This paper discusses the nonlinear acoustic wave propagation problem in atmosphere, well-posedness of the problem establishes the regularity requirement of the forcing function. This property plays an important role in numerical simulation.
Mathematical formulation is based on a two-dimensional model for the acoustic wave propagation in an isothermal atmosphere. Regular perturbation method is used to expand the wave propagation problem generated by a point source from an instantaneous release of small but finite amount of energy. Solution is retained to the second order. Since the perturbation approach is not uniformly valid in the vicinity of the source where shock discontinuity exists, the problem addresses only to region away from the source.
The linear systems formed by the perturbation scheme are transformed into symmetric systems of partial differential equations and are shown to be hyperbolic but not strongly hyperbolic. Well posedness of the problem is established from the inequality deduced from the energy integral. Well- posedness of the problem requires that the forcing functions be square integrable, i.e., \(\in L_ 2(\Omega)\). Since the forcing function of the second order problem contains the derivatives of the solution to the first order problem, this leads to the requirement that both the solution and the forcing function be in H 1(\(\Omega)\), Sobolev space of order 1. Thus a sufficient condition to ensure enough smoothness on the forcing function, such as \(C^{\infty}\), rather than a step by step consideration.
For numerical computation, it is necessary to truncate the half space into a finite region. Radiation boundary condition is discussed and derived for a finite boundary in the vertical direction.