Traveling Waves of Arbitrary Amplitude in Compressible Hydrodynamics under Gravity: An Exact Solution

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Abstract
Exact traveling-wave solutions to the hydrodynamic equations for an isothermal atmosphere stratified by a uniform gravitational field are obtained in two Cartesian spatial coordinates and one temporal coordinate and represent simple Riemannian waves of arbitrary amplitude and wave form which reduce directly into acoustic waves in the absence of gravity. While a number of characteristics are similar to the theory of small amplitude acoustic-gravity waves, these solutions exhibit features such as shock formation and phase dispersion which cannot be treated in small-amplitude theory. Solutions are given in terms of relations between physical wave characteristics and hydrodynamic variables. Comparison with traveling atmospheric disturbances are proposed.