Secular Equation Of Rayleigh Waves Research Articles

Method of first integrals and interface, surface waves

The method of first integrals (MFI) based on the equation of motion for the displacement vector, or based on the one for the traction vector was introduced recently in order to find explicit secular equations of Rayleigh waves whose characteristic equations (i.e the equations determining the attenuation factor) are fully quartic or are of higher order (then the classical approach is not applicable). In this paper it is shown that, not only to Rayleigh waves, the MFI can be applicable also to other waves by running it on the equations for mixed vectors. In particular: (i) By applying the MFI to the equations for the displacement-traction vector we get the explicit dispersion equations of Stoneley waves in twinned crystals (ii) Running the MFI on the equations for the traction-electric induction vector and the traction-electrical potential vector provides the explicit dispersion equations of SH-waves in piezoelastic materials. The obtained dispersion equations are identical with the ones previously derived using the method of polarization vector, but the procedure of driving them is more simple.

Explicit secular equations of Rayleigh waves in elastic media under the influence of gravity and initial stress

The problem of Rayleigh waves in an orthotropic elastic medium under the influence of gravity and initial stress was investigated by Abd-Alla [A. M. Abd-Alla, Propagation of Rayleigh waves in an elastic half-space of orthotropic material, Appl. Math. Comput. 99 (1999) 61–69], and the secular equation of the wave in the implicit form was derived. However, due to the uncorrect representation of the solution, the secular equation is not right. The main aim of the present paper is to reconsider this problem. We find the secular equation of the wave in explicit form. By considering some special cases, we obtain the exact explicit secular equations of Rayleigh waves under the effect of gravity of some previous studies, in which only implicit secular equations were derived.

Explicit secular equations of Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity

The problem of the Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity is investigated. Using an appropriate representation of the solution we derive the secular equation of the wave motion in the explicit form. Moreover, following the same approach, we obtain the explicit secular equations for a number of previously investigated Rayleigh wave problems whose dispersion equations were obtained only in the implicit form.

Uniqueness of the solution to the secular equation for viscoelastic surface waves

A technique of complex analysis is employed to show that the secular equation for Rayleigh waves in viscoelastic half-spaces always admits only one complex root. This solution corresponds to an admissible surface wave which is the counterpart of the well-known elastic Rayleigh wave and exists for arbitrary values of the viscoelastic moduli.

A note on Rayleigh wave velocity in saturated soils with compressible constituents

The problem of Rayleigh waves in a semi-infinite saturated soil medium is reconsidered in this study, with the purpose of clarifying existing confusion and limitations of available studies. By employing Biot's general formulation, which takes into account not only the compressibility of the solid and fluid constituents but also the viscous dissipation due to fluid flow, the secular equation for Rayleigh waves is rigorously derived and the velocity of Rayleigh waves is computed for several typical types of saturated soils. The results show that the velocity of Rayleigh waves in general is independent of frequency in the frequency range actually employed in engineering practice and is only slightly less than the shear wave velocity. The results confirm that current understanding of Rayleigh wave velocity achieved based on the classical theory of elasticity is acceptable and indicate that some results in the literature are incorrect.Key words: Rayleigh wave velocity, saturated soil, porous media, wave propagation, analytical method.

Rayleigh waves on a viscoelastic solid half-space

It is shown that the secular equation for Rayleigh waves propagating on a viscoelastic half-space always admits only one complex root corresponding to a surface wave. This result is proved in those cases in which the displacement field can be obtained analytically, including the isotropic case. The root is obtained in terms of complex integrals extending to the viscoelastic case, a result by Nkemzi [Wave Motion 26, 199–205 (1997)]. The wave solution is shown to represent an admissible surface wave for any viscoelastic relaxation kernel compatible with thermodynamics. A correspondence is then established between elastic and viscoelastic Rayleigh waves and their propagation properties are pointed out by an approximated analysis of the solution. Illustrative results are given by a numerical evaluation of the complex root for anisotropic viscoelastic half-spaces.

A study of the secular equation for Rayleigh waves using the root locus method

In a recent note Rahman and Barber [Trans. ASME 62 (1995) 250] proposed an exact expression for the roots of the secular equation for the Rayleigh waves in an isotropic half-space. Using the root locus method, we describe in a very simpler manner the evolution of the roots versus the Poisson’s ratio and we derive easily the critical value for which the nature of the roots changes.

Exact Expressions for the Roots of the Secular Equation for Rayleigh Waves

M. Rahman t9'21 and J. R. Barber 2°'21 (1958) for a line load moving steadily over the surface of the half-plane. Both these results can be used as Green's functions to generate convolution integrals for more general boundary value problems and the availability of an explicit factorization of the rationalized form (3) of R ( V ) permits such integrals to be broken down into simpler terms by partial fractions.

Dynamics of Saturated Rocks. III: Rayleigh Waves

The propagation of plane harmonic Rayleigh waves in a half-space elastic, fully saturated, rock medium characterized by two degrees of porosity—one due to the pores and the other due to the fissures—is studied analytically. With the aid of the Helmholtz method of decomposition of displacement and velocity vectors, the application of the boundary conditions, and the usual assumption of exponential attenuation of waves with depth, the governing equations of motion finally reduce to the secular equation for Rayleigh waves. The solution of this equation helps to determine both the phase velocity and attenuation coefficient of the Rayleigh waves as functions of frequency. A parametric numerical study reveals that, for certain ranges of frequencies, velocity and attenuation of the two-porosity model show small and large differences, respectively, from their values for the single-porosity model.

THE SECULAR EQUATION FOR RAYLEIGH WAVES ON ELASTIC CRYSTALS II: CORRECTIONS AND ADDITIONS

Using the methods developed by Currie (1), a corrected version is derived of the secular equation for Rayleigh waves on the surface of an anisotropic elastic half-space. The earlier equation suggested by Currie turns out to be an identity. The corrected version is also a secular equation for pseudo-surface waves.

A note on the secular equation for Rayleigh waves

In der klassischen Elastizitatstheorie wird die Sakulargleichung fur Rayleighsche Oberflachenwellen fur diejenigen Werte der Poissonschen Zahl im einzelnen besprochen, fur welche die Quadrate der Geschwindigkeiten der divergenzfreien und der irrotationalen Wellen positiv ausfallen.Rayleigh vermutete, dass komplexen Wurzeln der Sakulargleichung unannehmbare Verschiebungskomponenten entsprechen, welche nicht im Unendlichen verschwinden. Er bewies diese Vermutung fur ein unzusammendruckbares Material. In der vorliegenden Arbeit wird diese Vermutung allgemein bestatigt.