Expressions For Phase Velocity Research Articles

Generation mechanism of rogue waves for the discrete nonlinear Schrödinger equation

In this paper, we analyze the generation mechanism of rogue waves for the discrete nonlinear Schrödinger (DNLS) equation from the viewpoint of structural discontinuities. First of all, we derive the analytical breather solutions of the DNLS equation on a new nonvanishing background through the Darboux transformation (DT). Via the explicit expressions of group and phase velocities, we give the parameter conditions for existence of the velocity jumps, which are consistent with the derivation of rogue waves via the generalized DT. Furthermore, to verify such statement, we apply the Taylor expansion to the breather solutions and find that the first-order rogue wave can be obtained at the velocity-jumping point. Our analysis can help to enrich the understanding on the rogue waves for the discrete nonlinear systems.

Propagation of Love waves in an isotropic poroelastic layer bounded between two viscous fluids

The present paper is devoted to the study of Love wave propagation in an infinite isotropic poroelastic layer bounded between two compressible viscous liquids. The equations of motion of an isotropic poroelastic solid are formulated in the framework of Biot’s theory. An analytical solution for the complex frequency equation of Love waves is derived. The analytical expressions for phase velocity of Love waves are obtained and solved numerically for two materials, namely, sandstone saturated with kerosene and sandstone saturated with water. Further, effect of viscosity is observed by considering several particular cases.

Stationary rotary force waves on the liquid–air core interface of a swirl atomizer

A one-dimensional wave equation, applicable to the waves on the surface of the air-core of a swirl atomizer is derived analytically, by analogy to the similar one-dimensional wave equation derivation for shallow-water gravity waves. In addition an analogy to the flow of water over a weir is used to produce an analytical derivation of the flow over the lip of the outlet of a swirl atomizer using the principle of maximum flow. The principle of maximum flow is substantiated by reference to continuity of the discharge in the direction of streaming. For shallow-water gravity waves, the phase velocity is the same expression as for the critical velocity over the weir. Similarly, in the present work, the wave phase velocity on the surface of the air-core is shown to be the same expression as for the critical velocity for the flow at the outlet. In addition, this wave phase velocity is shown to be the square root of the product of the radial acceleration and the liquid thickness, as analogous with the wave phase velocity for shallow water gravity waves, which is the square root of the product of the acceleration due to gravity and the water depth. The work revisits the weirs and flumes work of Binnie et al. but using a different methodology. The results corroborate with the work of Binnie. High speed video, Laser Doppler Anemometry and deflected laser beam experimental work has been carried out on an oversize Perspex (Plexiglas) swirl atomizer. Three distinctive types of waves were detected: helical striations, low amplitude random ripples and low frequency stationary waves. It is the latter wave type that is considered further in this article. The experimentally observed waves appear to be stationary upon the axially moving flow. The mathematical analysis allows for the possibility of a negative value for the phase velocity expression. Therefore the critical velocity and the wave phase velocity do indeed lead to stationary waves in the atomizer. A quantitative comparison between the analytically derived wave phase velocity and that measured experimentally, for this stationary pulsating wave, show very good agreement within a few percent.

Torsional vibrations of circular poroelastic plates

Torsional vibrations of annular poroelastic circular plates are studied in the framework of Biot's theory of wave propagation in porous solids. The frequency equation of torsional vibrations is same for pervious and impervious surfaces. The frequency equation of torsional vibrations is obtained by using the traction free boundaries and edges of the annular circular plate. Non-dimensional frequency of annular plate is computed as a function of aspect ratio. The frequency equation is discussed for first two modes. Frequency equation of torsional vibrations of uniform circular poroelastic plate is obtained as a particular case of the annular plate. Resonant frequency of infinite poroelastic plate is obtained as a limiting case of annular plate. The expressions for phase velocity and attenuation are obtained and these are computed for two different poroelastic materials as a function of frequency in presence of dissipation. Phase velocity is almost same for the considered poroelastic materials and dissipations. Results of previous study are obtained as a particular case of the present investigation.

