# An approximate formula for the H/V ratio of Rayleigh waves in compressible pre-stressed elastic half-spaces coated with a thin layer

## Author affiliations

## DOI:

https://doi.org/10.15625/0866-7136/10417## Keywords:

Rayleigh waves, the H/V ratio, compressible, pre-stressed, approximate formula for the H/V ratio## Abstract

This paper is concerned with the propagation of Rayleigh waves in a compressible pre-stressed elastic half-space coated with a thin compressible pre-stressed elastic layer. The main purpose of the paper is to establish an approximate formula for the H/V ratio (the ratio between the amplitudes of the horizontal and vertical displacements of Rayleigh waves at the traction-free surface of the layer). First, the relations between the traction amplitude vector and the displacement amplitude one of Rayleigh waves at two sides of the interface between the layer and the half-space are created. From the continuity condition at the interface and these relations, the displacement amplitude vector of Rayleigh waves at the interface is determined. Then, a third-order approximate formula for the H/V ratio has been derived by expanding the layer displacement amplitude vector at its traction-free surface into Taylor series at the interface. It is shown numerically that the obtained formula is a good approximate one.

### Downloads

## References

M. Junge, J. Qu, and L. J. Jacobs. Relationship between Rayleigh wave polarization and state of stress. Ultrasonics, 44, (3), (2006), pp. 233–237. doi:10.1016/j.ultras.2006.03.004.

P. C. Vinh and R. W. Ogden. On formulas for the Rayleigh wave speed. Wave Motion, 39, (3), (2004), pp. 191–197. doi:10.1016/j.wavemoti.2003.08.004.

M. A. Dowaikh and R. W. Ogden. On surface waves and deformations in a compressible elastic half-space. Stability and Applied Analysis of Continuous Media, 1, (1), (1991), pp. 27–45.

R. W. Ogden. Non-linear elastic deformations. Ellis Horwood, Chichester, (1984).

P. C. Vinh and R. W. Ogden. Formulas for the Rayleigh wave speed in orthotropic elastic solids. Archives of Mechanics, 56, (3), (2004), pp. 247–265.

R. W. Ogden and P. C. Vinh. On Rayleigh waves in incompressible orthotropic elastic solids. The Journal of the Acoustical Society of America, 115, (2), (2004), pp. 530–533. doi:10.1121/1.1636464.

P. C. Vinh and R. W. Ogden. On the Rayleigh wave speed in orthotropic elastic solids. Meccanica, 40, (2), (2005), pp. 147–161. doi:10.1007/s11012-005-1603-6.

P. C. Vinh and P. G. Malischewsky. An improved approximation of Bergmann’s form for the Rayleigh wave velocity. Ultrasonics, 47, (1), (2007), pp. 49–54. doi:10.1016/j.ultras.2007.07.002.

P. C. Vinh and P. G. Malischewsky. An approach for obtaining approximate formulas for the Rayleigh wave velocity. Wave Motion, 44, (7), (2007), pp. 549–562. doi:10.1016/j.wavemoti.2007.02.001.

P. C. Vinh and P. G. Malischewsky. Improved approximations of the Rayleigh wave velocity. Journal of Thermoplastic Composite Materials, 21, (4), (2008), pp. 337–352. doi:10.1177/0892705708089479.

P. C. Vinh. On formulas for the velocity of Rayleigh waves in prestrained incompressible elastic solids. Journal of Applied Mechanics, 77, (2), (2010). doi:10.1115/1.3197139.

P. C. Vinh and P. T. H. Giang. On formulas for the Rayleigh wave velocity in pre-strained elastic materials subject to an isotropic internal constraint. International Journal of Engineering Science, 48, (3), (2010), pp. 275–289. doi:10.1016/j.ijengsci.2009.09.010.

P. C. Vinh. On formulas for the Rayleigh wave velocity in pre-stressed compressible solids. Wave Motion, 48, (7), (2011), pp. 614–625. doi:10.1016/j.wavemoti.2011.04.015.

P. C. Vinh and P. G. Malischewsky. Improved approximations for the rayleigh wave velocity in [-1, 0.5]. Vietnam Journal of Mechanics, 30, (4), (2008), pp. 347–358.

P. G. Malischewsky and F. Scherbaum. Love’s formula and H/V-ratio (ellipticity) of Rayleigh waves. Wave motion, 40, (1), (2004), pp. 57–67. doi:10.1016/j.wavemoti.2003.12.015.

A. E. H. Love. Some problems of geodynamics. Cambridge University Press, (1911).

P. G. Malischewsky, F. Wuttke, and A. Ziegert. The use of surface acoustic waves for nondestructive testing. SchriftenreiheWerkstoffwissenschaffen, 17, (2002), pp. 135–140. (in German).

L. M. Munirova and T. B. Yanovskaya. Spectral ratio of the horizontal and vertical Rayleigh wave components and its application to some problems of seismology. Izvestiia Physics of the Solid Earth, 37, (9), (2001), pp. 709–716.

F. Scherbaum, K. G. Hinzen, and M. Ohrnberger. Determination of shallow shear wave velocity profiles in the Cologne, Germany area using ambient vibrations. Geophysical Journal International, 152, (3), (2003), pp. 597–612. doi:10.1046/j.1365-246x.2003.01856.x.

A. N. Stroh. Steady state problems in anisotropic elasticity. Studies in Applied Mathematics, 41, (1-4), (1962), pp. 77–103. doi:10.1002/sapm196241177.

P. C. Vinh and N. T. K. Linh. An approximate secular equation of generalized Rayleigh waves in pre-stressed compressible elastic solids. International Journal of Non-Linear Mechanics, 50, (2013), pp. 91–96. doi:10.1016/j.ijnonlinmec.2012.11.004.

R. W. Ogden and D. A. Sotiropoulos. The effect of pre-stress on guided ultrasonic waves between a surface layer and a half-space. Ultrasonics, 34, (2-5), (1996), pp. 491–494. doi:10.1016/0041-624x(95)00102-9.

D. G. Roxburgh and R. W. Ogden. Stability and vibration of pre-stressed compressible elastic plates. International Journal of Engineering Science, 32, (3), (1994), pp. 427–454. doi:10.1016/0020-7225(94)90133-3.

## Downloads

## Published

## Issue

## Section

## License

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.