For water waves on an incompressible fluid and irrotational flow in the  plane, the boundary conditions at the free surface elevation  are:
where:
* is the horizontal flow velocity component: ,
* is the vertical flow velocity component: ,
* is the acceleration by gravity.
Now the Boussinesq approximation for the velocity potential , as given above, is applied in these boundary conditions. Further, in the resulting equations only the linear and quadratic terms with respect to  and  are retained (with  the horizontal velocity at the bed ). The cubic and higher order terms are assumed to be negligible. Then, the following partial differential equations are obtained:
;set A – Boussinesq (1872), equation (25)
This set of equations has been derived for a flat horizontal bed, i.e. the mean depth  is a constant independent of position . When the right-hand sides of the above equations are set to zero, they reduce to the shallow water equations.
Under some additional approximations, but at the same order of accuracy, the above set A can be reduced to a single partial differential equation for the free surface elevation :
;set B – Boussinesq (1872), equation (26)
From the terms between brackets, the importance of nonlinearity of the equation can be expressed in terms of the Ursell number.
In dimensionless quantities, using the water depth  and gravitational acceleration  for non-dimensionalization, this equation reads, after normalization:
with: