Stability of a Limiting Case of Plane Couette Flow

A fluid is supposed to be viscous and incompressible. The flow examined is $u=gy$; $v=0\mathrm{}$ ($0<~y<\mathrm{}\infty$; $\infty <x<\mathrm{}\infty$). Its stability is investigated by the method of small vibrations. Accepting the legitimacy of the usual expansions, the problem is reduced to the solution of a transcendental equation containing one parameter apart from the unknown. It is rigorously shown that the solutions of this equation are such as to make all modes of vibration of the flow damped.