Nonlinear effects in trapped modes of standing waves on the surface of shallow water

It was shown that traveling waves may coexist with standing waves in a plan ar infinitely long channel filled by ideal liquid with a free surface. The standing waves are localized near a dynamic inclusion-a massive die on an e lastic base. The amplitude of the traveling waves may be turned to zero by appropriately selecting the vibration frequency of the die. The standing wa ves arise because the vibration eigenfrequencies have a mixed spectrum; tha t is, the discrete and continuous spectra superpose. Nonlinear effects were observed for the first time when standing waves form in shallow water. In particular, a relationship between the die weight necessary to excite trapp ed modes, die dimensions, and vibration frequency was derived. It was shown that the nonlinear effects cause double-frequency traveling waves with amp litudes of the next order of smallness. These traveling waves vanish if the die geometry is properly chosen, as for the waves of the zeroth order. (C) 2000 MAIK "Nauka/Interperiodica".