Unsteady ballistic heat transport in a 1D harmonic crystal due to a source on an isotopic defect

In the paper we apply asymptotic technique based on the method of stationary phase and obtain the approximate analytical description of thermal motions caused by a source on an isotopic defect of an arbitrary mass in a 1D harmonic crystal. It is well known that localized oscillation is possible in this system in the case of a light defect. We consider the unsteady heat propagation and obtain formulae, which provide continualization (everywhere excepting a neighbourhood of a defect) and asymptotic uncoupling of the thermal motion into the sum of the slow and fast components. The slow motion is related to ballistic heat transport, whereas the fast motion is energy oscillation related to transformation of the kinetic energy into the potential one and in the opposite direction. To obtain the propagating component of the fast and slow motions we estimate the exact solution in the integral form at a moving point of observation. We demonstrate that the propagating parts of the slow and the fast motions are "anti-localized" near the defect. The physical meaning of the anti-localization is a tendency for the unsteady propagating wave-field to avoid a neighbourhood of a defect. The effect of anti-localization increases with the absolute value of the difference between the alternated mass and the mass of a regular particle, and, therefore, more energy concentrates just behind the leading wave-front of the propagating component. The obtained solution is valid in a wide range of a spatial co-ordinate (i.e. a particle number), everywhere excepting a neighbourhood of the leading wave-front.