On the periodical solutions for single-dimensional gas dynamics

Abstract

In the theory of one-dimensional non stationary adiabatic gas dynamics the motion of a perfect ideal gas with plane waves is considered. With the help of Lagrangian mass variable the equations can be reduced to the system with quadratic nonlinearity. The solutions for the motion law and pressure, in particular, are sought in the form of series in powers of the sine of time with coefficients depending on the mass. For the series construction it is necessary to define the first three coefficients according to initial conditions. All the other coefficients are calculated recurrently without solving any differential equations or integration. Numerical calculations of series coefficients also show a regular behavior of the constructed solutions under certain restrictions on the derivatives of predetermined functions. The advantage of this approach over the Fourier series expansions of motion law and pressure is precisely the finiteness of algebraic recurrence relations associated only with the computation of derivatives with respect to the mass. As a result a new class of gas dynamics solutions is obtained. The examples with no any singularity are presented. The way to generalize the constructed solution is discussed.