Radar received field no spreading

[tray84]

Consider a received radar signal of the form

$$s(t) = \frac{p e^{i 2 \pi f ( \frac{2 R(t)}{c} )}}{ [4 \pi R(t)]^2}$$

where p is the reflectivity value, f is the carrier frequency, and R(t) is the range. In some cases I have seen this written as

$$s(t) = p e^{ i 2 \pi f ( \frac{2 R(t)}{c} )}$$

(That is in many cases the geometric spreading is ignored). My question is can the $[4 \pi R(t)]^2$ be eliminated?

Note that I am a math grad student working on a research project in radar, so I am not sure about the specifics reasons this is done.

Additional Details
It is assumed that the antenna is an isotropic point source and the target is a point scatter. Also, the incident wave is assumed to be a complex sinusoidal.

 

[jsgruszynski]

The geometric spreading can't be ignored strictly. It's inherent to the geometry of propagation. It's like asking "can I ignore 1/r^2 in gravitational force?".

It is separable because the waveform frequency spectrum doesn't depend on distance, only the magnitude changes and the phase shift in time (it is delayed because it propagates over a distance at a finite speed). Basically the pulse maintains its shape but it's amplitude decays with distance. This because air and vacuum are linear media.

The reason for dropping it is because you may want to look at the dynamics of reflection or propagation already knowing the amplitude must decrease. This is particular true if there is some nonlinear media involved or if you want to look at the information content with regard to coding of the pulse for digital signal processing.

 

[johnsmi]

If I may ask, what kind of project would a math grad. student be working on? (Still, yet unsuccessfuly, trying to find the beauty in math)