Acoustic scattering in dispersions: Improvements in the calculation of single particle scattering coefficients

Abstract

Measurements of ultrasound speed and attenuation can be related to the properties of dispersed systems by applying a scattering model. Rayleigh's method for scattering of sound by a spherical object, and its subsequent developments to include viscous, thermal, and other effects ͑known as the ECAH model͒ has been widely adopted. The ECAH method has difficulties, including numerical ill-conditioning, calculation of Bessel functions at large arguments, and inclusion of thermal effects in all cases. The present work develops techniques for improving the ECAH calculations to allow its use in instrumentation. It is shown that thermal terms can be neglected in some boundary equations up to ϳ100 GHz in water, and several simplified solutions result. An analytical solution for the zero-order coefficient is presented, with separate nonthermal and thermal parts, allowing estimation of the thermal contribution. Higher orders have been simplified by estimating the small shear contribution as the inertial limit is approached. The condition of the matrix solutions have been greatly improved by these techniques and by including appropriate scaling factors. A method is presented for calculating the required Bessel functions when the argument is large ͑high frequency or large particle size͒. The required number of partial wave orders is also considered.

Advances in ultrasonic monitoring of oil-in-water emulsions

“…. This scaling is convenient for numerical stabilization of the calculations of the coefficients, as discussed by the author previously (Pinfield, 2007).…”

Thermo-elastic multiple scattering in random dispersions of spherical scatterers

“…Assumption (2) is supported by previous investigations into the effects of thermal scattering which demonstrate that it is predominantly incorporated in the zero-order coefficient for an incident compressional wave. 1,26 However, assumption (3) relates to the scattering of an incident thermal wave, a problem which has not previously been studied. The use of only the zero-order coefficient for the scattered compressional wave is based on the dominance of the zero order in the converse scattering case (incident compressional mode, scattered thermal mode) since the boundary conditions in both cases differ only in the incident field.…”

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

“…Our expansion is based on the analytical solution for A 0 derived by Pinfield 15 [Eqs. (55-56) and (61-62)], which is not restricted to small k c a and is conveniently separated into non-thermal and thermal parts.…”

“…[10][11][12][13][14] In addition, an analytical solution for the lowest order coefficients was derived by one of the present authors, which is not restricted to low frequencies, but which does retain some complex functions of the wavenumber-radius products. 15 The early analytical solutions for the scattering coefficients, both from the ECAH formulation 3 and from the Isakovich formulation 7 have also been adopted in studies which have incorporated further physical effects into the model, such as the thermal overlap which occurs in relatively concentrated emulsions. 8,9,16 The analytical or "explicit" solution to the scattering coefficients in the ECAH formulation has some application a) Author to whom correspondence should be addressed.…”

“…(56) of Ref. 15, taking series expansions of the numerator and denominator independently and neglecting terms of order (k c a) 3 and higher, which are small relative to unity. In the numerator, the dominant term is of order (k c a) 2 and is predominantly real.…”