Realization of nonlinear Cherenkov frequency down-conversion by phase velocity modulation

By using ultrathin nonlinear media, the phase velocity of nonlinear polarization waves can be continuously adjusted. We realized nonlinear Cherenkov radiation (NCR) in an anomalous dispersion medium experimentally, which breaks the minimum speed limit of NCR. This modified nonlinear Cherenkov radiation has two new special states with respect to its prior, i.e., the degenerate state and extreme state. Also, periodically arranged ultrathin nonlinear media can enhance NCR and realize the nonlinear Smith-Purcell effect.

By use of the frequency response method with a radioactive tracer, it was possible to determine fluid dispersion and distribution in a natural consolidated and an unconsolidated medium. Measurements were made in a linear flow system at oleic saturations of 69 per cent in the consolidated medium, and 100 per cent in both media. Dispersion and distribution were obtained by measuring the amplitude attenuation and the phase velocity of sinusoidal waves with a dual monitor apparatus. The gamma ray emissions permitted in situ measurements at any distance along the porous samples. One result of importance was that the effective diffusivity increased as the wave length increased. As a consequence, a dispersion coefficient appropriate for the injection of large slugs might exceed the value measured by use of small slugs. Since flow models based solely on fluid velocity and an effective diffusivity coefficient imply that the diffusivity should be independent of frequency, such representations were not adequate for the data of this study. A comparison was made with a capacitance model of porous media with dead-end PV's, but even this model was not completely adequate. By using attenuation and phase velocity data, fluid dispersion can be predicted without postulating a differential equation satisfied by the tracer concentration, thereby eliminating the need of a complicated model to represent dispersion. Introduction The flow of similar miscible fluids through a porous medium can be fairly adequately described by two parameters: the average fluid velocity and the effective diffusivity.1-3 It has been pointed out recently, however, that significant discrepancies exist between this representation and the experimental data.4-7 An improved agreement can be obtained by introducing additional parameters based on the concept of dead-end pores. The purpose of the present investigation was to find out whether the frequency response method could be used to measure the relevant parameters. The method was used in the following form. A stream of fluid was flowed at a constant rate through a sample of porous material and the concentration of a radioactive tracer in the fluid was varied sinusoidally at a fixed frequency. The effects of flow through a porous medium are a decrease in the amplitude of the concentration wave and an increase in the velocity of the peaks of the waves above the average velocity. Attenuation and phase velocity of the waves were measured as a function of frequency and fluid velocity. The simple two-parameter model implies that the diffusivity should be independent of frequency. Data reported in this paper show that the diffusivity decreases as the frequency increases. Hence, as shown also by many others, the two-parameter model is not completely adequate. Coats and Smith5 used two additional parameters in their model: the volume of the dead-end pores and the rate of mass transfer between dead-end pores and the flowing stream. Their capacitance model of a porous medium containing some stagnant fluid, to which transfer occurs by molecular diffusion, did not explain the dispersion results of either the present study or of theirs. Instead, the capacitance effect can be better described as the result of extreme velocity variations within the pores of the medium, with transfer between the velocity zones by convection.

Phase velocities of Love waves from five major earthquakes are measured over six great circle paths in the period range of 50 to 400 seconds. For two of the great circle paths the phase velocities of Rayleigh waves are also obtained. The digitized seismograph traces are Fourier analyzed, and the phase spectra are used in determining the phase velocities. Where the great circle paths are close, the phase velocities over these paths are found to be in very good agreement with each other indicating that the measured velocities are accurate and reliable. Phase velocities of Love waves over paths that lie far from each other are different, and this difference is consistent and much greater than the experimental error. From this it is concluded that there are lateral variations in the structure of the earth's mantle. One interpretation of this variation is that the mantle under the continents is different from that under the oceans, since the path with the highest phase velocities is almost completely oceanic. This interpretation, however, is not unique and variations under the oceans and continents are also possible. Group velocities are computed from the phase velocities and are also directly measured from the seismograms. The group-velocity curve of Love waves has a plateau between periods of 100 and 300 seconds with a shallow minimum at about 290 seconds. The sources of error in both Fourier analysis and direct time domain methods of phase velocity measurement are discussed.