The edge waves on a Kirchhoff plate bilaterally supported by a two-parameter elastic foundation

In this paper, the bending waves propagating along the edge of a semi-infinite Kirchhoff plate resting on a two-parameter Pasternak elastic foundation are studied. Two geometries of the foundation are considered: either it is infinite or it is semi-infinite with the edges of the plate and of the foundation coinciding. Dispersion relations along with phase and group velocity expressions are obtained. It is shown that the semi-infinite foundation setup exhibits a cut-off frequency which is the same as for a Winkler foundation. The phase velocity possesses a minimum which corresponds to the critical velocity of a moving load. The infinite foundation exhibits a cut-off frequency which depends on its relative stiffness and occurs at a nonzero wavenumber, which is in fact hardly observed in elastodynamics. As a result, the associated phase velocity minimum is admissible only up to a limiting value of the stiffness. In the case of a foundation with small stiffness, asymptotic expansions are derived and beam-like one-dimensional equivalent models are deduced accordingly. It is demonstrated that for the infinite foundation the related nonclassical beam-like model comprises a pseudo-differential operator.

Green's Functions for Surface Waves in a Generic Velocity Structure

Methodologies for calculating surface‐wave velocities and the associated displacement/stress eigenfunctions and Green’s functions have been well established for many decades. However, to our knowledge, no one has ever documented a quantitative evaluation of these properties for commonly used empirical scalings. For example, it is currently not possible to take a given power‐law dependence of shear‐wave velocity on depth and look up the corresponding dependence of phase velocity on frequency, or Green’s function surface displacement. We address this gap in the literature and here provide explicit quantitatively accurate expressions for phase velocities and Green’s function amplitudes for a few commonly used empirical formulas for near‐surface velocity structure. These exact expressions are found to be immediately useful in applications that use shallow phase velocities and also in applications that interpret seismic amplitudes or amplitude ratios from near‐surface processes such as fluvial transport, icequakes, landslides, and volcanic tremor.

Phase Velocity and Attenuation of Longitudinal Shear Vibrations of Hollow Poroelastic Cylinders

The present paper is devoted to the study of phase velocity and attenuation of longitudinal shear vibrations of hollow poroelastic circular cylinders in the presence of dissipation. The explicit expressions for phase velocity and attenuation of longitudinal shear vibrations are derived. The frequency equation of longitudinal shear vibrations and modes obtained in a previous paper are used to compute the phase velocity and attenuation for different dissipations for thin and thick poroelastic cylindrical shells and poroelastic solid cylinder. The physical parameters of sandstone saturated with kerosene and sandstone saturated with water are used for the purpose of computation. It is found that the phase velocity is linear beyond certain frequency. Phase velocity is smaller for a typical anti-symmetric mode compared to the flexural mode. It is greater for the second mode than that of the first mode. Also the phase velocity is larger for a thin poroelastic cylindrical shell than that of a thick poroelastic cylindrical shell. The same is true for attenuation also. Attenuation is very high for the considered dissipations and it increases with the increase in dissipation.

Comparisons of visco-acoustic wave equations

In this paper, we compare the dissipation and dispersion characteristics of six visco-acoustic wave equations using numerical methods. We derive the expressions for spectral ratio and phase velocity for the six wave equations, which are used in the analysis. We then compute wave fields by numerically solving the six wave equations for the same model and compare their frequency spectrums, waveforms and quality factors, as calculated by the spectral ratio method. Finally, we conclude that four of the wave equations have analogous dissipation and dispersion characteristics, whereas the other two wave equations are quite different.

