A nonlinear plane longitudinal elastic displacement wave is studied theoretically and numerically using the Murnaghan model for two forms of initial profile: harmonic and bell-shaped. A major novelty is that the evolution of waves is analyzed by approximate methods taking into account the first three approximations. The harmonic wave is analyzed only to compare with the new results for the bell-shaped wave. Some significant differences between the evolution of waves are shown. The initially symmetric profiles transform differently due to distortion: symmetrically (for the harmonic profile) and asymmetrically (for the bell-shaped profile). The third approximation introduces the fourth harmonic for the harmonic wave when this wave is analyzed by the method of successive approximations, while the bell-shaped wave is characterized in the third approximation differently when using the method of constraints on the displacement gradient. At relatively long distances from the beginning of the propagation, the one-hump bell-shaped wave transforms into a two-hump one. These humps adjoin each other halving their lengths. The third approximation allows us to observe new wave effects: the asymmetry of the left and right humps about their peaks and the asymmetry of the humps about each other; the lowering of the left hump and the rise of the right one. The results obtained are analyzed.
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Translated from Prikladnaya Mekhanika, Vol. 56, No. 5, pp. 65–77, September–October 2020.
This study was sponsored by the budget program “Support for Priority Areas of Scientific Research” (KPKVK 6541230).
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Rushchitsky, J.J., Yurchuk, V.N. Effect of the Third Approximation in the Analysis of the Evolution of a Nonlinear Elastic P-wave. Part 1*. Int Appl Mech 56, 581–589 (2020). https://doi.org/10.1007/s10778-020-01036-4
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DOI: https://doi.org/10.1007/s10778-020-01036-4