The solution to the problem of surface gravity waves is often sought in the form of a series whose first term corresponds to the shallow-water theory. Such series have been previously studied numerically and analytically but their structure remains in general unclear due to the complicated initial formulation of the problem. In this article, instead of the full boundary value problem with free boundary and with several unknown functions, the author solves an ordinary quadratically nonlinear differential-difference equation of the first order containing only one unknown function.