Bona, Pritchard and Scott showed that solutions of these two evolution equations agree to the neglected order of approximation over a long time scale, if the initial disturbance in question is genuinely of small-amplitude and long-wavelength. The same formal argument that allows one to infer (0.2) from (0.1) in small-amplitude, long-wavelength regimes also produces a third equation, namelyu_t+u_x+uu_x+u_xtt=0.Kruskal, in a wide-ranging discussion of modelling considerations, pointed to this equation as an example that might not accurately describe water waves. Its status has remained unresolved. It is our purpose here to show that the initial-value problem for the latter equation is indeed well posed. Moreover, we show that for small-amplitude, long waves, solutions of this model also agree to the neglected order with solutions of either (0.1) or (0.2) provided the initial data is properly imposed.¹