Harmonic vibration heat transfer

Equations 12-4a,b define the harmonically inhomogeneous medium. Bear (1972), however, does not provide flow solutions to these formation models, it turns out, though, that simple solutions are possible. We now explain how these equations can be solved using methods in heat transfer and structural vibrations. 

In 1911, Einstein proposed a model for heat conduction in amorphous solids. In this model, he assumed that all the atoms vibrate as harmonic oscillators at the same frequency co . In addition, he also assumed that a particular oscillator (or atom) is coupled to only first, second, and third nearest neighbors. Hence, the vibrational energy of the oscillator can only be transferred to these atoms. A further assumption was that the phases of these oscillators were uncorrelated and were completely random. Using these assumptions, he derived the thermal conductivity to be... 

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