Wave equation with cubic nonlinearity and excitation of oscillations in the “medium-source” system

A new accurate solution of the wave equation with a source depending on the required function-time has been obtained. The source function has a polynomial (third degree) nonlinearity, as well as two complementary additive members, which include the second and third degrees of the required function and an explicit sine dependence on time. The constructed relations describe namely the process of excitation of oscillations in the “medium - nonlinear rheonomic source” system and therefore do not contain, as a special case, the solution of the wave equation with common cubic nonlinearity. Physical interpretation of the results of work is explained by the properties of the effect of the periodic external force on the medium. The solution has been obtained on the plane the “required function-time” and gives analytical expressions of the partial spatial and time derivatives of the required function. This allows to study the rheonomic properties of isolines of the required function: their velocity and the conditions at which this velocity is alternating. An important influence on the isolines behavior is exerted by the slope of the source function in a small neighborhood of the zero value of the required function. Specifically: its sign determines the mode of (subsonic or supersonic) isoline movement, and its module serves as the scale when calculating the non-dimensional frequency of the exciting oscillations. The intervals of low and high frequencies are considered in this work. At every instant, the gradient properties of the required function are characterized by the monotone profile, located in the semi-infinite domain on the plane of the “coordinate - required function”. The conditions are indicated, at which time-periodic kink-pulsations occur: at separate moments, the source monotone profile transforms into a kink, which corresponds to two equilibrium states of the system. The rheonomic properties of the monotone profiles curvature have been studied: the appearance of points of inflection and rectification points. Two monotone profiles have been considered: the left and the right branches, located in the semi-infinite domains, respectively, to the left and right of the origin of coordinates. These branches move in oscillation, occasionally approaching each other and moving apart. At the moments when the branches adjoin the origin of coordinates, they form an immobile discontinuity, which is either weak or strong depending on whether the branch slopes are respectively either of different sign or of the same sign. It has been revealed that in the course of such oscillation process, a transonic transition is possible in the high frequency interval: the isoline velocity changes from the subsonic value to the supersonic one. A wave type formation has been constructed: the left and the right branches, forming either weak or strong discontinuity, move in time-periodic oscillation along the coordinate axis.