Secular equation of Rayleigh surface wave in a dry sandy half-space coated by a thin self-reinforced layer

Abstract

The propagation of Rayleigh surface wave in a dry sandy half-space coated by a thin self-reinforced medium of an arbitrary uniform thickness is investigated. The layer and half-space are in smooth contact with each other. The main aim of this work is to derive the exact and approximate secular equation of Rayleigh wave using effective boundary condition method. Taking an assumption of a thin layer, explicit secular equations are derived by replacing the entire effect of a thin layer on the half-space approximately, so-called effective boundary conditions which relate the displacement and stresses of the half-space at its surface. The approximate formulas of third-order velocity have been calculated numerically. Mathematica software has been applied to make the graphical results.

Appendix

$$ \begin{array}{@{}rcl@{}} M_{3}&=&\!\sqrt{-2c_{1}-c_{1}c_{3}-\frac{1}{c_{2}}-c_{1}-\frac{c_{1}}{c_{2}}+y^{2}\!+\frac{y^{2}}{c_{2}}-\!2\sqrt{\left( 1-y^{2}\right)\left( \frac{c_{1}}{c_{2}}-\!\frac{y^{2}}{c_{2}}\right)}}\\ M_{4}&=&\frac{\frac{-2y\left( 1-y^{2}\right)}{c_{2}}-2y}{\sqrt{\left( 1-y^{2}\right)\left( \frac{c_{1}}{c_{2}}-\frac{y^{2}}{c_{2}}\right)}},~~ M_{5}=\frac{\left( c_{1}-y^{2}\right)M_{3}}{\left( 1+c_{3}\right)\sqrt{\left( 1-y^{2}\right)\left( \frac{c_{1}}{c_{2}}-\frac{y^{2}}{c_{2}}\right)}} \\ M_{6}&=&\frac{\left( c_{1}-y^{2}\right)\left( 2y+\frac{2y}{c_{2}}+M_{4}\right)}{2\left( 1+c_{3}\right)\sqrt{\left( 1-y^{2}\right)\left( \frac{c_{1}}{c_{2}}-\frac{y^{2}}{c_{2}}\right)M_{3}}},\\ M_{7}&=&\frac{2yM_{3}}{{\left( 1+c_{3}\right)\sqrt{\left( 1-y^{2}\right)\left( \frac{c_{1}}{c_{2}}-\frac{y^{2}}{c_{2}}\right)}}}\\ M_{8}&=-&\frac{2y\left( 1-y^{2}\right)}{c_{2}}-2y\left( \frac{c_{1}}{c_{2}}-\frac{y^{2}}{c_{2}}\right)\\ \end{array} $$