Can you please explain to me why we can define eq 1.8 the way it is? It makes sense to me but only in the way that if you replace v1 with Vf+Vr, it makes equation 1.7 stay true.
Yes, exactly, but I can see its a little confusing and I can add a some more comment.
VF is a definition (which means it can be anything we want). It means the forward voltage wave (like a1) and indicates a kind of power flow (really power flow in the forward direction is |a1|^2). VR is the reverse power flow. Equation 1.4 defines the relationship between VF and VR, which is exactly what you propose. What we do here is decompose the total voltage at any point on the line (which is what you would measure with a perfect voltmeter from that point to ground) into a foward voltage wave (which generally you can't measure) and a reverse voltage wave (which in general you also can't measure) .With these definitions we can get to a1 and b1, and from that to S11 and from that we can get the formula that S11=(Z-Z0)/(Z+Z0).
But, if you add a device called a directional coupler, it produces a voltage at a third port that is a percentage (like 10%) of the forward voltage (it can separate forward and reverse waves), so in that manner you can get a measure of the foward (or reverse, with a reverse coupler) voltage waves.
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