Acoustic-gravity quantum tunnelling analogy

We present a mathematical solution for the two-dimensional linear problem involving acoustic-gravity waves interacting with rectangular barriers at the bottom of a channel containing a slightly compressible fluid. Our analysis reveals that, below a certain cutoff frequency, the presence of a barrier inhibits the propagation of acoustic-gravity modes. However, through the coupling with evanescent modes existing in the barrier region, we demonstrate the phenomenon of ‘tunnelling’ where the incident acoustic-gravity wave energy can leak to the other side of the barrier, creating a propagating acoustic-gravity mode of the same frequency. Notably, the amplitude of the tunnelling waves exponentially decays with the width of the barrier, analogous to the behaviour observed in quantum tunnelling phenomena. Moreover, a more general solution for multi-barrier and multi-modes is discussed. It is found that tunnelling energy tends to transform from an incident mode to the lowest neighbouring modes. Resonance due to barrier length results in more efficient energy transfer between modes.