The complex amplitude of a wave may be defined in three ways. It can be a voltage amplitude, a current amplitude, or a normalised amplitude whose squared modulus equals the power conveyed by the wave. In each case we represent the wave amplitude by a complex phasor whose length is proportional to the size of the wave and whose phase angle tells us the relative phase with respect to the origin or zero of the time variable.
Waves travelling from generator to load have complex amplitudes usually written V+ (voltage) I+ (current) or a (normalised power amplitude).
Waves travelling from load to generator have complex amplitudes usually written V- (voltage) I- (current) or b (normalised power amplitude).
The phase angle of the complex amplitude varies as we move along the transmission line. In the case of positive travelling waves, the phase decreases as (-j beta x) with increasing distance x from the generator; whereas for negative travelling waves the phase advances as (+ j beta x) with increasing distance x from the generator.
The complex reflection coefficient (gamma), or scattering parameter (s), or s parameter, for reflections from a load, is defined as gamma = s = b/a = V-/V+ = -I-/I+. These three ratios have the same value. The phase angle of gamma or s depends on where along the line we measure it; it is usually taken to be the value at the load impedance attachment point. On lossless lines the magnitude of gamma is independent of the point of measurement as the magnitudes of the waves do not depend on distance.
At the load therefore, for a general complex gamma, the phasors (vector representations of amplitudes and phases of the forward and backward waves) will lie in different directions in the complex plane. We remember the Argand diagram; the vector length represents the amplitude of the wave and the angle between the real axis and the vector represents the phase ange of the wave.
As we move back along the transmission line towards the generator, the value of the distance variable x is decreasing (x is the distance from the generator) the phase of the backward travelling wave advances with distance x, whereas the phase of the forward travelling wave is decreasing with distance x. If we could observe the phasors of the forward and backward waves, at any fixed point along the line they would have a fixed angle to each other. As we move in the positive x direction along the line the phasors contra-rotate (the forward wave has phase decreasing with x and the backward wave has phse increasing with x), and there will be two special points where they lie in the same straight line; the first where they lie in the same direction and add vectorially to give a maximum value ("standing wave maximum") and the second where they subtract ("standing wave minimum"). A little thought shows that if the resultant vector describing the standing wave maximum lies along say the X axis, then the vector describing the standing wave minimum lies along the Y axis in the Argand diagram.
At any intermediate point the resultant vector describing the amplitude of the oscillating voltage on the line has length intermediate between the maximum and the minimum. If you take two vectors of appropriate lengths and rotate them in opposite directions at a constant angular rotation rate, plotting the length of the resultant in terms of the phase angle of one of the component vectors, you will trace out a curve which corresponds to the standing voltage wave pattern along the transmission line. Note that this standing wave pattern is not sinusoidal with distance, in general.
In the case of a complex reflection coefficient gamma, the phase angle of gamma determines where along the line the first standing wave minimum lies, in terms of the wavelength and the position of the load. The magnitude of gamma determines the "voltage standing wave ratio" or VSWR, which is clearly given by the formula
1 + |gamma|
VSWR = ---------------
1 - |gamma|
for, remembering |gamma| = |V-|/|V+|
and substituting, we find
|V+| + |V-|
VSWR = ---------------
|V+| - |V-|
as expected.