Editing Heaviside condition (section)

==Background==
A signal on a transmission line can become distorted even if the line constants, and the resulting [[Propagation constant|transmission function]], are all perfectly linear.  There are two mechanisms: firstly, the attenuation of the line can vary with frequency which results in a change to the shape of a pulse transmitted down the line.  Secondly, and usually more problematically, distortion is caused by a frequency dependence on [[phase velocity]] of the transmitted signal frequency components.  If different frequency components of the signal are transmitted at different velocities the signal becomes "smeared out" in space and time, a form of distortion called [[dispersion (optics)|dispersion]].

A transmission line is ''dispersionless'', if the velocity of signals is independent of frequency.  Mathematically <math> \frac {d} {d \omega} v  = 0 </math>.  

A transmission line is ''distortionless'' if it is dispersionless and the [[attenuation coefficient]] is independent of frequency.  Mathematically <math> \frac {d} {d \omega} \alpha  = 0 </math>.  

This was a major problem on the first [[transatlantic telegraph cable]] and led to the theory of the causes of dispersion being investigated first by [[Lord Kelvin]] and then by Heaviside who discovered in 1876 how it could be countered.  Dispersion of [[electric telegraph|telegraph]] pulses, if severe enough, will cause them to overlap with adjacent pulses, causing what is now called [[intersymbol interference]].  To prevent intersymbol interference it was necessary to reduce the transmission speed of the transatlantic telegraph cable to the equivalent of {{Frac|1|15}} [[baud]].  This is an exceptionally slow data transmission rate, even for human operators who had great difficulty operating a morse key that slowly. 

For voice circuits (telephone) the frequency response distortion is usually more important than dispersion whereas digital signals are highly susceptible to dispersion distortion.  For any kind of analogue image transmission such as video or facsimile both kinds of distortion need to be mitigated.

An analogous Heaviside condition for dispersionless propagation in left-handed transmission line [[metamaterial]]s cannot be derived, since no combination of reactive and resistive elements would yield a constant group velocity.<ref>{{cite journal |last1=Caloz |first1=C. |last2=Itoh |first2=T. |author1-link=Christophe Caloz |author2-link=Tatsuo Itoh |title=Transmission line approach of left-handed (LH) materials and microstrip implementation of an artificial LH transmission line |journal=[[ IEEE Transactions on Antennas and Propagation]] |date=May 2004 |volume=52 |issue=5 |pages=1159-1166 |doi=10.1109/TAP.2004.827249}}</ref>