Linear Waves on Shallow Water Slowing Down near the Shore over Uneven Bottom

The exact solutions to the system of equations of the linear theory of shallow water that represent travelling waves with some specific properties on the time propagation interval are discussed. These solutions are infinite when approaching the shore and finite when leaving for deep water. The solutions are obtained by reducing one-dimensional equations of shallow water to the Euler-Poisson-Darboux equation with negative integer coefficient ahead of the lower derivative. An analysis of the wave field dynamics is carried out. It is shown that the shape of a wave approaching the shore will be differentiated a certain number of times. This is illustrated by a number of examples. When the wave moves away from the shore, its profile is integrated. The solutions obtained within the framework of linear theory are valid only on a finite interval of variation in the depth.