The low frequency-high frequency transition

What can we consider a low frequency and what a high frequency? The preceding little study of diffracted waves suggests that we have the low-frequency case as soon as the diffracted spherical wave and the transmitted plane wave interfere constructively. In terms of distances, this condition can be written as  

$$\left\vert d_s - d_p \right\vert < \frac{\lambda}{4} ,$$ (5)

where $\lambda$ is the wavelength of the waves, and d_s and d_p are the distances traveled by the spherical and planar waves, respectively, during a time t. The expression of d_s is given by equation (4), and the distance d_p is simply

 

d_p = t v_w . (6)

Then the condition (5) becomes  

$$2r \left\vert 1 - \frac{v_w}{v_g} \right\vert < \frac{\lambda}{4} .$$ (7)

Finally, substituting $\lambda f = v_w$ into the latter equation, we obtain the following low-frequency condition:  

$$f < \frac{v_w}{8r} \left\vert 1 - \frac{v_w}{v_g} \right\vert .$$ (8)