## 1 Introduction  
It is well-known that for an appropriately designed wing the presence of an underlying solid boundary enhances its lifting properties and the corresponding phenomenon is known as _ground effect_. It turns out that a wing oscillating near a wall similarly manifests enhancement of its propulsive characteristics, such as thrust and efficiency, as compared to an unbounded fluid case. Therewith, decrease in relative distance to the wall results in further improvement of the propulsion. An investigation of this _propulsion ground effect_ from the viewpoint of behavior of thrust and efficiency can be carried out with the use of an asymptotic approach, directly accounting for the smallness of relative average distance from the wall. Developed therefrom is an _asymptotic theory of a flapping wing in immediate proximity to the ground_, leading to analytical results even for a wing of the finite aspect ratio. The proposed mathematical model enables analysis of the efficiency of different modes of operation. It can also be used as an instrument of control of results obtained in this limiting case computationally. _The paper includes_: a short review of relevant publications; mathematical formulation of the flow problem for oscillations of a flat wing based on the concept of _extreme ground effect_; derivations of general expressions for lift, moment and thrust coefficients as well as for propulsive efficiency for separate motions and combination of motions; an illustration of nonplanar unsteady ground effect; discussion of the case of a wing oscillating in a narrow canal between parallel walls and conclusions. The purpose of this work consists in using substantial simplification of the relevant mathematics which opens an interesting and practical parametric space for investigation of flapping wing propulsion close to solid boundaries with visible extension to elastic propulsion systems of hydro-bionic type. Propulsive characteristics of oscillating wing out of ground effect have been investigated in the works of [PERSON] [1], [PERSON] [2], [PERSON] [3] and [PERSON] [4]. [PERSON] and [PERSON] [5] provided one of the first reviews of research and development of flapping wing propulsors and some representative vehicles equipped with them. In particular, they analyzed natural bionic propulsors of insects, flying warm-blooded vertebrates, also touching upon fish and cetaceans. The authors discussed an engineering classification of propulsive systems with flapping wing elements and some 2D and 3D mathematical models thereof comprising effects of the flow boundaries, unsteadiness and optimal modes of operation. Examples of vehicles making use of flapping wings included: a cutter on underwater flapping and lifting wings developed at Central Aerohydrodynamic Institute named after [PERSON] in the 1970s, see [PERSON] and [PERSON] [6] (These ideas have been further implemented by designers of _pumping hydrofoil_ and O-foil); a human-powered submarine \"Afalina\" of Saint Petersburg State Marine Technical University (1992), a 200-ton Russian fishing vessel with a spring-loaded bow wing, see [PERSON], [PERSON] and [PERSON] [7]; and MIT laboratory robot \"Robotuna\", see [PERSON], [PERSON] and [PERSON] [8]. In [PERSON] [9], a simplified mathematical model of a wave glider (WG) was introduced leading to a mass-spring type equation of forced motions for the WG upper displacement unit with account of the inertial and damping influence of the winged lower module. The motions thus found were input into [PERSON] flapping foil theory to determine the thrust of the winged module propelling the WG. The speed of the wave glider was then estimated by equating the thrust available for a given length of the oncoming wave to the drag of the WG, the latter including viscous drag of the glider elements and wave drag of the upper module. [PERSON] et al.