Unsteady thermal transport in an instantly heated semi-infinite free end Hooke chain

We consider unsteady ballistic heat transport in a semi-infinite Hooke chain with free end and arbitrary initial temperature profile. An analytical description of the evolution of the kinetic temperature is proposed in both discrete (exact) and continuum (approximate) formulations. By comparison of the discrete and continuum descriptions of kinetic temperature field, we reveal some restrictions to the latter. Specifically, the far-field kinetic temperature is well described by the continuum solution, which, however, deviates near and at the free end (boundary). We show analytically that, after thermal wave reflects from the boundary, the discrete solution for the kinetic temperature undergoes a jump near the free end. A comparison of the descriptions of heat propagation in the semi-infinite and infinite Hooke chains is presented. Results of the current paper are expected to provide insight into non-stationary heat transport in the semi-infinite lattices.