Considering the physical and mechanical properties change continuously along the depth in non-homogeneous foundation

the linear transformative relationship is analyzed between the phase velocity and group velocity of the inhomogeneous elastic body waves and the solution of corresponding homogeneous elastic foundation. Firstly

based on the basic equations of elastodynamics

the analytical expressions for the phase velocity and group velocity of the planar elastic body waves in any non-uniformly changing foundation are obtained. Compared with the classical solution of the elastic wave phase velocity of the homogeneous foundation

it is found that the solution of the non-homogeneous foundation can be expressed linearly with the solution of the homogeneous foundation

thus the similarity conversion between the two solutions is obtained. At last

as a numerical example

considering the power law relationship of material properties change continuously along the thickness in the non-homogeneous foundation

the influence of foundation non-uniformity

Poisson's ratio and phase angle on the similarity conversion coefficients is analyzed. The calculation results show that

with the help of the classical solution of elastic wave velocity in homogeneous foundation

the solution of wave velocity in non-homogeneous foundation can be converted into the calculation of the similarity conversion coefficients

thereby simplifying the solution process.