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The complex mechanical behavior of materials are characterized by fluid and solid models with fractional calculus differentials to relate stress and strain fields. Fractional derivatives have been shown to describe the viscoelastic stress from polymer chain theory for molecular solutions [Rouse and Sittel, J. Appl. Phys. 24, 690 (1953)]. Here the propagation of infinitesimal waves in one dimensional horns with a small cross-sectional area change along the longitudinal axis are examined. In particular, the linear, conical, exponential, and catenoidal shapes are studied. The wave amplitudes versus frequency are solved analytically and predicted with mathematical computation. Fractional rheology data from Bagley [J. Rheol. 27, 201 (1983); Bagley and Torvik, J. Rheol. 30, 133 (1986)] are incorporated in the simulations. Classical elastic and fluid ‘‘Webster equations’’ are recovered in the appropriate limits. Horns with real materials that employ fractional calculus representations can be modeled to examine design trade-offs for engineering or for scientific application.

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The hydrodynamic or continuum approach is utilized to examine sound wave propagation through a dilute suspension of spherical particles in a viscous, heat-conducting fluid. The acoustical theory accounts for mechanical particle–fluid interactions such as Stokes drag, as well as coupled phase phenomena, collectively called phoresis effects due to gradients of temperature, density, or concentration (e.g., processes of thermophoresis, pcynophoresis, and diffusion phoresis). Linearized volume-averaged balance equations for mass, linear momentum, and energy are solved for a plane wave of arbitrary frequency. Approximations are provided to enable better physical interpretation of the results and to compare to the earlier treatment by Temkin and Dobbins [J. Acoust. Soc. Am. 40, 317–324 (1966)] for an inviscid fluid phase, but with a Stokes drag force on each particle. The investigation also considers several generalizations for the case when the phoresis terms can be neglected. For example, a distribution of particle sizes is accounted for by developing a frequency-dependent function that weights the drag forces by a particle-size distribution function. Furthermore, by invoking the correspondence principle, the drag force function for a Newtonian fluid is extended to a viscoelastic particle-laden material by using complex viscosities for shear and compressional relaxation functions. In the limit that the concentration of particles goes to zero, and the viscosity is Newtonian, the classical Kirchhoff–Langevin equation is obtained. Several calculated results are provided for comparison to available experimental measurements and a viscoelastic fluid suspension simulation illustrates attenuation and dispersion relationships versus particle size and concentration.