THEORETICAL ANALYSIS OF LOVE AND RAYLEIGH SURFACE ELASTIC WAVES IN EXPLORATION SEISMOLOGY

International Journal of Research and Development Studies                 Volume 8, Number 1, 2017

ISSN: 2056 – 2121

© 2017 McEvans Publishing Company

THEORETICAL ANALYSIS OF LOVE AND RAYLEIGH SURFACE ELASTIC WAVES IN EXPLORATION SEISMOLOGY

Ibim, D.F. and Womuru, E.N.

Department of Physics, Ignatius Ajuru University of Education

P.M.B. 5047, Port Harcourt, Nigeria

E-mail: dagogofranklin@gmail.com

ABSTRACT

This paper is written with the aim of presenting the basics of seismic surface waves theory that can be applied in both earthquake and exploration seismology. Seismologist measures the seismic waves near the free-surface, and it is important to understand the near surface effects. At the surface, both incident and reflected waves coexist, and the total amplitude is the sum of the two. Shear horizontal (SH) waves do not interact with the P and shear vertical (SV) waves at the free-surface. The interaction between P and SV waves with the free-surface gives rise to an interference waves that travel along the surface as Rayleigh waves. Love waves have no vertical displacements and travel with a slower velocity than P- or S-waves, but faster than Rayleigh waves.

Key words: Exploration seismology, Free-surface, Homogeneous, Isotropic medium, Love-wave,             Poissonian, Rayleigh-wave.

INTRODUCTION

Only longitudinal and transverse waves can propagate in a homogeneous, isotropic and unlimited medium. If the medium is bounded, another type of waves, surface waves, can be guided along the surface of the medium which usually form the principal phase of seismograms (Burg, el al., 1999). There are two types of surface elastic waves namely Love waves and Rayleigh waves. 

Love waves are seismic surface waves in which the particle motion is transverse and parallel to the surface. As opposed to Rayleigh waves, Love waves cannot propagate in a homogeneous half-space. They can propagate only if the S-wave velocity generally increases with the distance from the surface of the medium (Cramppin and Taylor, 2011). The waves travel by multiple reflections in the surface low velocity layer (LVL). They are surface waves consisting of SH motion parallel to an interface. They exist only when a semi-interface medium is overlaid by an upper layer of finite thickness terminating at a free surface. Their motion is essentially the same as that of the s-waves that has no vertical displacement. Augustus Edward Hough Love predicted the existence of Love waves mathematically in 1911. Rayleigh waves also known as ground rolls are like rolling ocean waves whose motion is both vertical and horizontal in a vertical plane pointing in the direction of propagation of the wave. They are responsible for the entire disturbance caused by an earthquake (Eslick, et al.,2008). Rayleigh waves are efficient in the transport of seismic energy and are responsible for ground rolls (Bolt and Butcher, 1996). The most important surface wave in exploration seismology is the Rayleigh wave which is propagated along a free surface of a semi-infinite elastic half-space (Ash and Paige, 2005). Near the surface of a homogeneous half-space, the particle motion is a retrograde vertical ellipse (anticlockwise for a wave travelling to the right). The existence of Rayleigh waves was predicted in 1885 by Lord Rayleigh, after whom they were named.

Surface waves travel more slowly than body waves (p- and s-waves) and of the two surface waves, Love waves generally travel faster than Rayleigh waves, i.e. v_R < v_L < β (Keillis-Borock, 2000).

Love Waves in a Layer on a Half-Space

Looking at Love waves mathematically, consider a medium which consists of a homogeneous and isotropic layer of a constant thickness H" sitting on a homogeneous and isotropic half-space.

International Journal of Research and Development Studies                 Volume 8, Number 1, 2017

Assume the layer and the half-space to be perfectly elastic and a welded contact to exist between them (Fig. 1). Denote the velocity of shear waves by β', the density by ρ' and the shear modulus by μ' = ρ'β'² in the layer and β, ρ, and μ = ρβ² the corresponding parameters in the half-space. Assume the velocity in the layer to be lower than that in the half-space, i.e. β' < β. Introduce again a Cartesian coordinate system whose (x, y)-plane coincides with the surface of the medium, and the z-axis is oriented into the medium (downwards). We wish to find out whether surface waves of the SH type can propagate in this medium. In other words, we are seeking surface waves which are polarized in the horizontal plane perpendicularly to the direction of propagation.

