The initial value problem for acoustic gravity waves
The initial value problem for acoustic-gravity waves in atmospheres composed of one or two isothermal layers is solved. For the case of two layers the problem is formulated in an indefinite space. Introduction. Acoustic-gravity waves are small-amplitude disturbances which propagate through a perfect, compressible fluid stratified in a gravitational field. A discussion of the physical properties of these waves including their properties in various parameter regimes, such as the acoustic limit at high frequencies, etc., can be found in reference [1], which also includes an extensive bibliography. In this paper we shall solve the initial value problem for such waves by formulating it as an eigenvalue problem in a Hilbert space, or, in the case of two layers, in an indefinite space. The form of the solution is convenient for application to problems involving a disturbance which is initially strong enough to require nonlinear techniques; if the nonlinear problem can be solved until the disturbance is weak, the end of the nonlinear calculation can serve as an initial condition for the present technique. The method is also of particular value for problems where one is interested in calculating the coupling of an initial configuration to one particular type of mode, such as ducted modes; since these modes appear in our formulation as eigenvectors orthogonal to other modes, the coupling can be calculated without solving the full problem; for example, in the case of an atmosphere composed of two isothermal layers one can calculate the coupling to the ducted modes without solving the full problem. We have found the formulation convenient to use for practical problems and think others may find use for it, but another reason for presenting the results is the possibility that mathematical physicists interested in indefinite spaces may find the space used here to solve a physics problem useful as an example of indefinite spaces. The case of one isothermal layer. Hydrodynamic variables and equations. To begin with, we consider the case of one isothermal layer. The hydrostatic solution about which we wish to perturb is given ♦Received May 2, 1988. ©1989 Brown University 295