Phase velocities of Rayleigh waves in Finland have been measured by the Fourier analysis method, using records of three stations which are on almost the same great circle as several Greek earthquakes. Phase velocities of Love waves of two of these earthquakes have also been determined. The wave groups were divided by numerical filtering into two frequency channels, which were analyzed separately. The resulting phase velocity curves were then smoothed in order to remove rapid oscillations, and mean values were calculated. The phase velocities were found to be slightly lower than those found for the Canadian Shield (Brune and Dorman, 1963), the maximum deviation for Rayleigh waves being about 0.07 km/sec at a period of 28 seconds and for Love waves about 0.15 km/sec at a period of 32 seconds. At periods shorter than 40 seconds the phase velocities of Rayleigh waves were found to be equal in Northern and Southern Finland within the accuracy of measurement.

PurposeThis mathematical analysis has been accomplished for the purpose of understanding the propagation behaviour like phase velocity and attenuation of Love-type waves through visco-micropolar composite Earth’s structure.Design/methodology/approachThe considered geometry of this problem involves a micropolar Voigt-type viscoelastic stratum imperfectly bonded to a heterogeneous Voigt-type viscoelastic substratum. With the aid of governing equations of motion of each individual medium and method of separation of variable, the components of micro-rotation and displacement have been obtained.FindingsThe boundary conditions of the presumed geometry at the free surface and at the interface, together with the obtained components of micro-rotation, displacement and mechanical stresses give rise to the determinant form of the dispersion relation. Moreover, some noteworthy cases have also been extrapolated in detail. Graphical interpretation irradiating the impact of viscoelasticity, micropolarity, heterogeneity and imperfectness on the phase velocity and attenuation of Love-type waves is the principal highlight of the present study.Practical implicationsIn this study, the influence of the considered parameters such as micropolarity, viscoelasticity, heterogeneity, and imperfectness has been elucidated graphically on the phase velocity and attenuation of Love-type waves. It has been noticed from the graphs that with the rising magnitude of micropolarity and heterogeneity, the attenuation curves shift upwards, that is the loss of energy of these waves takes place in a rapid way. Hence, from the outcomes of the present analysis, it can be concluded that heterogeneous micropolar stratified media can serve as a helpful tool in increasing the attenuation or in other words, loss of energy of Love-type waves, thus reducing the devastating behaviour of these waves.Originality/valueTill date, the mathematical modelling as well as vibrational analysis of Love-type waves in a viscoelastic substrate overloaded by visco-micropolar composite Earth’s structure with mechanical interfacial imperfection remain unattempted by researchers round the globe. The current analysis is an approach for studying the traversal traits of surface waves (here, Love-type waves) in a realistic stratified model of the Earth’s crust and may thus, serves as a dynamic paraphernalia in various domains like earthquake and geotechnical engineering; exploration geology and soil mechanics and many more, both in a conceptual as well as pragmatic manner.

The relationship between the experimentally obtained attenuation coefficient and dispersion of the phase velocity of longitudinal ultrasonic waves in polymer composite materials has been analyzed. The laser optoacoustic method has been used to measure the ultrasonic attenuation and velocity in a wide frequency range. Verification of the Kramers–Kronig relations for ultrasound-scattering and absorbing carbon fiber reinforced plastic (CFRP) composites has been performed using the analysis of the relationship between the frequency dependences of the attenuation coefficient and phase velocity of longitudinal acoustic waves in samples. We have shown that the Kramers–Kronig relations between the ultrasonic attenuation and phase velocity are valid within the 1–10 MHz frequency range without regard to a particular mechanism of the decay in the energy of an initial acoustic wave during its propagation in a composite.

