Photothermal waves in a periodically modulated structure

The generation of thermal waves in a one-dimensional structure with a sinusoidal variation in thermal diffusivity can be described by an inhomogeneous Mathieu equation. An exact solution to the frequency-domain Mathieu equation can be found by using a variation of parameters method to give the alternating component of the temperature along the structure. From the theory of Mathieu functions a dispersion relation for the thermal waves is found, which does not display band gaps, but which shows damping that departs from that for thermal waves in a uniform medium.