Diagonalization of a three-dimensional system of equations in terms of displacements of the linear theory of elasticity of transversely isotropic media

A dynamic three-dimensional system of linear equations in terms of displacements of the theory of elasticity of transversely isotropic media is given explicit expressions for phase velocities and polarization vectors of plane waves. All the longitudinal normals are found. For some values of the elasticity moduli, the system of equations is reduced to a diagonal shape. For static equations, all the conditions of the system ellipticity are determined. Two new representations of displacements through potential functions that satisfy three independent quasi-harmonic equations are given. Constraints on elasticity moludi, at which the corresponding coefficients in these representations are real, different, equal, or complex, are determined. It is shown that these representations are general and complete. Each representation corresponds to a recursion (symmetry) operator, i.e., a formula of production of new solutions.

Surface wave propagation in an elastic laser‐excited half‐space with small‐scale effects

The propagation of Rayleigh waves in an isotopic elastic half‐space with small length scale effects whose surface is free from stress and subjected to laser irradiation is investigated. Using the nonlocal form of constitutive equations the balance of moment and laser‐induced defect‐concentration field equations are obtained. The frequency equation that governs the Rayleigh surface wave propagation in the considered medium has been derived. It is observed that the waves are dispersive in nature and that the dispersion is caused by the atomic defect generation. An approximate procedure is applied to obtain explicit expressions for phase velocity and attenuation (amplification) coefficients which characterize surface waves. It is shown that if the pump parameter is above the critical value, due to concentration–elastic instability coupled strain‐defect nanosized ordered structures on the surface of solids arises. The period of these structures is proportional to the characteristic length of defect‐atom interaction and increases with the increase of the temperature of the medium.

Size Effects on Surface Elastic Waves in a Semi-Infinite Medium with Atomic Defect Generation

The paper investigates small-scale effects on the Rayleigh-type surface wave propagation in an isotopic elastic half-space upon laser irradiation. Based on Eringen’s theory of nonlocal continuum mechanics, the basic equations of wave motion and laser-induced atomic defect dynamics are derived. Dispersion equation that governs the Rayleigh surface waves in the considered medium is derived and analyzed. Explicit expressions for phase velocity and attenuation (amplification) coefficients which characterize surface waves are obtained. It is shown that if the generation rate is above the critical value, due to concentration-elastic instability, nanometer sized ordered concentration-strain structures on the surface or volume of solids arise. The spatial scale of these structures is proportional to the characteristic length of defect-atom interaction and increases with the increase of the temperature of the medium. The critical value of the pump parameter is directly proportional to recombination rate and inversely proportional to deformational potentials of defects.

Ion-acoustic cnoidal wave and associated non-linear ion flux in dusty plasma

Using reductive perturbation method with appropriate boundary conditions, coupled evolution equations for first and second order potentials are derived for ion-acoustic waves in a collisionless, un-magnetized plasma consisting of hot isothermal electrons, cold ions, and massive mobile charged dust grains. The boundary conditions give rise to renormalization term, which enable us to eliminate secular contribution in higher order terms. Determining the non secular solution of these coupled equations, expressions for wave phase velocity and averaged non-linear ion flux associated with ion-acoustic cnoidal wave are obtained. Variation of the wave phase velocity and averaged non-linear ion flux as a function of modulus (k2) dependent wave amplitude are numerically examined for different values of dust concentration, charge on dust grains, and mass ratio of dust grains with plasma ions. It is found that for a given amplitude, the presence of positively (negatively) charged dust grains in plasma decreases (increases) the wave phase velocity. This behavior is more pronounced with increase in dust concentrations or increase in charge on dust grains or decrease in mass ratio of dust grains. The averaged non-linear ion flux associated with wave is positive (negative) for negatively (positively) charged dust grains in the plasma and increases (decreases) with modulus (k2) dependent wave amplitude. For given amplitude, it increases (decreases) as dust concentration or charge of negatively (positively) charged dust grains increases in the plasma.

