A nonlinear eigenvalue problem in hydrodynamics

Abstract

In this note we use a result of M. Shinbrot [l] to investigate the existence and completeness of eigenfunctions for a boundary value problem that arises in the theory of small amplitude motions of a stratified polytropic gas. It has been noted before, in some other problems of hydrodynamic stability, that the “eigenvalue” parameter may be present in a nonlinear way in the differential operator, or may be present in the boundary condition. (See, for example, [l-5].) The particular problem discussed here has not been con- sidered before. (The references mentioned above are concerned with incom- pressible fluids.) Thus let us consider an inviscid, nonheat-conducting, polytropic, ideal gas under gravity. (A polytropic gas is one in which the internal energy is proportional to the temperature.) We assume that in equilibrium (no motion) the gas is stratified as a result of the action of gravity; that is, the equilibrium values of pressure, entropy and density depend only on the coordinate, say x, along the axis antiparallel to the gravitational force. The equations are the equations of conservation of momentum, of mass, and of energy; and the equation of state for a polytrope. As these are well known we do not write them down here. These equations, when linearized about the equilibrium values, have solutions of the form exp(ot + i& ‘2) multiplied by a function of x. Here t is time and y is the horizontal coordinate. We call R = 1 b 1 the wave-number. Let the equilibrium value of entropy at x be denoted s(x). Then as in [6], we introduce a new variable where c,, is the specific heat at constant pressure and x, is arbitrary. The above equation can be solved for x in terms of z (uniquely), say x = g(x). It is convenient to introduce J(Z) = exp( - s(g(x))/c,). 616