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An analytical approach for calculating radiation force and torque on a sphere or spheroid near a rigid or pressure release boundary due to an arbitrary incident acoustic field was developed recently by the authors [Proc. Meet. Acoust. 48, 045004 (2022)]. In that approach, the linear scattering problem is solved using the method of images along with expansions of the acoustic fields in spherical or spheroidal wave functions. The expansion coefficients can then be substituted into an existing expression to obtain the radiation force and torque on the object. In the present work, the aforementioned theory is extended to include penetrable interfaces and impedance boundaries using an approximation of the interfacial boundary conditions. For fluid-fluid interfaces, the object can be placed on either side of the interface with respect to the incident field. This approximate analytical approach is computationally efficient and compares well with the results from an independent finite element model. In addition, an experiment was conducted with a plastic sphere submerged in water and positioned near a boundary. Results from the experiment are discussed in relation to the analytical model. [BES is supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]

Linear theory for quasi-longitudinal surface wave propagation along an elastic interface coupled to a viscous, incompressible liquid was first developed by Lucassen [Trans. Faraday Soc. 1968]. Lucassen waves are modeled with a fractional diffusion-wave equation in which the order of the fractional time derivative is 3/2. Nonlinearity in the elastic interface was taken into account recently by Kappler et al. [Phys. Rev. Fluids 2017]. Nonlinear Lucassen interface waves exhibit certain features associated with the mechanical disturbance that accompanies the electric action potential in the biological membranes of nerve axons, such as the “all-or-none” principle in which wave speed and pulse shape change dramatically above some amplitude threshold. While Lucassen waves are highly damped, for nonlinear propagation the attenuation described by the fractional time derivative provides insufficient energy loss near regions where shocks form in the waveform, resulting in the failure of conventional numerical algorithms such as Runge-Kutta schemes. Presented here is a modified model equation for nonlinear Lucassen waves that includes viscoelastic effects in the interface. The inclusion of viscosity in the interface results in greater losses at shock fronts and increased stability for numerical calculations. [B.E.S. is supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]

Fractional calculus models used for biomedical ultrasound are associated with attenuation proportional to ωy, where y is typically in the range 1 < y < 2. To determine whether the attenuation and accompanying dispersion are sufficient to stabilize shock formation, the models are formulated as a Burgers equation with the traditional loss term replaced by a fractional derivative of order y. For y < 1 the resulting equation predicts unphysical solutions beyond the shock-formation distance. The second example pertains to nonlinear Lucassen interface waves, a model equation for which has been proposed to describe mechanical perturbations that accompany the transmission of nerve impulses. Linear Lucassen waves are defined by a second-order space derivative and a fractional time derivative of order 3/2, which falls between order 2 in the wave equation and order 1 in the diffusion equation. The resulting attenuation is proportional to ω3/4, and the corresponding nonlinear “fractional diffusive waves,” while strongly attenuated on the scale of a wavelength, may be lacking essential physics beyond the predicted shock-formation distance. Calculations are presented that determine wave amplitudes and propagation distances for which these two fractional calculus models may be of questionable physical significance due to nonlinearity. [B.E.S. is supported by the ARL:UT McKinney Fellowship in Acoustics.]