Polar Decomposition of Frequency‐Dependent Green's Functions in Fractional Viscoelastic Media

Understanding the behavior of wave propagation in viscoelastic materials is essential across numerous scientific and engineering fields. It improves our ability to predict and interpret the dynamics of viscoelastic materials and wave interactions in complex environments. This paper presents a tensor‐based frameSwork for analyzing and visualizing frequency‐dependent wave propagation in three‐dimensional linear viscoelastic media. Fractional‐order damping is incorporated through combining elastic and viscous tensors, leading to complex stiffness models governed by fractional calculus. These models are integrated into the frequency‐wavenumber domain formulation of the Green's function, which captures anisotropic elasticity and directional attenuation. While it is well established that damping shifts wave poles into the complex plane, the present framework embeds this effect directly into a frequency‐dependent Green's function formulation extended to fractional viscoelasticity. It demonstrates that viscoelastic damping regularizes the Green‐Christoffel operator by removing real‐direction singularities. Furthermore, polar decomposition is applied to the Green's function tensors, uniquely separating them into stretch (amplitude, damping) and rotation (phase, direction) components. This dual spectral‐geometric representation separates amplitude (stretch) from direction‐dependent phase evolution encoded in the unitary rotation tensor that reflects complex polarization. Overall, this framework offers a structured method for visualizing viscoelastic wave propagation and supports anisotropic material characterization.