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.
A complex number is an ordered pair of real numbers. These can be the magnitude ("size") and phase ("timing") or real and imaginary parts. Two numbers are needed in general to specify an alternating voltage or quantity unambiguously, assuming the frequency is already known.
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).
On a lossless transmission line the modulus or size of the wave complex amplitude is independent of position along the line; the wave is neither growing not attenuating with distance and time.
On the other hand, the phase angle of the wave complex amplitude varies as we move along the transmission line. In the case of positive-travelling waves, the phase decreases with increasing distance from the generator; whereas for negative-travelling waves the phase advances with increasing distance 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. However it can be easily calculated at other points along the line, and from the value at another point the local "normalised impedance" z presented (by the load with the length of line attached to it) at the measurement point, may be calculated from z = (1+s)/(1-s).
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.
The "wavelength" of a wave is the distance we have to move along the transmission line for the sinusoidal voltage to repeat its pattern. It is the "spatial period" of the wave.
The "propagation constant" beta or "wave-vector" k of the wave is the equivalent of the angular frequency, but with respect to the distance variable. Thus the angular frequency is defined as (2 pi)/period, and the propagation constant is defined as (2 pi)/wavelength. Thus the phase angle of a forward wave depends on distance x as -(beta x). In some texts the concept of "propagation constant" is extended to the case of complex beta. In this case, the real part of the complex beta is the quantity equivalent to (2 pi)/wavelength, and the imaginary part of the complex beta represents the attenuation. However, you cannot define a "period" for an attenuating wave as the wave voltage waveform does not repeat as you move along the transmission line, so this extension is not as obvious as it appears at first sight.
The "velocity factor" times the speed of light in vacuum equals the speed of waves on the transmission line. Typically for coaxial cable the velocity factor is about 2/3, or 0.67-0.7. It is higher in semi-air-spaced coax. That puts the wave velocity at about 20 cm per nanosecond on this cable.
The "direction of propagation" is the direction of power flow; either from generator to load (forward) or from load to generator (backward or reflected).
The "characteristic impedance" (Zo) is the ratio of voltage to current in a forward travelling wave, assuming there is no backward wave. It is a "lossless real impedance" because any power delivered to it can be recovered later in time. A lossy real impedance is a resistance; power delivered to it appears as heat and cannot then be recovered.
The "normalised impedance", a dimensionless number z=Z/Zo, is the ratio of the actual impedance Z in ohms to the characteristic impedance in ohms.
Similarly, the "characteristic admittance" Yo = 1/Zo Siemens, and the "normalised admittance" = 1/z = (Zo)/Z.
The "capacitance per unit length" (C) is the total capacitance of a cable, measured at low frequency, divided by the total length of the cable. Similarly for the "Inductance per unit length" (L). The characteristic impedance turns out to be equal to sqrt(L/C).
The "wave velocity" divides into two types; "phase velocity" and "group velocity". These are the same on a coaxial or two wire line. The phase velocity is the speed with which a point of constant phase in the wave appears to move. Thus, for example, if we find a point at which V+=0 in a forward travelling wave and track it in time it moves at the wave phase velocity. The energy in the wave is moving at the "group velocity" which can be different in waveguide from the phase velocity. The modulations on a signal travel at the group velocity.
On coaxial cable or two wire line, the velocity has numerical value and units (SI) equal to 1/sqrt(LC).
The "return loss" in dB is the amount by which the returned signal power from a load reflection is less than the incident signal power. Numerically it is equal to -20 log modulus(gamma) where the log is to base 10.
The "Voltage Standing Wave Ratio" (VSWR) is the ratio of maximum ac voltage amplitude, as we move along the line, to the minimum ac voltage amplitude (which occurs 1/4 wavelength away from the maximum) and is related to the magnitude of the reflection coefficient |gamma| by VSWR = (1+|gamma|)/(1-|gamma|) or (1+|s|)/(1-|s|). It is a sensitive indicator of mismatch on a line. For example, if 1% of the incident power is reflected from a certain load, then |s|^2 = 1/100 and |s| = 1/10 so that the VSWR = 1.1/0.9 = 1.2 about. Of course if no power is reflected at all then |s| = 0 and the VSWR = 1.