Internal gravity waves near to the sources of disturbances at the   critical modes of generation
  The paper presents the description of the structure of the nearest field of the internal gravity waves at the critical modes of their excitation. Studied are the exact solutions both for the elevation component and the vertical component of the speed describing the structure of the wave field in the direct vicinity of the source. At that the single mode of the elevation is expressed through the full elliptic integral of the first order, and the single mode of the vertical speed - through McDonald function and the logarithmic functions. As the result of the study it was possible to obtain expressions for the full field representing the sum of the wave modes and expressed through the derivatives of the gamma function. The obtained asymptotic and exact representations of the solution allow to describe the critical modes of generation of the internal gravity waves near to the sources of excitations - for the wide ranges of the sources movement velocity . 
Vitaly V. Bulatov (1), Yuriy V. Vladimirov (1) ((1) Institute for   Problems in Mechanics of Russian Academy of Sciences)