Ultrasonic Scattering by a Viscoelastic Fiber of Elliptic Cross‐Section Suspended in a Viscous Fluid Medium

Abstract

This study treats the two‐dimensional interaction of a plane compressional sound wave with a viscoelastic fiber of elliptic cross‐section submerged in a boundless viscous nonheat‐conducting compressible fluid medium. The classical method of eigen‐function expansion along with the novel features of Havriliak‐Negami model for viscoelastic material behavior and the pertinent boundary conditions are used to develop a closed form series solution involving Mathieu and modified Mathieu functions of complex argument. The complications that arise because of the nonorthogonality of angular Mathieu functions corresponding to different wave numbers, as well as problems associated with the appearance of additional angular dependent terms in the boundary conditions, are all avoided by expansion of the angular Mathieu functions in terms of transcendental functions and subsequent integration, leading to a linear set of independent equations in terms of the unknown scattering coefficients. The presented solution demonstrates that acoustic characteristics of fiber suspensions are strongly influenced by cross‐sectional ellipticity of the fibers in addition to the dynamic viscoelastic properties of the fiber material. It also shows that the common rigid circular fiber approximations used to model fiber suspensions can not capture the important resonance and damping effects associated with the highly viscoelastic noncircular fibers. The proposed model is valid for a wide range of cross‐sectional geometries (aspect ratios) and incident wave frequencies where most numerical methods fail. Limiting case involving an elastic elliptic cylinder in an ideal fluid is considered and fair agreement with a recent solution is established.