A floating ice sheet under compressive stress is modeled as a laterally compressed thin linearly elastic plate, floating on a compressible liquid of constant depth over a rigid bottom. This system is analyzed as a wave guide for plane waves. Sinusoidal traveling waves having vertical planes of constant phase are sought for the impulse response of the system. Two bands of gravity waves are found: (1) flexural and (2) floating-membrane. The two bands join at the critical frequency of the system, at which the impuse response is unbounded. The long-period floating-membrane gravity waves exist at all frequencies below the flexural gravity wave band. For long-period gravity waves the coupling force between plate and liquid is found to be a Hooke's Law force; the effective foundation modulus is directly proportional to the difference between the square of the phase velocity of a wave in the floating-plate system and that of a free-surface gravity wave at the same wavelength. The analogy of an elastic plate supported on an elastic foundation is apt for these long-period waves. It is concluded that the impulsive release of stored elastic energy when the compressed ice breaks supplies adequate energy to produce measurable long-period waves in floating ice. The results suggest that long-period wave motion accompanies lateral compression on the ice sheet; hence wave measurements may provide a means of detecting the buildup of widespread compressive stress in the ice.