Figure 1: Reflection and transmission at a medium which has the free surface

            and a finite layer.

Consider vertical displacement only and recall that the general equation of motion for SH is

Also,

Applying boundary conditions such as

BC1- On the free surface,

z = H' and vertical stress component σ_zy = 0                    (2)

BC2- On common boundary,

By condition 2 and 1, putting equations (2) and (3) into equation (1) give

 

and

These are 3 equations in 3 unknown C, C' and D'. Eliminating C, C' and D' and substituting the values for s and s' will give

Theoretical Analysis of Love and Rayleigh Surface                                Ibim, D.F. and Womuru, E.N.

Elastic Waves in Exploration Seismology

And

Recall:

The dispersion relation shows that Love-wave velocities at low frequencies tend toward the half-space velocity β, while observations at high frequencies give the layer velocity β'. Equation 5 relates the Love-wave velocity c_L to its frequency f, layer thickness H', layer and half-space shear-wave velocities β', β, and densities ρ', ρ.

Rayleigh Waves in a Layer on a Half-Space

Consider a half space medium ‘m’ whose top part is a vacuum as shown below. Denote by α' the longitudinal wave velocity, β' the transverse wave velocity, ρ' the density, λ' and μ', the Lame’s constants, and H the thickness of the layer. Denote the corresponding parameters in the half space by α, β, ρ, λ and μ, respectively. Assume that α' < α and β' < β.

 

Since the SH wave does not interact         with the P and          SV waves at the surface (Eslick, et al., 2008), it is proper to disregard the former. The P and SV potentials in ‘m’ (the solutions for wave equation in ‘m’) now become:

where

and

At the boundary, stresses vanish at the free surface

But

International Journal of Research and Development Studies                 Volume 8, Number 1, 2017

                  

Similarly,

Substituting equations (6) and (7) into equations (9) and (10) respectively gives

and

Equations (11) and (12) can be solved using algebra or matrix. Using matrix implies that

To find the non-trivial solutions, set the determinant to zero

If the medium is Poissonian, then α²/β² = 3 and the determinant becomes

The equation (13) gives the velocity c of the wave that propagates on a free surface.

There are 4 roots to this polynomial equation (Stein and Wyssesion 2003):

Only the last solution satisfies the requirement that 0 < c_R < β and it can be concluded that (for Poissonian solid) the Rayleigh wave speed is slightly less than the shear wave speed (~0.92β) (Bullen and Bolt, 2003).

Rationalizing and factorizing , it will follow that

Substituting c_R = β and c_R = 0, we obtain one and  respectively.

Theoretical Analysis of Love and Rayleigh Surface                                Ibim, D.F. and Womuru, E.N.

Elastic Waves in Exploration Seismology

DISCUSSION

Elastic surface waves do not represent principally new types of waves, but only interference phenomena of body waves. Therefore, in principle, attempt could be made to construct the wave field of surface waves (and of other guided waves) by summing body waves (Burg, et al., 1999). However, this approach would be inconvenient if a large number of waves are to be taken into account (thin layers, large distances from the source). Therefore, a more appropriate mathematical description must be sought for surface waves.

The simplest medium in which Rayleigh waves can propagate is a homogeneous isotropic half-space. The velocity of Rayleigh waves in this medium, c_R, is slightly less than the transverse wave velocity, c_R = 0.9β, and is independent of frequency (Ash and Paige, 2005). Thus, Rayleigh waves in this simple model of the medium are non-dispersive. The simplest model in which Love waves can propagate consists of a homogeneous isotropic layer on a homogeneous isotropic half-space (Eslick, et al., 2008). Both the Rayleigh and Love waves in this model are already dispersive, i.e. their velocities are dependent on frequency. The velocity of Rayleigh waves also depends upon the elastic constants in the vicinity of the surface Love waves travel with a slower velocity than P- or S-waves, but faster than Rayleigh waves. Love waves have velocities intermediate between the s-wave velocity at the surface and that in deeper layers, and exhibits dispersion. 

