So you're right that the two antennas being the same doesn't really matter. What I meant was that if the two antennas are the same, the entire part of the Friis equation that deals with frequency dependence goes away (the lambda / (4piR) part). If the antennas are different, the frequency dependence still goes away, but there's some new scale factor.
There are two interesting things that you want to know about your antenna. The gain, which measures directivity, and the effective area, which roughly corresponds to the cross section of sky that the antenna can listen to. Going from the transmitter to the receiver, you have some transmit power going into the antenna. You now want to figure out what the power density is in the vicinity of the receiver. You get this by spreading the power over a sphere and multiplying by the antenna gain of the transmitter.
Now you need to know how much of that power density is seen by the receiver. This is slightly more complicated than the transmit case, because you now have to take into account the antenna gain of the receiver (i.e. where it's pointing), which the Friis equation considers, as well as how big a chunk of sky it's listening to (i.e. effective area), which the Friis equation does not consider.
It turns out that the effective area is a function of lambda^2 (an antenna of some size and ideal frequency will have an easier time collecting higher frequency signals, and a harder time collecting lower frequency signals). So the lambda^2 from the effective area cancels the 1/(lambda^2) from the Friis equation.
So you're right that the two antennas being the same doesn't really matter. What I meant was that if the two antennas are the same, the entire part of the Friis equation that deals with frequency dependence goes away (the lambda / (4piR) part). If the antennas are different, the frequency dependence still goes away, but there's some new scale factor.
There are two interesting things that you want to know about your antenna. The gain, which measures directivity, and the effective area, which roughly corresponds to the cross section of sky that the antenna can listen to. Going from the transmitter to the receiver, you have some transmit power going into the antenna. You now want to figure out what the power density is in the vicinity of the receiver. You get this by spreading the power over a sphere and multiplying by the antenna gain of the transmitter.
Now you need to know how much of that power density is seen by the receiver. This is slightly more complicated than the transmit case, because you now have to take into account the antenna gain of the receiver (i.e. where it's pointing), which the Friis equation considers, as well as how big a chunk of sky it's listening to (i.e. effective area), which the Friis equation does not consider.
It turns out that the effective area is a function of lambda^2 (an antenna of some size and ideal frequency will have an easier time collecting higher frequency signals, and a harder time collecting lower frequency signals). So the lambda^2 from the effective area cancels the 1/(lambda^2) from the Friis equation.