Radar sensors make use of the Doppler effect to measure the radial velocity of an object. The radar's Tx antennas emit an electromagnetic wave which travels to the moving objects, is reflected and the radar's Rx antennas detect the incident wave. Due to the movement of the measured object, the received wave has a different frequency.
According to wikipedia (https://en.wikipedia.org/wiki/Doppler_effect#Radar), the shift in frequency $\Delta f$ is given by: $$\Delta f = \frac{2 \Delta v_r}{c} f_0,$$ where $\Delta v_r$ is the relative radial velocity between radar sensor and object, $c$ is the speed of light and $f_0$ is the radar's frequency. A derivation of this equation is for example given in this script about radar sensors (page 21 and following): https://www.ei.ruhr-uni-bochum.de/media/ei/lehrmaterialien/39/a715b063167d904ec4a9a5cea2a1a54d4defc115/RuhrUni_Scriptum.pdf
What bugs me know is the following: The derivation is entirely classifcal. No time dilation or relativistic effects enter the picture. My understanding is that for electromagnetic waves (which do not need a medium for propagation), the equation for the relativistic Doppler effect has to be used: $$f_r = \sqrt{ \frac{1-\beta}{1+\beta} } f_s$$ with $\beta = v/c$.
But this equation (with its square root) does not match at all with the mentioned formula for the Doppler radar. For me it appears like that a classical method is used for the derivation of the radar's frequency shift, just like it would have been done for a sonar sensor (which uses sound waves and not EM waves). Why is this correct?