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The phenomenon of reflection of attenuated waves is considered at the boundary of a transversely isotropic piezothermoelastic composite. Identical horizontal slowness for all waves at the boundary defines Snell's law, which ensures the continuation of wave-field at a scattering boundary. Relations from this law are solved to relate the propagation characteristics of reflected waves to those of the incident wave. Any outcome from these relations should conform to the relevant mathematical principles. Else, the phenomenon of reflection wanders in incorrect domain and does not make a true contribution to existing knowledge. Any study with incorrect reflection procedure should be marked for its lapses and corrected so as to restrict the spread of wrong information on wave propagation in real materials. [ABSTRACT FROM AUTHOR]

Mathematical model for mechanical and electrical dynamics is solved for three‐dimensional propagation of harmonic plane waves in a piezoelectric medium with arbitrary anisotropy. A system of modified Christoffel equations is derived to explain the existence and propagation of bulk waves or decaying phases in the considered medium. At the free plane boundary, a superposition of decaying phases form a generalised Rayleigh wave, which is governed by a linear system of four homogeneous equations. A complex determinantal secular equation ensures a solution to this system. True surface wave at the boundary of the considered medium demands a real solution of this complex secular equation. The linear system of four equations is then transformed to replace the complex secular equation with a real one, which can be solved by standard numerical methods. A real solution of this real secular equation provides the phase velocity for generalised Rayleigh wave at the boundary of piezoelectric medium with arbitrary anisotropy. This phase velocity defines a complex slowness vector, which is used to calculate the motion of material particles and the wave‐induced electric field. A numerical example is considered to compute the phase velocity as well as group velocity for given (arbitrary) propagation directions of Rayleigh wave at the boundary. Variations in particle motion and electric field, induced by Rayleigh wave, are analysed at different depths for different propagation directions at the boundary. [ABSTRACT FROM AUTHOR]

This study considers to solve the complex dispersion equation for propagation of Rayleigh waves in a dissipative porous composite. This porous medium is a viscoelastic porous solid frame saturated with a viscous fluid. Seepage of fluid at the boundary is controlled through pore-fluid pressure and its normal gradient. A complex irrational equation governs the propagation, attenuation and dispersion of the Rayleigh waves. A complex analysis technique is used to solve this equation into an explicit expression for complex phase velocity. Effects of porosity and seepage from surface pores are analyzed numerically on the velocity of Rayleigh wave in a saturated poroelastic material. [ABSTRACT FROM AUTHOR]

Propagation of Rayleigh waves is considered in an isotropic thermoelastic semi-infinite medium with isothermal or insulated boundary. This requires to solve an irrational complex equation for an implicit complex velocity. A complex analysis technique is used to solve this implicit equation so as to derive an explicit expression for the velocity of the Rayleigh wave in thermoelastic materials. The same expression allows calculation of the velocity of Rayleigh wave in thermo-viscoelastic medium as well. [ABSTRACT FROM AUTHOR]

This study considers the propagation of surface waves along all directions on the plane boundary of piezo-poroelastic half-space with arbitrary anisotropy. This generalised propagation is characterized through an anisotropic phase velocity, which should ensure the decay of wave-field with depth into the medium. A linear homogeneous system of six equations with complex coefficients governs the existence and propagation of surface waves in the considered medium. The real phase velocity of surface waves lies implicit in a complex determinantal equation, which ensures a non-trivial solution to the system of equations. Through a specific transformation, the system of complex equations is modified to yield a real secular equation, with phase velocity being the only unknown. This equation can always be solved numerically for phase velocity of surface wave along any direction on the plane boundary of anisotropic piezo-poroelastic medium. The phase velocity has been used further to calculate the components of energy flux at the boundary. Horizontal components of energy flux define the group velocity and ray direction for the surface wave. A numerical example is solved to analyse the phase/group velocity curves at the boundary of the medium. [ABSTRACT FROM AUTHOR]

