Linear waves in two-layer fluids over periodic bottoms

A new, exact Floquet theory is presented for linear waves in two-layer fluids over a periodic bottom of arbitrary shape and amplitude. A method of conformal transformation is adapted. The solutions are given, in essentially analytical form, for the dispersion relation between wave frequency and generalized wavenumber (Floquet exponent), and for the waveforms of free wave modes. These are the analogues of the classical Lamb's solutions for two-layer fluids over a flat bottom. For internal modes the interfacial wave shows rapid modulation at the scale of its own wavelength that is comparable to the bottom wavelength, whereas for surface modes it becomes a long wave carrier for modulating short waves of the bottom wavelength. The approximation using a rigid lid is given. Sample calculations are shown, including the solutions that are inside the forbidden bands (i.e. Bragg resonated).