In a recent study, we showed that the large ( x , t ) \left(x,t) behavior of a class of physically relevant solutions of Boussinesq's equation for water waves is described by ten main asymptotic sectors. In the sector adjacent to the positive x x -axis, referred to as Sector I, we stated without proof an exact expression for the leading asymptotic term together with an error estimate. Here, we provide a proof of this asymptotic formula.