Phenomenon of reflection and refraction is considered at the plane interface between a thermoelastic medium and thermo-poroelastic medium. Both the media are isotropic and behave dissipative to wave propagation. Incident wave in thermo-poroelastic medium is considered inhomogeneous with deviation allowed between the directions of propagation and maximum attenuation. For this incidence, four attenuated waves reflect back in thermo-poroelastic medium and three waves refract to the continuing thermoelastic medium. Each of these reflected/refracted waves is inhomogeneous and propagates with a phase shift. The propagation characteristics (velocity, attenuation, inhomogeneity, phase shift, amplitude, energy) of reflected and refracted waves are calculated as functions of propagation direction and inhomogeneity of the incident wave. Variations in these propagation characteristics with the incident direction are illustrated through a numerical example. [ABSTRACT FROM AUTHOR]

A secular equation governs the propagation of Rayleigh wave at the surface of an anisotropic poroelastic medium. In the case of anisotropy with symmetry, this equation is obtained as a real irrational equation. But, in the absence of anisotropic symmetries, this secular equation is obtained as a complex irrational equation. True surface waves in non-dissipative materials decay only with depth. That means, propagation of Rayleigh wave in anisotropic poroelastic solid should be represented by a real phase velocity. In this study, the determinantal system leading to the complex secular equation is manipulated to obtain a transformed equation. Even for arbitrary (triclinic) anisotropy, this transformed equation remains real for the propagation of true surfacewaves. Such a real secular equation is obtained with the option of boundary pores being opened or sealed. A numerical example is solved to study the existence and propagation of Rayleigh waves in porous media for the top three (i.e. triclinic, monoclinic and orthorhombic) anisotropies. [ABSTRACT FROM AUTHOR]

This study considers the propagation of harmonic plane waves in a double-porosity dualpermeability solid saturated with single viscous fluid. Christoffel system is obtained to explain the existence of three longitudinalwaves and a transversewave in the medium considered. Each wave is identified with a complex velocity, which is resolved for inhomogeneous propagation to calculate the phase velocity and attenuation of the wave. Pore-fluid pressures are expressed in terms of velocities of solid particles corresponding to the propagation of three longitudinal waves. Then, transfer rate of pore-fluid between two porosities induced by each longitudinal wave is calculated as a function of its complex velocity. Numerical example is solved to study the dispersion in phase velocity and attenuation for each of the four waves. Effects of porefluid viscosity, wave-inhomogeneity and composition of double porosity on inhomogeneous propagation are analysed graphically. Transfer rate of pore-fluid, induced by each of the three longitudinal waves, is calculated as a periodic waveform. Variations in the fluid-flow profile are exhibited for different values of pore-fluid viscosity, skeleton permeability, wave-frequency and wave-inhomogeneity. [ABSTRACT FROM AUTHOR]

This study considers the propagation of elastic waves in a fluid-saturated double-porosity solid. Constitutive relations are derived for the double porosity medium having the transverse isotropy with vertical axis of symmetry. The equations of motion are solved for the propagation of harmonic waves. Then, the amount of wave-induced fluid flow at any point is expressed in terms of normal strain components in the composite medium. For propagation in a plane, the equations of motion decouple into two independent systems. One of these represents the particle motion normal to the plane and identifies the SH-type motion for solid particles. The other system represents the in-plane motion of particles and governs the propagation of four coupled waves. Among these four waves, three are quasi-longitudinal (qP1, qP2, qP3) waves and one is a quasi-transverse (qSV) wave. A numerical example is solved to calculate the velocity and attenuation anisotropy of the five waves. Effect of local fluid flow is observed on the wave velocities as well as the polarization of the constituent particles in double porosity medium. Effects of wave-frequency, propagation direction, pore-fluid viscosity, permeability and radius of spherical inclusions are analysed on the velocities and attenuation. [ABSTRACT FROM AUTHOR]

Biot's theories on wave propagation in saturated porous solid consider the existence of three waves, i.e., two longitudinal waves and one transverse wave. This is true when the porous solid is saturated with a non-viscous fluid. But for the presence of viscosity in interstitial fluid, the propagation of an additional transverse wave is expected. So, in a porous solid saturated with viscous fluid, four attenuated waves should be propagating. Two of these are longitudinal/dilatational waves and other two are transverse/rotational waves. The additional fourth wave has a larger attenuation as compared to other three waves in the medium. Of the two transverse waves, it propagates with a smaller phase velocity and termed as slow S wave, analogous to the slow P wave of Biot's theories. The propagation velocities and attenuation coefficients of all the four waves vary, differently, with wave frequency as well as pore-fluid viscosity. The slow transverse wave travels faster with the increase of frequency and viscosity. However, this weak transverse wave slows down for a stronger fluid–solid coupling (both, bulk and shear). [ABSTRACT FROM AUTHOR]

