## 2 Problem Formulation  
Let us consider a thin wing with _straight trailing edge_, moving in ideal incompressible fluid in immediate proximity to the flat rigid wall with constant speed \(U_{0}\) and, simultaneously performing small-amplitude unsteady motions normal to the underlying boundary, see Figure 1. These can be, in particular, vertical and angular oscillations of a wing as a solid body as well as prescribed deformations of the wing surface. Restricting our analysis to a linear formulation and with reference to [PERSON] [32] one can show that for very small relative clearances under the wing the formulation of the flow problem can be essentially simplified and reduced to considering the following relationships (Note that here and further on all quantities and functions have been rendered non-dimensional with use of the wing chord \(b\) and speed of translational motion \(U_{0}\)):  
* Equation \[h\left(\frac{\partial^{2}\Gamma}{\partial x^{2}}+\frac{\partial^{2}\Gamma}{ \partial z^{2}}\right)=\frac{\partial y_{s}}{\partial x}-\frac{\partial y_{ s}}{\partial t},\,(x,z)\in S;\] (1)Boundary conditions  
\[\Gamma=0\text{ at the leading edge }l_{1}, \tag{2}\]  
\[\frac{\partial\Gamma}{\partial x}-\frac{\partial\Gamma}{\partial t}=0\text{ at the trailing edge }l_{2}. \tag{3}\]  
Note that the governing equation of extreme ground effect can be interpreted as mass (volume) conservation in a narrow channel between the wing and the ground. Correspondingly, the boundary condition (2) states the requirement of no vorticity in front of the wing, and the boundary condition (3) reflects the requirement of zero pressure jump across the wake behind the trailing edge (This requirement can be interpreted as a Kutta-Zhukovsky condition for a linearized unsteady liftin flow). It is worth mentioning that the mathematical model introduced in this paper belongs to a category of _limiting_ models. It allows to substantially simplify mathematics of flapping wing propulsion in close proximity to the ground (wall) enabling to obtain results and predictions for wings of finite aspect ratio both rigid and elastic in a wide range relative frequencies (Strouhal numbers) and to explore available margins for optimization. Due to unsteady 3D nature of the problem and smallness of gaps between lifting and bounding surfaces the CFD solution of this problem is quite difficult. It should be mentioned that the extreme-ground effect lifting and propulsive surface theory which reduces the flow model to 2D in horizontal plane \((x0z)\) parallel to the ground methodologically can be viewed as a complement to [PERSON] lifting line theory of high-aspect-ratio wing which is _almost_ 2D in a vertical plane \((x0y)\), and [PERSON] small-aspect-ratio wing theory deployed in \((y0z)\) plane. In the Formulas (1)-(3): \(\Gamma(x,z,t)=\varphi^{-}-\varphi^{+}\) is a jump of the velocity potential across the wing surface at a point with coordinates \((x,z)\) at a moment of time \(t\textit{y}_{s}(x,z,t)\) is a function, describing instantaneous position of the wing surface relative to the plane \(y=0\textit{h}\)-average distance of the trailing edge of the root chord as fraction of the chord; \(S,l_{1},l_{2}\) are correspondingly projections of the wing and its leading and trailing edge contours upon the interface plane. In linearized formulation the transition points separating the leading edge \(l_{1}\) and trailing edge \(l_{2}\) parts of the piece-wise smooth planform contour coincide with side tips of the wing. Note that the jump of the pressure coefficient can be calculated using the formula  
\[[p]=p^{-}-p^{+}=2\bigg{(}\frac{\partial\Gamma}{\partial x}-\frac{\partial \Gamma}{\partial t}\bigg{)}. \tag{4}\]  
In the case of harmonic oscillations, we can assume  
\[y_{w}(x,z,t)=h+q_{i}(t)f_{q_{i}}(x,z); \tag{5}\]  
Figure 1: A lifting surface oscillating in close proximity to the ground, \(S\),\(W\) are areas of the wing and wake.