Diffraction grating with space-time modulation

We present a theoretical and numerical analysis of the diffraction of acoustic waves by space-time modulated gratings with rigid-type modulations. This is done by deriving the form of the modes which are exact, uncoupled, solutions of the problem in the unbounded regions, inside and outside the grating. The dispersion of the modes is studied as a function of the ratio of the modulation speed to the speed of sound which shows that each spatial diffraction order is associated with a single temporal diffraction order. For a grating of finite extend, the solution is obtained as a superposition of these modes, which couple at the grating interfaces. This provides a numerical, multimodal, method when considering a truncated version of the solution. We provide analysis of the solutions in the harmonic and in the transient regimes.