This study considers the propagation of Rayleigh waves on the stress-free surface of a viscoelastic, porous solid saturated with viscous fluid. The surface pores have the option of being either sealed or fully opened. The dispersion equation is obtained in the form of a complex irrational expression due to the presence of radicals. After rationalizing to an algebraic form, the dispersion equation is solved numerically, for exact complex roots. These roots are resolved for velocity and attenuation of the inhomogeneous propagation of Rayleigh waves in the poro-viscoelastic solid half-space. Effects of frame anelasticity, pore-fluid viscosity, frequency, and pore characteristics are observed numerically on the velocities of existing Rayleigh waves. Behavior of the elliptical particle motion for these waves is studied inside and at the surface of the porous solid. With the sealing of surface pores, these waves should be more sensitive to the changes in porosity and tortuosity. An exclusive contribution of dissipation due to the anelastic frame and/or the viscous pore-fluid is the existence of an additional inhomogeneous wave that satisfies the radiation condition for surface waves. This additional surface wave is more likely when surface pores of the dissipative porous medium are sealed. [ABSTRACT FROM AUTHOR]

SUMMARY Propagation of Rayleigh waves is studied in a porous medium, which is not fully saturated. The porous medium is assumed to be a continuum consisting of a solid skeletal with connected void space occupied by a mixture of two immiscible viscous fluids. This model also represents a case when liquid fills only a part of the pore space and gas bubbles span the remaining void space. In this isotropic medium, potential functions identify the existence of three dilatational waves coupled with a shear wave. For propagation of plane harmonic waves restricted to a plane, these potentials decay with depth from the plane boundary of the medium. Rayleigh wave in this dissipative medium is an inhomogeneous wave which decays with depth and ensures the vanishing of stresses at the plane boundary of the medium. The existence and propagation of such a wave is represented by a secular equation, which happens to be complex and irrational. This irrational equation is resolved into a polynomial form so as to find its exact roots and hence to analyse the existence and propagation of Rayleigh wave. The velocity and amplitude of Rayleigh wave are used further to calculate the averaged polarization of aggregate displacement in the medium. Existence of Rayleigh wave in a particular porous medium depends on the values of various parameters involved in secular equation. Hence, a numerical example is studied to find pertinent saturation levels for which Rayleigh waves exist in the considered numerical model of the porous medium. Variations in valid saturation range, velocity, quality factor and polarization of Rayleigh waves are studied with the changes in wave frequency, capillary pressure, liquid viscosity and frame anelasticity. [ABSTRACT FROM AUTHOR]

The mathematical model for wave motion in a porous solid saturated by two immiscible fluids is solved for the propagation of harmonic plane waves along a general direction in 3-D space. The solution is obtained in the form of Christoffel equations, which are solved further to calculate the complex velocities and polarisations of three longitudinal waves and one transverse wave. For any of these four attenuated waves, a general inhomogeneous propagation is considered through a particular specification of complex slowness vector. Inhomogeneity of an attenuated wave is represented through a finite non-dimensional parameter. For an arbitrarily chosen value of this inhomogeneity parameter, phase velocity and attenuation of a wave are calculated from the specification of its slowness vector. This specification enables to separate the contribution from homogeneous propagation of attenuated wave to the total attenuation. The phenomenon of reflection is studied to calculate the partition of wave-induced energy incident at the plane boundary of the porous solid. A parameter is used to define the partial opening of pores at the surface of porous solid. An arbitrary value of this parameter allows to study the variations in the energy partition with the opening of surface pores from fully closed to perfectly open. Another parameter is used to vary the saturation in pores from whole liquid to whole gas. Numerical examples are considered to discuss the effects of propagation direction, inhomogeneity parameter, opening of surface pores and saturating pore-fluid on the partition of incident energy. [ABSTRACT FROM AUTHOR]