Małgorzata Seredyńska

Diffraction of pulses in the vicinity of simple caustics and caustic cusps

Abstract The objective of this paper is an extension of the Geometrical Theory of Diffraction (GTD) taking account of caustics. A closed list of those cuspoid caustics which are stable in the absence of symmetry constraints in a space of at most 5 dimensions is given along with integral expressions for the associated pulse shapes. Ludwigs formulae for harmonic fields are rederived by a new method. Universally valid time-domain expressions are associated with each caustic type. For the simple caustic and caustic cusp time-domain expressions are obtained in closed form. Finally, algorithms for the computation of time-domain expressions for a given wave field based on real and complex ray tracing are discussed.

Spatially fractional-order viscoelasticity, non-locality, and a new kind of anisotropy

A class of non-local viscoelastic equations of motion including equations of fractional order with respect to the spatial variables is studied. It is shown that space-fractional equations of motion of an order strictly less than 2 allow for a new kind of anisotropy, associated with azimuthal dependence of non-local interactions between stress and strain at different material points. Constitutive equations of such viscoelastic media are determined. Relaxation effects are additionally accounted for by replacing second-order time derivatives by lower-order fractional derivatives. Explicit fundamental solutions of the Cauchy problem for scalar equations with isotropic and anisotropic non-locality are constructed. For some particular choices of the parameters, numerical solutions are constructed.

Relaxation, dispersion, attenuation, and finite propagation speed in viscoelastic media

Dispersion and attenuation functions in a linear viscoelastic medium with a positive relaxation spectrum are given by integral representations in terms of a positive Radon measure satisfying a growth condition. Kramers–Kronig dispersion relations with one subtraction can be derived from the integral representations of the dispersion and attenuation functions. The dispersion and attenuation functions have sublinear growth in the high frequency range. The wave number vector can have a linear component in addition to the dispersion function. In this case the viscoelastic waves propagate with a bounded speed. In the other cases viscoelastic wave propagation has a diffusion-like character.

Cones of material response functions in one-dimensional and anisotropic linear viscoelasticity

Viscoelastic materials have non-negative relaxation spectra. This property implies that viscoelastic response functions satisfy certain necessary and sufficient conditions. These conditions can be expressed in terms of each viscoelastic response function ranging over a cone. The elements of each cone are completely characterized by an integral representation. The 1:1 correspondence between the viscoelastic response functions is expressed in terms of cone-preserving mappings and their inverses. The theory covers scalar- and tensor-valued viscoelastic response functions.