In order to discuss water wave problems in unlimited waters, it becomes very important what kind of wave radiation should be placed on a virtual surface corresponding to infinity. For this kind of problem, Sommerfeld radiation condition is well known. In the present paper, the condition is extended to treat a case with an incident wave. Furthermore, a more general wave radiation condition are introduced from a different point of view. The above mentioned wave radiation conditions are included into the variational principles of Kelvin, Hellinnger-Reissner and Dirichlet type. Then, the Dirichlet type variational principles are used in numerical calculations for bending waves of a bar, and the effectiveness of the wave radiation conditions and the variational principles are shown.