Ion acoustic solitary waves in an electron–ion–positron plasma

Large amplitude ion-acoustic solitary waves have been theoretically studied in a non-collisional plasma consisting of warm positive ions, positrons and isothermal electrons using pseudo-potential (effective potential) method. The expressions of phase velocities of slow- and fast- modes of the wave have been derived and graphically analyzed. The critical values of densities of the positrons and the ions, Mach number and positron temperature for the excitation of ion-acoustic solitary wave have also been obtained. The solutions of the first-order and next higher order solitary waves have been obtained and the profiles of the solitary waves have been drawn taking different values of the plasma parameters. It has been observed that both compressive and rarefactive solitary waves are being excited in presence of positrons in the plasma. The variation of amplitudes of the solitary waves has been shown graphically for different values of density and temperature of positrons, drift velocity and temperature of the ions.

Recovery of airflow resistivity of poroelastic beams submitted to transient mechanical stress

The airflow resistivities of air-saturated poroelastic slender beams submitted to transient mechanical stress are recovered using fluid and solid borne compressional mode phase velocity expressions drawn from a modified Biot theory. A point where the two dilatational modes intersect and their phase velocities equal is first sought. This point also corresponds to the Biot transitional frequency indicating the frequency at which the solid and the pore fluid start disassociating due to the weakening of the viscous forces by the thinning of the viscous boundary layer in the pores. A bilinear time–frequency (TF) distribution is used to represent on the time–frequency plane, the captured transient mechanical stress waves from which the point of intersection/separation of the two modes is located. The projection of the Eigenfrequencies obtained from a simple 3D finite element modeling of the thin poroelastic beam, on a (TF) diagram, facilitates the identification of the modes. The transition frequencies for the poroelastic beams thus retrieved are verified through the use of variable frequency, single cycle sine wave bursts. The anisotropy of the foams are also revealed by analyzing the transient responses of the poroelastic beam specimens cut from the same panel but in two perpendicular directions in orientation to each other.

An Approximate Acoustic Wave Equation for VTI Media

AbstractAcoustic approximation is an effective way to simplify modeling and migration of quasi‐P waves in the presence of anisotropy. To generate a new acoustic wave equation for VTI media (transversely isotropic media with a vertical symmetric axis) from the accurate phase velocity expression, we make simplification to the square root term by an undetermined coefficients method to get an approximate phase velocity formula, from which we derive a second‐order temporal and fourth‐order spatial partial differential equation through inverse Fourier transform. Quantitative computations show that the phase velocity specified by the new acoustic wave equation is accurate when Thomsen parameter ε is equal to δ (elliptical anisotropy) and the errors of the phase velocity become bigger as the difference between ε and δ gets bigger. The maximum relative error of the phase velocity is 0.13% in the case of ε = 0.1 and δ = 0.2, and becomes –1.65% in the case of ε = 0.8 and δ = 0.3. The shear‐wave‐free snapshots from finite difference modeling demonstrate that the new acoustic wave equation is a pure quasi‐P wave equation.

Analysis of Phase Velocity and Polarization Features for Elastic Waves in TTI Media

Phase velocity and polarization direction are the important parameters for studying seismic wave propagation and describing medium properties, which play essential roles in theory and application. We assume that the symmetry axis of tilted transverse isotropy (TTI) media lies in XOZ plane in the observation coordinates system and directly derive an analytic expression for phase velocity and polarization direction in TTI media in this system. Based on the Thomson dimensionless anisotropic parameters and weak anisotropy approximation theory, we also derive the approximate expressions of phase velocity for elastic waves, polarization direction for qP wave and qSV wave. Theoretical analysis and numerical examples indicate that the change of velocity for qP wave and qSH wave is gentle, and the change of velocity for qSV wave is dramatic. The errors of approximate values of phase velocity are small, so the approximate expression is a good approximation for precise phase velocity. The polarization direction of SH wave is the function of propagation angle and tilted angle of symmetry axis instead of anisotropic parameters. The polarization directions of SH wave is not only perpendicular to propagation direction, but also perpendicular to the direction of symmetry axis. The polarization directions of qP wave and qSV wave are neither parallel nor perpendicular to propagation direction except some special directions, and changes with the angle of symmetry axis in TTI media. For both precise and approximate expressions, the polarization directions of qP wave and qSV wave are always perpendicular. In a given range of errors, the weak anisotropy approximation and analytic expressions match well.

