Properties of a Transmission Line, or;
Proof that only one type of wave-front pattern can be propagated down a two-wire system
[1].

In order to discover how we characterise a transmission line we shall consider an observer watching a step passing him along a two-wire line (Fig.36).

The observer knows (a) Faraday's Law of Induction and (b) that electric charge is conserved.

Use Faraday's law  around the loop AA'B'B.
Define l as the inductance per unit length of the wire pair, then

In a time  the step will advance a distance  such that

and the change of flux will be (from Eqn.1)

Substitution into (a) Faraday's law gives the input voltage v across AB needed to equal and overcome the back e.m.f.

From (2) and (3),

Now we consider the conservation of charge. In a capacitor in general, q=cv. In our case, the charge  entering the line in time  equals the charge trapped in charging up the next segment  of the line, , where c is the capacitance per unit length between the pair of wires, and  is the capacitance of our section.

Combining (4) and (5);

Thus we see that, knowing only Faraday's Law and that charge is conserved, the observer in fig.36 concludes that any step passing him must have a single velocity C and a single voltage-current relationship given by an 'Ohm's Law' type relation

where  is a property of (1) f, the geometry of a cross-section of the wires and (2) of   , characteristics of the medium in which the wires are embedded.

Crosstalk in digital systems, or;

Proof that only two types of wave-front pattern can be propagated down a system of two similar wires and ground plane[2].

In Fig.37, the method of images is used; it is assumed that ib=ia, iq=ip.

The following terms are defined for steady state conditions:

l   =   Magnetic flux per unit length between AA' and BB' when unit current flows down AA' and back on BB'.

m = Magnetic flux per unit length between AA' and BB' when unit current flows down PP' and back on QQ'.

c = Charge per unit length on AA' and BB' which produces unit voltage drop between AA' and BB'=l/(coefficient of capacitance).

d = Charge per unit length on AA' and BB' which produces unit voltage drop between PP' and QQ'=1/(coefficient of induction).
This could well be called "Mutual Capacitance".

In order to discover how we characterise the four wire system we shall consider an observer watching a step passing him (Fig.38).

The observer knows (a) Faraday's Law of  Induction and (b) that electric charge is conserved.

Now assume that the wave front passing him involves current steps ia and ip travelling down the lines with a velocity C.

From  between AA' and BB', we get {as in (4)}

Similarly  between PP' and QQ', so

Also, from v=q/c {as in (5)},

First find C. Eliminate voltages from (9) thru (12).

From (9) and (11) we get

Therefore:

Similarly, from (10) and (12) we get

Eliminate ia and ip from (13) and (14) to get

So in the forward direction there are two possible velocities of propagation,

or

Returning to (13) and using the results for C, we find that the following two wave fronts are possible:

The EM, or Even Mode, wave (Fig.39) is like a TEM step travelling down between two wires made up of A shorted to P and B shorted to Q. It has the higher  and (in the case of surface, or stripline, conductors) the lower propagation velocity.

(15)

2) OM wave.

The OM, or Odd Mode, wave (Fig.40) is like a TEM step travelling down between two wires made up of A shorted to Q and P shorted to B.

.                                         (16)

Our initial assumption was that a stable waveform passed the observer; that is, a TEM wave which was in equilibrium[3]. Following that assumption, we concluded from our calculations that no other waveform may pass the observer. However, superposed combinations of EM and OM are permissible, as are seen in photographs in the literature[4]. For instance, a step travelling between AA' and BB' with no voltage visible between P and Q must be a combination of equal amplitudes of EM and OM, which cancel at P (for instance if P has been shorted to ground). As another example, if P is open circuit so that no electric current enters P, then the sum of currents in the EM and OM must be zero. .... [Riposte] .... [Riposte]

[1]
Published more thoroughly as Ref.15, Appendix I

[2]
Published more thoroughly as Ref.15, Appendix II

[3]
This observed and photographed phenomenon (see Ref.15, Fig.7 , Ref.3a, Fig.9.4 and Ref.6a, p57) contradicts the starting point of Einstein's theory of relativity. Einstein dismissed such a possibility as absurd (Ref.19, Ref.6a,); ".... If I pursue a beam of light with the velocity c (velocity of light in a vacuum), I should observe such a beam of light as a spatially oscillatory electromagnetic field at rest. [This is what I assume the observer to see, Fig.38.]  However, there seems to be no such thing, either on the basis of experience or according to Maxwell's equations."
Notice that in addition to my observing and photographing such a "spatially  oscillatory electromagnetic field at rest", my calculations towards the same conclusion are based only on Maxwell's equations. Of course, Einstein never used a high speed sampling oscilloscope. It is less clear why he avoided the imperatives of Maxwell's equations. (However, see Ref.9.)

[4]
Photographs in Ref.15, Fig.7 and Ref.3a, Fig.9.4