Flexural-gravity wave characteristics are analysed, in the presence of a compressive force and a two-layer fluid, under the assumption of linearized water wave theory and small amplitude structural response. The occurrence of blocking for flexural-gravity waves is demonstrated in both the surface and internal modes. Within the threshold of the blocking and the buckling limit, the dispersion relation possesses four positive roots (for fixed wavenumber). It is shown that, under certain conditions, the phase and group velocities coalesce. Moreover, a wavenumber range for certain critical values of compression and depth is provided within which the internal wave energy moves faster than that of the surface wave. It is also demonstrated that, for shallow water, the wave frequencies in the surface and internal modes will never coalesce. It is established that the phase speed in the surface and internal modes attains a minimum and maximum, respectively, when the interface is located approximately in the middle of the water depth. An analogue of the dead water phenomenon, the occurrence of a high amplitude internal wave with a low amplitude at the surface, is established, irrespective of water depth, when the densities of the two fluids are close to each other. When the interface becomes close to the seabed, the dead water effect ceases to exist. The theory developed in the frequency domain is extended to the time domain and examples of negative energy waves and blocking are presented.