The anti-localization of non-stationary linear waves and its relation to the localization. The simplest illustrative problem

Shishkina, Ekaterina V.; Гаврилов, С. Н.; Mochalova, Yulia

doi:10.1016/j.jsv.2023.117673

Cited by 4 publications

(10 citation statements)

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“…On the other hand, for a system where the hodograph of P(0, T ) transversely crosses the boundary of L where Ω 0 → Ω * , we expect fast decaying of the oscillation amplitude W. Indeed, due to the anti-localization, the rate of decay near the discrete sub-system for the corresponding system with constant parameters lying outside L is τ −3/2 [20]. In Fig.…”

Section: Systems With Parameters Approaching and Crossing The Boundar...mentioning

confidence: 85%

“…Remark 7.4 For the particular case of the discrete subsystem in the form of pure inertial inclusion, the result, which shows that the decay rate for the vanishing component is τ −3/2 was obtained by Kaplunov in [27], see also earlier studies [102][103][104][105][106] concerning infinite discrete systems. In paper [20], we first time have demonstrated that the emergence and the intensity of the antilocalization are related not with its non-uniformity itself,…”

Section: Systems With Parameters Approaching and Crossing The Boundar...mentioning

confidence: 99%

“…Later, we have shown that at least for ǫ = 0 the displacement U(t) in the composite system vanishes asymptotically faster than in the corresponding pure continuum system. This happens due to the existence of the wave phenomenon, which we have called the anti-localization of non-stationary linear waves [20,[58][59][60].…”

Section: Remark 23mentioning

confidence: 99%

“…Actually, in such a system with constant parameters, the order of decay is τ −1/2 [20]. Thus, apparently, the approximate description for U can be obtained as a matched asymptotics, which stages correspond to the single-mode approximation for the system with time-varying parameters and asymptotics for the same system with constant parameters, respectively.…”

Section: Systems With Parameters Approaching and Crossing The Boundar...mentioning

confidence: 99%

“…The particular case of such a system is a uniform pure continuum system. We have called this wave phenomenon the anti-localization of non-stationary linear waves [20,[58][59][60]. Thus, just the existence of the anti-localization makes the single-mode asymptotics to be so good approximation for a system with parameters inside the localization domain far from its boundaries.…”

Section: Systems With Parameters Approaching and Crossing The Boundar...mentioning

confidence: 99%

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Universal formal asymptotics for localized oscillation of a discrete mass-spring-damper system of time-varying properties, embedded into a one-dimensional medium described by the telegraph equation with variable coefficients

Gavrilov,

Poroshin,

Shishkina

et al. 2024

Preprint

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We consider a quite general problem concerning a linear free oscillation of a discrete mass-spring-damper system. This discrete subsystem is embedded into a one-dimensional continuum medium described by the linear telegraph equation. In a particular case, the discrete subsystem can move along the continuum one at a sub-critical speed. Provided that the dissipation in both discrete and continuum subsystems is absent, if parameters of the subsystems are constants, under certain conditions (the localization conditions), a non-vanishing oscillation localized near the discrete subsystem can be possible. In the paper we assume that the dissipation in the damper and the medium is small, and all discrete-continuum system parameters are slowly varying functions in time and in space (when applicable), such that the localization condition is fulfilled for the instantaneous values of the parameters in a certain neighbourhood of the discrete subsystem position. This general statement can describe a number of mechanical systems of various nature. We derive the expression for the leading-order term of a universal asymptotics, which describes a localized oscillation of the discrete subsystem. In the non-dissipative case, the leading-order term of the expansion for the amplitude is found in the form of an algebraic expression, which involves the instantaneous values of the system parameters. In the dissipative case, the leading-order term for the amplitude, generally, is found in quadratures in the form of a functional, which depends on the history of the system parameters, though in some exceptional cases the result can be obtained as a function of time and the instantaneous limiting values of the system parameters. In previous studies, several non-dissipative particular cases of the problem under consideration are investigated using a similar approach, provided that only one parameter of the discrete-continuum system is a slowly time-varying non-constant quantity, whereas all other parameters are constants. We show that asymptotics obtained in previous studies are particular cases of the universal asymptotics. The existence of a universal asymptotics in the form of a function is a non-trivial fact, which does not follow from the summarization of the results obtained for particular cases. Finally, we have justified the universal asymptotics by numerical calculations.

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Section: Systems With Parameters Approaching and Crossing The Boundar...mentioning

confidence: 85%

Section: Systems With Parameters Approaching and Crossing The Boundar...mentioning

confidence: 99%

Section: Remark 23mentioning

confidence: 99%

Section: Systems With Parameters Approaching and Crossing The Boundar...mentioning

confidence: 99%

Section: Systems With Parameters Approaching and Crossing The Boundar...mentioning

confidence: 99%

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Universal formal asymptotics for localized oscillation of a discrete mass-spring-damper system of time-varying properties, embedded into a one-dimensional medium described by the telegraph equation with variable coefficients

Gavrilov,

Poroshin,

Shishkina

et al. 2024

Preprint

Self Cite

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Localized Modes in a 1D Harmonic Crystal with a Mass-Spring Inclusion

Shishkina,

Gavrilov

2023

Advanced Structured Materials

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Non-stationary elastic wave scattering and energy transport in a one-dimensional harmonic chain with an isotopic defect

Gavrilov,

Shishkina

2024

Continuum Mech. Thermodyn.

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The anti-localization of non-stationary linear waves and its relation to the localization. The simplest illustrative problem

Cited by 4 publications

References 50 publications

Universal formal asymptotics for localized oscillation of a discrete mass-spring-damper system of time-varying properties, embedded into a one-dimensional medium described by the telegraph equation with variable coefficients

Universal formal asymptotics for localized oscillation of a discrete mass-spring-damper system of time-varying properties, embedded into a one-dimensional medium described by the telegraph equation with variable coefficients

Localized Modes in a 1D Harmonic Crystal with a Mass-Spring Inclusion

Non-stationary elastic wave scattering and energy transport in a one-dimensional harmonic chain with an isotopic defect

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