Abstract. Disorder of size (polydispersity) and mass of discrete elements or particles in randomly structured media (e.g., granular matter such as soil) has numerous effects on the materials' sound propagation characteristics. The influence of disorder on energy and momentum transport, the sound wave speed and its low-pass frequency-filtering characteristics is the subject of this study. The goal is understanding the connection between the particle-microscale disorder and dynamics and the system-macroscale wave propagation, which can be applied to nondestructive testing, seismic exploration of buried objects (oil, mineral, etc.) or to study the internal structure of the Earth. To isolate the longitudinal P-wave mode from shear and rotational modes, a one-dimensional system of equally sized elements or particles is used to study the effect of mass disorder alone via (direct and/or ensemble averaged) real time signals, signals in Fourier space, energy and dispersion curves. Increase in mass disorder (where disorder has been defined such that it is independent of the shape of the probability distribution of masses) decreases the sound wave speed along a granular chain. Energies associated with the eigenmodes can be used to obtain better quality dispersion relations for disordered chains; these dispersion relations confirm the decrease in pass frequency and wave speed with increasing disorder acting opposite to the wave acceleration close to the source.
We employ numerical simulations to understand the evolution of elastic standing waves in disordered frictional disk systems, where the dispersion relations of rotational sound modes are analyzed in detail. As in the case of frictional particles on a lattice, the rotational modes exhibit an "optical-like" dispersion relation in the high frequency regime, representing a shoulder of the vibrational density of states and fast oscillations of the autocorrelations of rotational velocities. A lattice-based model describes the dispersion relations of the rotational modes for small wave numbers. The rotational modes are perfectly explained by the model if tangential elastic forces between the disks in contact are large enough. If the tangential forces are comparable with or smaller than normal forces, the model fails for short wave lengths. However, the dispersion relation of the rotational modes then follows the model prediction for transverse modes, implying that the fast oscillations of disks' rotations switch to acoustic sound behavior. We evidence such a transition of the rotational modes by analyzing the eigen vectors of disordered frictional disks and identify upper and lower limits of the frequency-bands. We find that those are not reversed over the whole range of tangential stiffness as a remarkable difference from the rotational sound in frictional particles on a lattice. granular material | elastic wave | dispersion relation | disordered configuration | tangential force
A mechanical wave is propagation of vibration with transfer of energy and momentum. Understanding the spectral energy characteristics of a propagating wave through disordered granular media can assist in understanding the overall properties of wave propagation through inhomogeneous materials like soil. The study of these properties is aimed at modeling wave propagation for oil, mineral or gas exploration (seismic prospecting) or non-destructive testing of the internal structure of solids. The focus is on the total energy content of a pulse propagating through an idealized one-dimensional discrete particle system like a mass disordered granular chain, which allows understanding the energy attenuation due to disorder since it isolates the longitudinal P-wave from shear or rotational modes. It is observed from the signal that stronger disorder leads to faster attenuation of the signal. An ordered granular chain exhibits ballistic propagation of energy whereas, a disordered granular chain exhibits more diffusive like propagation, which eventually becomes localized at long time periods. For obtaining mean-field macroscopic/continuum properties, ensemble averaging has been used, however, such an ensemble averaged spectral energy response does not resolve multiple scattering, leading to loss of information, indicating the need for a different framework for micro-macro averaging.
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The reflection and transmission properties of electromagnetic waves obliquely incident on a relativistically moving uniaxial plasma slab are investigated theoretically and the maxima and minima of the power reflection coefficient as a function of the normalized incident wave frequency (Q) and the slab velocity (6) are discussed for slab motion parallel as well as normal to the slab boundaries.One can predict the nature of variation of the reflection coefficient with respect to any parameter from the variation of q' (the root of the dispersion equation) with that parameter. When q' is real and varies rapidly, the oscillations in the reflection coefficient are rapid and when q' dBers significantly from the free-space value, the amplitude will be large. In the imaginary range of q', the reflection coefficient is nonoscillatory.The discussion of the results is facilitated by introducing a parameter p which is a quadratic in p. For slab motion parallel to the interface, the two slab velocities for a given p give rise to same reflection coeficient whilst for normal motion the results are si@cantly different.
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