Suppose a radar is emitting a spherical wave with some wavelength $\lambda$ and there is a plane traveling with velocity $\vec{v}$ towards the radar (and assuming plane waves getting to the plane).
I know that this configuration would produce beatings.
What I am trying to understand is what would happen if we take into account the plane's speed.
How would you show that there are beatings and how to find their frequency from the velocity of the plane?
The Doppler radar beat-frequency between a tone emitted and the tone reflected is twice the Doppler shift. For radio waves and non-relativistic speeds, it is $2.f_c(v/c)$, where $f_c$ is the frequency of the emitted carrier, v is the velocity of the aircraft, and c is the speed of light.
You get the double-dose because the Doppler shift applies both to the signal seen by the aircraft and again to the signal seen in the reflected wave at the receiver. Or, equivalently, because the total path-length (there and back) is changing twice as fast as the aircraft is moving.
