The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. The interaction of a periodic solitary wave (cnoidal wave) with high frequency radiation of finite energy ($L^2$-norm) is studied. It is proved that the interaction
of low frequency component (cnoidal wave) and high frequency radiation is weak for finite time in the following sense: the radiation approximately satisfies Airy equation.
Dispersive averaging effects are used to show that KdV equation with periodic boundary conditions possesses high frequency solutions which behave nearly linearly. Numerical simulations are presented which indicate high accuracy of this approximation.
Furthermore, this result is applied to shallow water wave dynamics in the limit of KdV approximation, which is obtained by asymptotic analysis in combination with numerical simulations of KdV.
