Reflection of SV wave at the free boundary of unsaturated soil

Abstract

The propagation mechanism of seismic waves in unsaturated soils and their impact on engineering structures constitute a fundamental research topic in geotechnical earthquake engineering. Conventional single-phase elastic models and two-phase saturated theories fail to accurately characterize the solid-liquid-gas coupling effects, while extended Biot theory-based models exhibit limited prediction accuracy due to insufficient consideration of fluid-solid inertial coupling. To address this issue, this study introduces a fluid additional mass density parameter to establish a more precise three-phase wave equation for unsaturated soils, systematically revealing the reflection and propagation mechanisms of SV waves at the free surface. Through rigorous mathematical derivation, complete analytical solutions including reflected SV, P₁, P₂, and P₃ waves are obtained, along with expressions for free-field displacement, velocity, acceleration, and stress components. Parametric numerical analysis is conducted to quantitatively investigate the influence of gas content on body wave reflection characteristics under varying saturation levels. The results demonstrate that: (1) The amplitude reflection coefficient of SV waves decreases with incident angle, whereas P₁, P₂, and P₃ waves display inverse proportionality; (2) At low saturation levels (Sr < 0.9), the reflection coefficients of P₁ and P₂ waves exhibit minor variations, but increase significantly under high saturation levels (Sr ≥ 0.9); (3) The P₃ wave reflection coefficient exhibits positive incident angle dependence at low saturation levels, but diminishing to negligible magnitudes (1E-7) at high saturation levels; (4) Saturation variation has negligible effects on free-field displacement response. The proposed three-phase coupled wave theory model advances the quantitative characterization of wavefield energy distribution in multiphase media, providing a critical theoretical support for seismic response analysis of engineering sites and seismic design of underground structures.