Introduction
In geotechnical engineering, the Boussinesq and Westergaard theories are widely used to calculate the vertical stresses induced by surface loads on an elastic half-space. The vertical stresses calculated by these two theories are identical at a specific point when the r/Z ratio is equal to a certain value.

Boussinesq Theory
The Boussinesq theory assumes that the soil is homogeneous, isotropic, and semi-infinite, and the load is applied at the ground surface. The vertical stress induced by a surface load Q can be calculated using the following equation:

σz = (Q/(πZ^2))×(1+(r^2/Z^2))^-3/2

Where σz is the vertical stress at a depth of Z below the surface, and r is the radial distance from the load.

Westergaard Theory
The Westergaard theory is an extension of the Boussinesq theory that takes into account the effect of soil layers with different elastic properties. The vertical stress induced by a surface load Q can be calculated using the following equation:

σz = (Q/(πZ^2))×(1+(r/2Z)^2)^-1/2

Where σz is the vertical stress at a depth of Z below the surface, and r is the radial distance from the load.

Calculation of r/Z Ratio
The r/Z ratio is the ratio of the radial distance from the load to the depth of the point below the surface. It is an important parameter that determines the accuracy of the vertical stress calculation by the Boussinesq and Westergaard theories.

To calculate the r/Z ratio, we can use the following formula:

r/Z = (2πr)/(λ)

Where λ is the wavelength of the load, which is given by:

λ = (8Z^3/3Q)^1/2

Answer
The vertical stresses calculated by the Boussinesq theory and Westergaard theory at a particular point will be identical when r/Z ratio equals to 0.817. This means that when the radial distance from the load is 0.817 times the depth of the point below the surface, the vertical stresses calculated by the two theories will be the same.