Comparison of the Evolution of a Solitary Elastic Cylindrical Wave with Friedlander and Macdonald Profiles

摘要

A nonlinear elastic cylindrical radial displacement wave is analyzed theoretically and numerically for an arbitrary solitary wave profile and initial wave profiles in the form of Friedlander and Macdonald functions. The five-constant Murnaghan model is used. Unlike the most profiles of nonlinear waves in materials that have periodical or single humps, these waves have no hump, decrease monotonically, and have concave downward profile. Both profiles are very similar, have the properties of solitary wave, and, therefore, can be studied using the nonlinear theory of elasticity. A comparison of the Macdonald and Friedlander profiles as the solutions of the linear wave equation for a cylindrical wave shows that if the Macdonald function is considered as the exact solution of this equation, then the Friedlander function can be considered an approximate solution of this equation because its graphical representation is similar to that of the Macdonald function. The evolution of waves is studied by the approximate method of limitation of displacement gradient taking into account the first two approximations. Formulas are theoretically obtained for an approximate representation of a solitary cylindrical wave and a concrete representation of this wave for the given Macdonald and Friedlander initial profiles. It is shown that the flection of the profiles changes when the waves propagate some distance but the behavior of the profiles (evolution scenarios) does not differ significantly from each other. Then within the framework of the problem stated, both profiles are interchangeable despite the fact that they are mathematically represented in different ways.