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

TL;DR

This paper critically examines a recent derivation of the Green's function for the three-dimensional telegraph equation, identifying errors and providing a correct derivation using Fourier transforms to ensure mathematical accuracy.

Contribution

It corrects the previous derivation by explicitly including the Dirac delta term and fixing algebraic errors, clarifying the proper mathematical treatment of distributional derivatives.

Findings

Original derivation omitted the ballistic wavefront term.

Correct Green's function includes a Dirac delta representing ballistic waves.

Fourier transform method validates the corrected Green's function.

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.