Loss of phase coherence11/12/2023 Daraio, Discrete breathers in one-dimensional diatomic granular crystals. Fang, Interaction of a contact resonance of microspheres with surface acoustic waves. Daraio, Bifurcation-based acoustic switching and rectification. Deymier (ed.), Acoustic Metamaterials and Phononic Crystals, Chapter 7, Springer Series in Solid-State Sciences, vol 173 (Springer, Berlin, 2013) In this Chapter, we present a number of simple models of nonlinear media to illustrate some notions related to nonlinear waves, coherence, and decoherence. Furthermore, the strength and order of nonlinearity can be selectively amplified in composite media comprising linear and nonlinear constituents. These include: (a) geometrical nonlinearity associated with Hertzian contact in granular media (b) intrinsic nonlinearity of the constituent materials and components nonlinear rotational degrees of freedom in composite structures (c) hysteretic nonlinearity and (d) open system nonlinearity from exchanging matter or energy with an external reservoir. These responses extend over a range of nonlinear types, strengths and orders. Consequently, sound-supporting media, phononic structures and acoustic metamaterials offer a broader palette of nonlinear responses with possibility of control of the coherence of phonon propagation. The occurrence of ballistic transport over distances of many microns, with significant contributions to thermal conductivity and coherent propagation through phononic materials with numerous interfaces, have been reported. High frequency phonons have been shown to propagate over long distance coherently even at room temperature. For instance, granular materials are highly nonlinear prototypical phononic structures that have been extensively studied whereby strong localization of modes such as solitary waves enforce long-range coherent energy propagation. However, recently, extraordinary modes of phonon transport have been demonstrated. The figure below presents an illustration of the concept.Interactions in nonlinear elastic media cause multiple scattering and resonances of sound waves, which lead to the loss of phase coherence and acoustic wave degradation through amplitude reduction. Therefore, in such cases, the segments of time where the phase difference is constant are referred to as phase coherent segments. Situations may exist where the relative phase between the two waves may constantly vary with time. The figure below shows the scenario where the relative phases are not constant, or phase incoherent with respect to each other. In such scenarios, the relative phase or phase difference is not constant and hence the incident and disturbed waves are no longer coherent in phase – or become phase incoherent. In practical situations, waves (e.g., electromagnetic waves, acoustic waves) may face disturbances due to the surrounding environments (e.g., reflection, refraction, scattering, diffraction etc.,) and may lead to a change in phase. Phase coherent waves are particularly useful in producing stable interference patterns. Thus, it can be concluded that there exists a constant phase difference and hence a perfect phase coherence between the two waves. However, the phase of orange wave relative to the black wave (or vice-versa) does not change as a function of time and they produce crests and troughs at the same time. In the above figure, two sinusoidal waves of the same frequency, wavelength, and amplitude are generated and time-shifted with respect to each other. This means that the two waves are perfectly coherent in phase with changes in time. For instance, when two sinusoidal signals or sine waves are resonating at the same frequency and are time-shifted relative to each other, their relative phase does not change with respect to time. Phase Coherence is a phenomenon where a constant phase difference exists between any two signals or waves of the same frequency.
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