SH waves can exist if there are values of c (wave velocity) that can satisfy equation (5) and also make s and s' imaginary as required. For the wave velocity s to be imaginary will imply and mean that

From equation (12) the velocity c_R of the Rayleigh wave can be determined. It is important to note that equation (12) is cubic in c²/β², implying three (3) roots with three (3) solutions and the solution must satisfy requirements that r and s be imaginary. Also if we take note of these requirements, it is easy to show however that there is one such solutions, that is for c_R being between zero and β.

Only the last solution in equation (14) satisfies the requirement that 0 < c_R < β and it can be concluded that (for Poissonian solid) the Rayleigh wave speed is slightly less than the shear wave speed (~0.92β) (Kennett, 1998). The result that: c_R = 0.92β can now be used to find the coefficients of the potentials (A and B) and the displacements (u_x,u_z).        

Equation (5) illustrates that, opposed to Rayleigh waves, Love wave velocities do not depend on compressional p-wave velocities.

Records of a seismic events begin with longitudinal waves, followed by transverse waves, and finally by surface waves. Surface waves usually have larger amplitudes and longer periods. Surface waves display a characteristic dispersion and polarization (Crampin and Taylor, 2011). 

CONCLUSION

The conditions for Love Waves to exist are: and c_L the wave velocity must satisfy this equation, the velocity s must be imaginary, the velocity of S-body waves in the lower medium m must be greater than that in the medium m' and  (Takeuchi and Saito, 1997). Love waves result from the interaction of SH waves, require a velocity structure that varies with depth, i.e., cannot exist in a homogeneous half-space and that β > β'.

Love waves are horizontally polarized because they result from interaction of shear (SH) waves. As opposed to Rayleigh waves, Love waves exist in layered media only. For the one layer case, the Love wave represents the superposition of multiple, critically reflected down going SH waves from the bottom of the layer (Stein and Wysession, 2003). The layer of a thickness H' is considered as a wave guide and the Love-wave velocity c_L is in between the shear wave velocities of the layer and the half-space (Figure 1). In contrast, Rayleigh-wave velocities are always less than the layers shear-wave velocity (Watson, 1990). For surface waves to be trapped near the surface, the energy must decay with depth. Free-surface is traction free.

International Journal of Research and Development Studies                 Volume 8, Number 1, 2017

REFERENCES

Ash E. A. and Paige E. G. S. (Eds.) (2005). Rayleigh Wave Theory and Application. Springer   Verlag, Berlin.

Bolt B. A. and Butcher J. C. (1996). Rayleigh Wave Dispersion for a Single Layer on an Elastic         Half-space. Australian Journal of Physics 15, 418-424.

Bullen K. E. and Bolt B. A. (2003). An Introduction to the Theory of Seismology. Cambridge University Press, Cambridge. (Sixth edition).

Burg K. E., Ewing M., Press F. and Stulken E. J. (1999). A Seismic Wave Guide Phenomenon.          Geophysics 18, 504-522.

Crampin S. and Taylor D. B. (2011). The Propagation of Surface Waves in Anisotropic Media.          Geophys. J. R. Astr. Soc. 35, 61-77.

Eslick R., Tsoflias G. and Steeples D. (2008). Field investigation of Love waves in near-surface           seismology: Geophysics, 65, no. 5, G1– G6, http://dx.doi.org/10.1190/1.2901215.

Keilis-Borok V. I. (Ed.) (2000). Seismic Surface Waves in a Laterally Inhomogeneous Earth.   Kluwer Academic Publishers.

Kennett B. L. N. (1998): Seismic Wave Propagation in Stratzped Media. Cambridge University          Press, New York

Stein S. and Wyssesion M. (2003). An Introduction to Seismology, Earthquakes, and Earth    Structure: Blackwell publishing.

Takeuchi H. and Saito M. (1997). Seismic Surface Waves. In: B. A. Bolt (Ed.): Methods in     Computational Physics, Vol. 16, pp. 203-285. Academic Press, New York.

Watson T. H. (1990). A Note on Fast Computation of Rayleigh Wave Dispersion in the         Multilayered Elastic Half-space. Bull. Seism. Soc. Am. 67, 169-178.

Authors’ contributions

This paper was written in collaboration between the authors. Author IDF designed the study, performed the theoretical analysis, wrote the protocol and wrote the first draft of the manuscript. Author WEN managed the analysis of the study and the literature searches. All authors read and approved the final manuscript. The authors are Doctors of Philosophy in Geophysics.