Effect of magneto-elasticity, hydrostatic stress and gravity on Rayleigh waves in a hydrostatic stressed magneto-elastic crystalline medium over a gravitating half-space with sliding contact

The physical, gain–loss model for the stochasticity of the phase velocity of a wind-driven water wave

This paper deals with the propagation of elastic waves along a cylindrical borehole embedded in an anisotropic fluid-saturated porous solid of infinite extent. Biot's theory is used to derive the Christoffel equation for the propagation of cylindrical waves in an anisotropic fluid-saturated porous material. The saturated fluid is viscous. Waves of axial symmetry are considered here. The frequency equations of surface waves corresponding to the empty and liquid-filled borehole are obtained. The effects of transverse isotropy on the phase velocity of surface waves are studied. The phase velocity of surface waves increases with anisotropy. Surface modes in empty and liquid-filled boreholes in an isotropic poroelastic solid and Rayleigh waves propagating on the free surface of a transversely isotropic fluid-saturated porous solid are obtained as particular cases. The presence or absence of fluid in the borehole affects the phase velocity considerably. The phase velocity increases when the empty borehole is filled with liquid.

Dispersion curves of phase velocities of Rayleigh waves are determined by the method of frequency-time analysis in a range of periods of 10–200 s from data of 43 interstation traces in Central Asia. Because the joint use of phase and group velocities significantly decreases the uncertainty in the determination of S wave velocity structures, the same traces were used for calculating group velocities from tomographic reconstructions obtained in [Yanovskaya and Kozhevnikov, 2003, 2006] and determining average velocity structures along these traces. The velocity structures were calculated by the Monte Carlo and linear inversion methods, which gave consistent results. Using velocity values obtained at fixed depths by the 2-D tomography method, lateral variations in velocities at these depths were estimated, which allowed us to construct smoothed vertical velocity structures at some points in the region. The resulting structures were used as initial approximations for constructing local velocity structures solely from previously obtained local dispersion curves of group velocities in the area (32°–56°N, 80°–120°E). Based on these structures, we mapped the lateral distribution of velocity variations at upper mantle depths of 75–400 km and along three vertical profiles. The inferred velocity variations are in good agreement with data on the tectonics of the region.

In this paper, the variational inverse method for determining the depth of nonhomogeneous surface layers in elastic materials, from the measurements of the group velocity of surface shear horizontal (SH) waves, is developed. The direct problem for a given a priori type of profile of the coefficientc44in(x) (e.g. linear, Gaussian, etc.) is solved. The dispersion curves of phase and group velocity of surface SH waves in nonhomogeneous solids are calculated. Experimental verification of the inverse method has been performed for step profiles (structure of Cu on steel). It is stated that the inverse method based on the measurements of group velocity (for step profile) gives a smaller error in the unknown depth of the surface layer than that resulting from the inverse method based on the measurements of phase velocity of surface SH waves.

The question of how two known models of generalized continua describe a Rayleigh surface wave propagating along a stress-free surface of an elastic half-space is studied. The models of a gradient-elastic medium and a simplified (reduced) Cosserat medium are selected as the tested ones. Dispersion equations are obtained. The analysis of which showed that both models state that the dispersion properties of a Rayleigh surface wave in the wave number– frequency plane are described by two curves, one of which (the lower one) comes from the origin of coordinates, the origin of the second is shifted upward along the frequency axis. The surface wave is two-mode and each of its modes has dispersion, since the velocities of both modes depend on the frequency. Both models also state that the bulk shear wave has dispersion. For a gradient-elastic medium, as the frequency increases, the velocity of each mode of the surface wave increases, and if the velocity of the lower mode reaches a horizontal asymptote from below, then the velocity of the upper mode first reaches a maximum and only then reaches this horizontal asymptote from above. For any non-zero value of the wave number (or frequency), the phase velocity of the shear wave is greater than the velocity of the shear wave in a classical medium. The velocity of the surface wave cannot be greater than the phase velocity of the shear wave, and their equality is fulfilled only in a certain frequency range. In a reduced Cosserat medium, the velocity of the upper mode of the surface wave is improbably high, at high frequencies it increases indefinitely. The velocity of the lower mode of the surface wave decreases with increasing frequency, but in the entire frequency range it remains greater than the phase velocity of the shear wave.

