First eliminate the pressure p from the momentum flux S by use of the Bernoulli equation:
The streamfunction Ψ is expanded as a Maclaurin series around the bed at z = 0, and using that the impermeable bed is a streamline and the irrotationality of the flow: Ψ = 0 and ∂Ψ = 0 at z = 0:
with u the horizontal velocity at the bed z = 0. Because the waves are long, h ≫ λ, only terms up to z and ζ are retained in the approximations to Q and S. The momentum flux S then becomes:
The discharge Q becomes, since it is the value of the streamfunction Ψ at the free surface z = ζ:
As can be seen, the discharge Q is an O(ζ) quantity. From this, the bed velocity is seen to be
Note that Q / ζ is an order one quantity. This relation will be used to replace the bed velocity u by Q and ζ in the momentum flux S. The following terms can be derived from it:
Consequently, the momentum flux S becomes, again retaining only terms up to proportional to ζ:
Which can directly be recast in the form of equation ().