In this paper the authors are concerned theoretically with the forced oscillations of shallow water contained in a rectangular open tank. The nonlinear equations of motion are derived from the fundamental equations of hydrodynamics by the method of Friedrichs and Keller, and solved approximately after Chester who has treated the resonant oscillations of a gas in a closed tube. The solutions are applicable to any frequency including the cases of resonance and the leading results obtained are as follows : 1. The action of a rotatory oscillation of the tank about any horizontal axis upon the water in the tank can be expressed as a linear sum of a transverse horizontal oscillation and a rotatory oscillation about a horizontal axis in the water surface at rest. As the result of this fact, there exists a simple combination of these two oscillations which forms no wave on the water surface. 2. When the frequency ω of the oscillations of the tank is not near the natural frequency coo of the tank water the sinusoidal standing waves are formed, but as ω approaches ω_o the wave form is distorted and at near resonance there appears the “bore”, contrary to the standing waves of large amplitude in case of the linear theory. Even at the resonance ω=ω_o the bore keeps the finite wave height. Passing through the first resonance ω_o=ω the wave form becomes distorted sinusoidal again, but the phase is reversed. 3. During the period from the near resonance after ω= ω_oto the second resonance ω=2ω_o, the phenomena change as above-mentioned, and at ω=2ω_othe waves have the node at side wall. Thus near and at ω= n ω_o, according as n is odd or even the waves become bore or sinusoidal standing wave including still surface specifically.