On boundary conditions for the gravity wave equations

In this study, the reasons why mathematically well posed problems for gravity wave equations with quite natural initial and boundary conditions can produce physically meaningless solutions are examined. The mechanism of generating such solutions is analyzed and general conditions on initial and boundary functions are found under which the solutions have at least linear growth with respect to time. Different examples of smooth bounded input functions are given, which lead to unbounded growth of the respective solutions. The same problem can rise in numerical models for one- and two-dimensional gravity wave and shallow water equations, but origin of the problem is hard to be found without analysis of the primitive differential problems. Based on the performed analysis and numerical experiments, some recommendations for choosing the boundary conditions are given to avoid this unphysical behavior of differential and numerical solutions.