Impossibility of lossless matched 3-port combiner It is a well-known result of scattering matrix theory (acoustic or EM) that a lossless 3-port cannot be matched. In other words, one cannot combine (sum) two arbitrary coherent waves arriving at two separate ports into a lossless summing junction without reflection no matter how the 3rd port is terminated. The result follows trivially from the fact that a scattering matrix of a lossless junction is unitary. The word arbitrary describing the waves is needed for any particular pair of amplitudes and phases can always combined without reflection but when once those are changed we get imperfect summing.
Such a basic result should be explicable by some simple and intuitive physical argument without resorting to scattering matrices and their unitarity. Can somebody give an intuitive argument without resorting to reciprocity by which feeding the 3rd port leads to a fixed amplitude/phase relationship in the other two thereby excluding arbitrary amplitudes/phases?
 A: Imagine you have such a device. You have ports $A$, $B$, and $C$. Suppose $A$ and $B$ add to give $C$. 
Now put a coherent wave of amplitude $1$ and phase $0$ on both of $A$ and $B$. You would expect a coherent wave of amplitude $2$ and phase $0$ on port $C$. 
Now instead put a coherent wave of amplitude $2$ and phase $0$ on $A$ and put nothing on port $B$. You would again expect a coherent wave of amplitude $2$ and phase $0$ on $C$.
Now, reverse the system. Put a coherent wave of amplitude $2$ and phase $0$ on $C$. What do you expect to get on $A$ and $B$? Does it split evenly reversing into the first situation above or does all of the power go out one port? Or is it an unequal split? What fraction would it be?
This is the essential idea of unitarity. Unitary implies time reversible. Input states have a deterministic 1:1 mapping with output states meaning that given any output state you can figure out what input states it "came from". The lossless summing junction breaks this 1:1 mapping so cannot exist.
A: For about 50 years RF engineers have been combining and splitting wave energy using device's like:  3dB hybrids, directional couplers, Lang couplers and Wilkinson power dividers/ combiners. Signals must be frequency and phase coherent to maximize the combining effect.
