Abstract
The axially symmetric propagation of bending waves in a thin Timoshenko-type cylindrical shell, interacting with a nonlinear elastic Winkler medium, is herein studied. With the help of asymptotic integration, two analytically solvable models were obtained that have no physically realizable solitary wave solutions. The possibility for the real existence of exact solutions, in the form of traveling periodic waves of the nonlinear inhomogeneous Klein–Gordon equation, was established. Two cases were identified, which enabled the development of the modulation instability of periodic traveling waves: (1) a shell preliminarily compressed along a generatrix, surrounded by an elastic medium with hard nonlinearity, and (2) a preliminarily stretched shell interacting with an elastic medium with soft nonlinearity.
Highlights
IntroductionThe construction and study of analytically solvable models for nonlinear wave dynamics of deformable systems is a complicated multi-factorial problem, which has led to the development of non-classical mathematical physics, computational mathematics, and nondestructive testing methods in acoustics
Two cases were identified, which enabled the development of the modulation instability of periodic traveling waves: (1) a shell preliminarily compressed along a generatrix, surrounded by an elastic medium with hard nonlinearity, and (2) a preliminarily stretched shell interacting with an elastic medium with soft nonlinearity
The construction and study of analytically solvable models for nonlinear wave dynamics of deformable systems is a complicated multi-factorial problem, which has led to the development of non-classical mathematical physics, computational mathematics, and nondestructive testing methods in acoustics
Summary
The construction and study of analytically solvable models for nonlinear wave dynamics of deformable systems is a complicated multi-factorial problem, which has led to the development of non-classical mathematical physics, computational mathematics, and nondestructive testing methods in acoustics. The constructed model can be considered advantageous, if exact analytical solutions provide clear understanding of a physical phenomenon. It is the very case when computational simulations enable confirming, specifying, or even rejecting original hypotheses [8]. Asymptotics are introduced, which enable constructing models, supposing the existence of exact physically realizable solutions in the form of traveling periodic waves.
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