The present mathematical model is solved for the study of the effect of vertically aligned cracks present in the elastic half space on the propagation of elastic waves through the interface between double porosity solid and cracked elastic solid. Local fluid flow (LFF) between the inclusions (second fluid) and the host medium (first fluid) has been considered in double porosity solid which is possible due to wave propagation. This study follows Snell's law for reflection and refraction of an incident wave at the interface. Due to the presence of cracks in elastic half space, a loose bonding at the interface between these two media has been considered which is represented here as tangential slip. Final equation is framed in the form of Christoffel equation, which is solved analytically and found four reflected waves (three longitudinal body waves and one transverse body wave) and two refracted waves (one longitudinal body wave and one transverse body wave). The phase velocities and attenuation coefficients are calculated for each inhomogeneous wave. The expression of reflection and refraction coefficients, and energy shares of each of the reflected and refracted waves for a given incident waves is found in closed form. Numerical examples have been considered to discuss the effect of crack density, aspect ratio, local fluid flow, shear consolidation parameter and loose bonding parameter, etc. on energy shares by reflected and refracted waves.

本数学模型求解用于研究弹性半空间中垂直排列的裂纹对弹性波通过双孔隙固体和开裂弹性固体之间界面传播的影响。夹杂物（第二流体）和宿主介质（第一流体）之间的局部流体流动（LFF）已被考虑在双孔隙固体中，这可能由于波传播而成为可能。本研究遵循 Snell 定律对入射波在界面处的反射和折射。由于弹性半空间中存在裂缝，这两种介质之间的界面处的松散结合被认为在这里表示为切向滑移。最终方程以 Christoffel 方程的形式构成，解析求解得到4个反射波（3个纵向体波和1个横向体波）和2个折射波（1个纵向体波和1个横向体波）。计算每个非均匀波的相速度和衰减系数。反射和折射系数的表达式，以及给定入射波的每个反射波和折射波的能量份额以封闭形式存在。考虑了数值例子来讨论裂缝密度、纵横比、局部流体流动、剪切固结参数和松散结合参数等对反射波和折射波的能量份额的影响。计算每个非均匀波的相速度和衰减系数。反射和折射系数的表达式，以及给定入射波的每个反射波和折射波的能量份额以封闭形式存在。考虑了数值例子来讨论裂缝密度、纵横比、局部流体流动、剪切固结参数和松散结合参数等对反射波和折射波的能量份额的影响。计算每个非均匀波的相速度和衰减系数。反射和折射系数的表达式，以及给定入射波的每个反射波和折射波的能量份额以封闭形式存在。考虑了数值例子来讨论裂缝密度、纵横比、局部流体流动、剪切固结参数和松散结合参数等对反射波和折射波的能量份额的影响。