The ordinary differential equation $\epsilon^2 \ddot{y} + \epsilon^2 \delta \dot{y} = y^2 - \cos t - 1 - c$ which represents forced water waves on shallow water near resonance is considered when the dispersion ε and the constant c are small. Asymptotic and numerical methods are used to show that solutions which are bounded for all finite t can exist only if $c > - 2^{-\frac{2}{3}}\epsilon^{\frac{4}{3}}\times 1.4664 \ldots $.