These differences may be significant in the context of dispersed particle size estimates based on compression wave attenuation measurements. This paper gives an explanation of how these differences arise from approximations based on the significance of terms in the modulus of the complex zero-order partial wave coefficient, A0. It is proposed that a more accurate approximation results from considering the terms in the real and imaginary parts of the coefficient, separately

Multiple scattering by multiple spheres: a new proof of the Lloyd-Berry formula for the effective wavenumber

We provide the first classical derivation of the Lloyd-Berry formula for the effective wavenumber of an acoustic medium filled with a sparse random array of identical small scatterers. Our approach clarifies the assumptions under which the Lloyd-Berry formula is valid. More precisely, we derive an expression for the effective wavenumber which assumes the validity of Lax's quasicrystalline approximation but makes no further assumptions about scatterer size, and then we show that the Lloyd-Berry formula is obtained in the limit as the scatterer size tends to zero

A perturbation approach to acoustic scattering in dispersions

Ultrasound spectroscopy has many applications in characterizing dispersions, emulsions, gels, and biomolecules. Interpreting measurements of sound speed and attenuation relies on a theoretical understanding of the relationship between system properties and their effect on sound waves. At its basis is the scattering of a sound wave by a single particle in a suspending medium. The problem has a well-established solution derived by expressing incident and scattered fields in terms of Rayleigh expansions. However, the solution is badly conditioned numerically. By definition, in the long-wavelength limit, the wavelength is much larger than the particle radius, and the scattered fields can then be expressed as perturbation series in the parameter Ka (wave number multiplied by particle radius), which is small in this limit. In addition, spherical Bessel and Hankel functions are avoided by using alternative series expansions. In a previous development of this perturbation method, thermal effects had been considered but viscous effects were excluded for simplicity. Here, viscous effects, giving rise to scattered shear waves, are included in the formulation. Accurate numerical correspondence is demonstrated with the established Rayleigh series method for an emulsion. This solution offers a practical computational approach to scattering which can be embodied in acoustic instrumentatio