Comment on "Application of the three-dimensional telegraph equation to cosmic-ray transport" (arXiv:1606.08272)

Abstract

In a recent publication [R. C. Tautz and I. Lerche, Res. Astron. Astrophys. 16, 162 (2016); arXiv:1606.08272], the authors present a derivation of the Green's function for the three-dimensional telegraph equation (also known as the heat wave equation, or relativistic heat conduction equation). We demonstrate that the closed-form expression derived in their Appendix A is incorrect. Specifically, the published solution lacks a Dirac delta term representing the ballistic wavefront and contains an algebraic error in the prefactor of the wake term. These omissions arise from the neglect of distributional derivatives when differentiating a Heaviside step function. We provide a rigorous derivation of the Green's function using the Fourier transform method, verify the correct limiting behavior as the damping vanishes, and pinpoint the exact mathematical step where the original derivation failed.