We consider a simplified (reduced) dynamic model of a Cosserat medium, which occupies an intermediate position between the classical dynamic theory of elasticity and the proper Cosserat medium model, which has asymmetry in the stress tensor and the presence of moment stresses. In contrast to the latter, in the simplified model, three of the six elastic constants are zero and, as a result, there is no moment stress tensor. In the two-dimensional formulation for the model of a reduced medium, the problem of the propagation of an elastic surface wave along the half-space boundary was solved. The solution of the equations was described as the sum of the scalar and vector potentials, and only one component of the vector potential is nonzero. It is shown that such a wave, in contrast to the classical surface Rayleigh wave, has a dispersion. In the plane “phase velocity-frequency” for such waves there are two dispersion branches: the lower (acoustic) and upper (optical). With increasing frequency, the phase velocity of the wave related to the lower dispersion branch decreases. The phase velocity of the wave related to the upper dispersion branch increases with increasing frequency. The phase velocity of the surface wave in the entire frequency range exceeds the phase velocity of the bulk shear wave. The stresses and displacements arising in the zone of propagation of the surface wave are calculated.

In structural systems, impact-induced longitudinal elastic waves travel with finite speeds that depend on the material properties. Using Fourier method of analysis, the exact wave motion can be described as the sum of infinite number of harmonic waves which have the same phase velocity. In this case the medium is said to be nondispersive, since the phase velocities of the harmonic waves are equal and equal to the group velocity of the resulting wave motion. In mechanism systems with intermittent motion, on the other hand, elastic members undergo finite rotations. In this investigation, the effect of the finite rotation, coefficient of restitution, and impact conditions on the propagation of the impact-induced waves in costrained elastic systems is examined. The system equations of motion are developed using the principle of virtual work in dynamics. The jump discontinuities in the system variables as the result of impact are predicted using the generalized impulse momentum equations that involve the coefficient of restitution. It is shown that the phase velocities of different harmonic waves are no longer equal, that is, dispersion occurs in perfectly elastic mechanism members as the result of the finite rotation. The analysis presented in this paper shows that the finite rotation has more significant effect on the phase velocity of the low frequency harmonics as compared to the high frequency harmonics. A rotation-wave number that depends on the material properties and the wave length is defined for each harmonic wave. It is shown that if the angular velocity of the elastic member becomes large such that the rotation-wave number of a mode exceeds one, the associated modal displacement is no longer oscillatory.

An analytical framework is developed to explore the dispersion behavior of Rayleigh waves in a functionally graded piezoelectric (FGP) plate interfaced with a dry sandy foundation, accounting for nonlocal elasticity effects and interfacial imperfections. The governing equations are formulated using nonlocal elasticity theory and solved analytically via harmonic wave analysis approach. Interface effects are modeled using normal and tangential stiffness parameters, allowing for partial bonding at the interface. Numerical simulations are performed considering both electrically open and electrically short surface conditions. The influence of key parameters including material gradation, nonlocal scale effects, interfacial stiffness, and substrate properties on the phase velocity of Rayleigh waves is examined in detail. Parametric analysis reveals that variations in the plate thickness and nonlocal parameter considerably affect the phase velocity and dispersion behavior of Rayleigh waves. The 2D and 3D plots of the displacement and electric potential fields are generated using Mathematica , illustrating the spatial variation and confinement of wave energy within the FGP layer. The results not only validate classical trends under limiting conditions but also reveal important coupled effects that are absent in traditional local models. These findings contribute to a deeper understanding of wave mechanics in smart-layered structures and offer valuable guidance for the design and optimization of advanced piezoelectric sensors, actuators, and SAW devices.