Abstract
The equation with a nonzero generalized Laplace invariant which admits the maximal order motion group of a two-dimensional acoustics model in a vertically stratified medium is investigated by group analysis methods. All essentially different submodels of the mathematical model defined by this equation have been obtained. The simplest essentially different, i.e. non-connected by reversible point transformations, invariant solutions of this equation have been found. The use of the derived solution production formulas to these invariant solutions gives an independent 28-parametric set of essentially different exact solutions of the equation. The use of infinitesimal solution production formulas to these invariant solutions gives a denumerable set of exact solutions. A linear span of this set forms an infinite dimensional vector space (with a denumerable basis) of exact solutions. Thus, a database of exact solutions of this equation has been created. The form of the invariant solutions shows that the character of solutions of the equation depends strongly on the wave number. For low frequency vibrations solutions polynomially depend on the variable that defines the stratification of a medium. For high frequency vibrations solutions are periodic with an accuracy of the weighting factor. A fundamental solution of the equation is found by the method of doubling the admitted group, which makes it possible to construct Green's formulas and to obtain a generalized solution of the Poisson boundary value problem. The results obtained can be used in calculations related to geophysical exploration of oil and gas in stratified media and hydroacoustics.
Keywords: acoustic waves, stratified medium, invariant solutions, formulas of solution production.
