Long Waves Over Wavy Bottoms

The propagation of long waves over bottoms having sinusoidal undulations is investigated here within the confines of linearized shallow water theory. It is found that the presence of this irregularity in most cases impedes the propagation of the wave in keeping with the proper application of the Green-DuBoys formula. For wavelengths for which this formula is not valid, or those of the same order as the bottom wavelengths, it is found that there is a region for which the propagation is not retarded, and the travel time is less than that based upon the mean depth. Furthermore, the presence of regular undulations of the bottom of any amplitude prohibits the propagation of an infinite sequence of wavelengths on the surface, the most significant of which are those of the same order as the bottom. These waves are unstable, and through resonance with the bottom will grow without bound as they progress, or at least until the linearized theory is invalidated. An electrical analog is presented which exhibits the same instability, a subharmonic resonance, and can be used to determine the free surface profiles.