Explicit causal relations between material damping ratio and phase velocity from exact solutions of the dispersion equations of linear viscoelasticity

SUMMARY The theory of linear viscoelasticity is the simplest constitutive model that can be adopted to accurately predict the small-strain mechanical response of materials exhibiting the ability to both store and dissipate strain energy. An important result implied by this theory is the relationship existing between material attenuation and the velocity of propagation of a mechanical disturbance. The functional dependence of these important parameters is represented by the Kramers‐Kronig (KK) equations, also known as dispersion equations, which are nothing but a statement of the necessary and sufficient conditions to satisfy physical causality. This paper illustrates the derivation of exact solutions of the KK equations to provide explicit relations between frequency-dependent phase velocity and material damping ratio (or equivalently, quality factor). The assumptions that form the basis of the derivation are not beyond those established by the standard theory of viscoelasticity for a viscoelastic solid. The explicit expression for phase velocity as a function of damping ratio was derived by means of the theory of linear singular integral equations, and in particular by the solution of the associated Homogeneous Riemann Boundary Value Problem. It is shown that the same solution may be obtained also by using the implications of physical causality on the Fourier Transform. On the other hand, the explicit solution for damping ratio as a function of phase velocity was found through the components of the complex wavenumber. The exact solutions make it possible to obtain frequency-dependent material damping ratio solely from phase velocity measurements, and conversely. Hence, these relations provide an innovative and inexpensive tool to determine the small-strain dynamic properties of geomaterials. It is shown that the obtained rigorous solutions are in good agreement with well-known solutions based on simplifying assumptions that have been developed in the fields of seismology and geotechnical engineering. In addition, the theoretical results have been validated using recently published experimental material functions for fine-grained soils and for polymethyl methacrylate.

Extension of forward modeling phase-screen code in isotropic and anisotropic media up to critical angle

The computationally efficient phase-screen forward modeling technique is extended to allow investigation of nonnormal raypaths. The code is developed to accommodate all diffracted and converted phases up to critical angle, building on a geometric construction method. The new approach relies upon prescanning the model space to assess the complexity of each screen. The propagating wavefields are then divided as a function of horizontal wavenumber, and each subset is transformed to the spatial domain separately, carrying with it angular information. This allows both locally accurate 3D phase corrections and Zoeppritz reflection and transmission coefficients to be applied. The phase-screen code is further developed to handle simple anisotropic media. During phase-screen modeling, propagation is undertaken in the wavenumber domain where exact expressions for anisotropic phase velocities are available. Traveltimes and amplitude effects from a range of anisotropic shales are computed and compared with previous published results.

Phase velocity in 2D TTI media

We derive an expression for phase velocity in 2D tilted transverse isotropy (TTI) media. Snapshots of phase velocity in TTI and transverse isotropy (TI) model media are simulated and analyzed using the derived expression. In addition, the x-component character differences between the modeled phase velocities of the two media models are compared and analyzed.

Spatial dispersion of surface waves at the free surface of a general anisotropic elastic medium

A new technique relates the wave velocity of the surface waves in anisotropic elastic medium to its elastic constants. Anisotropic propagation of surface waves is studied in a half-space occupied by a general anisotropic elastic solid. The phase velocity expressions of quasi-waves, in three-dimensional space, are used to derive the secular equation of surface waves. The complex secular equation is resolved, analytically, into real and imaginary parts and is then solved, numerically, for phase velocity along a given phase direction on the surface. The complete procedure is thus analogous to the one used for conventional Rayleigh waves in isotropic medium. A non-linear equation relates the ray direction of the surface waves to its phase direction on the (plane) surface of the medium. The analytical differentiation of secular equation yields the directional derivative of phase velocity. This derivative is used to calculate the wave velocity of surface waves. Spatial variations of phase velocity, wave velocity and ray direction over the free plane surface are plotted for the numerical models of crustal rocks with orthorhombic, monoclinic and triclinic anisotropies.