Plane electromagnetic as well as plane elastodynamic waves in anisotropic media exhibit a different direction of their phase and energy propagation, resulting in slowness and group velocity surfaces. Of course, the availability of plane wave solutions gives rise to a spectral plane wave decomposition of point source excitations, i.e., Green's functions. Unfortunately, the coordinate-free closed-form solution of dyadic (electric) Green's functions in the Rω space is only known for electromagnetic (generalized) uniaxial media. Utilizing the relation between phase and group velocities of plane waves in uniaxial media we have been able to show that the phase and amplitude of the Green's function is related to the group velocity; i.e., time domain wave fronts reproduce group velocity surfaces. This has also been verified through numerical results obtained by the three-dimensional (3-D) electromagnetic finite integration technique code. In elastodynamics, where similar analytical results for anisotropic media are not available, we confirm this behavior with our numerical 3-D elastodynamic finite integration technique code. For electromagnetic uniaxial media, we present an analytic method to derive the dyadic far-field Green's function in Rω space from Kω space directly by utilizing the duality principle between wave vectors and ray vectors without performing the 3-D inverse Fourier transform from Kω space to Rω space analytically.