**Authors:** Yakov A. Iosilevskii

It is shown with complete logical and mathematical rigor that under the appropriate hypotheses of analytical extension and of asymptotic matching, which are stated in the article, the nonlinear problem of irrotational and incompressible gravity waves on an infinite water layer of a constant depth d reduces to an infinite recursive sequence of linear two-plane boundary value problems for a harmonic velocity potential with respect to powers of a dimensionless real-valued scaling parameter ‘ka’, where k>0 is the wave number and a>0 the amplitude of a priming (seeding) progressive, or standing, plane monochromatic gravity water wave (briefly PPPMGWW or PSPMGWW respectively). The method, by which the given nonlinear water wave problem is treated in the exposition from scratch, can be regarded as a peculiar instance of the general perturbation method, which is known as the Liouville-Green (LG) method in mathematics and as the Wentzel-Kramers-Brillouin (WKB) method in physics. In the framework of the recursive theory developed, the velocity potential and any bulk or surface measurable characteristic of the wave motion is represented by an infinite asymptotic power series with respect to ‘ka’, whose all coefficients are expressed in quadratures in accordance with a well-established an algorithm for their successive calculation. The theory developed applies particularly in the case where the depth d is taken to infinity. Besides the priming velocity potential of the first, linear asymptotic approximation in ka, the partial velocity potential and all relevant characteristics of wave motion of the second order with respect to ka are calculated in terms of elementary functions both in the case of a PPPMGWW and in the case of a PSPMGWW. Accordingly, the recursive theory incorporates the conventional Airy (linear) theory of water waves linear as its first non-vanishing approximation with the following proviso. In the Airy theory, the boundary condition at the perturbed free (upper) surface of a water layer is paradoxically stated at the equilibrium plane z=0, in spite of the fact that at any instant of time some part of the plane is necessarily located in air or in vacuum, and not in water. This and also a similar paradox arising in computing the time averages of bulk characteristics at spatial points close to the perturbed free surface are solved in the article.

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