In a plasma wake wave generated by a high power laser, modulations of the electron density take the shape of paraboloidal dense shells, moving almost at the speed of light. A counterpropagating laser pulse is partially reflected from the shells, acting as relativistic flying mirrors, producing a time-compressed frequency-multiplied pulse due to the double Doppler effect. The counterpropagating laser pulse reflection from the plasma wake wave accompanied by its frequency multiplication (with a factor from 50 to 114) was detected in our experiment.