The work focuses on the analytical description of unsteady thermal processes in low-dimensional structures. The object of study is an infinite one-dimensional harmonic crystal with interactions up to the third coordination sphere. The paper explains how a variation in bond stiffness between particles of different coordination spheres affects the behaviour of the system. The fundamental solution to the heat propagation problem has been constructed and investigated. It is shown that the initial thermal perturbation evolves into several consecutive thermal waves propagating with finite velocities. The number, the velocities, and the intensity coefficients of these waves are determined by the bond stiffnesses.