Using a Green's function method, we investigate theoretically the eigenmodes of a finite one-dimensional phononic crystal (superlattice) composed of N alternating layers of an elastic solid and an ideal fluid. If the finite superlattice is free of stress on both sides, we show that there are always N-1 modes in the allowed bands whereas there is one and only one mode corresponding to each band gap. This mode is either a surface mode in the band gap or a constant-frequency confined band-edge mode. If the finite superlattice is bounded from one side by a homogeneous fluid whereas the other surface is kept free, then an incident phonon from the fluid is perfectly reflected, however this reflection takes place with a large delay time if the frequency of the incident phonon coincides with the eigenfrequency of a surface mode