text: The essential idea in the Boussinesq approximation is the elimination of the vertical coordinate from the flow equations, while retaining some of the influences of the vertical structure of the flow under water waves. This is useful because the waves propagate in the horizontal plane and have a different (not wave-like) behaviour in the vertical direction. Often, as in Boussinesq's case, the interest is primarily in the wave propagation.
  This elimination of the vertical coordinate was first done by Joseph Boussinesq in 1871, to construct an approximate solution for the solitary wave (or wave of translation). Subsequently, in 1872, Boussinesq derived the equations known nowadays as the Boussinesq equations.
  The steps in the Boussinesq approximation are:
  *a Taylor expansion is made of the horizontal and vertical flow velocity (or velocity potential) around a certain elevation, 
  *this Taylor expansion is truncated to a finite number of terms, 
  *the conservation of mass (see continuity equation) for an incompressible flow and the zero-curl condition for an irrotational flow are used, to replace vertical partial derivatives of quantities in the Taylor expansion with horizontal partial derivatives.
  Thereafter, the Boussinesq approximation is applied to the remaining flow equations, in order to eliminate the dependence on the vertical coordinate.
  As a result, the resulting partial differential equations are in terms of functions of the horizontal coordinates (and time).
  As an example, consider potential flow over a horizontal bed in the  plane, with  the horizontal and  the vertical coordinate. The bed is located at , where  is the mean water depth. A Taylor expansion is made of the velocity potential  around the bed level :
  where  is the velocity potential at the bed. Invoking Laplace's equation for , as valid for incompressible flow, gives:
  since the vertical velocity  is zero at the – impermeable – horizontal bed . This series may subsequently be truncated to a finite number of terms.