LECTURE VIII. 
TRAVELING WAVES. 

33. In a stationary oscillation of a circuit having uniformly 
distributed capacity and inductance, that is, the transient of a 
circuit storing energy in the dielectric and magnetic field, current 
and voltage are given ^by the expression 

i = iQe~ut cos (0 T co - 7), ) 
e = e0e~ut sin (</> T co — 7), ) 

where 0 is the time angle, co the distance angle, u the exponential 
decrement, or the "power-dissipation constant," and i0 and eQ the 
maximunl current and voltage respectively. 

The power flow at any point of the circuit, that is, at any dis- 
tance angle co, and at any time t, that is, time angle <£, then is 

p = ei, 

= e0ioe~2ut cos (</> T co — 7) sin (0 =F co — 7), 

= ^|V2«<sin2(c/>=Fco-7), (2) 

and the average power flow is 

Po = avg p, (3) 

= 0. 

Hence, in a stationary oscillation, or standing wave of a uni- 
form circuit, the average flow of power, p0, is zero, and no power 
flows along the circuit, but there is a surge of power, of double 
frequency. That is, power flows first one way, during one-quarter 
cycle, and then in the opposite direction, during the next quarter- 
cycle, etc. 

Such a transient wave thus is analogous to the permanent wave 
of reactive power. 

As in a stationary wave, current and voltage are in quadrature 
with each other, the question then arises, whether, and what 


TRAVELING WAVES. 


89 


physical meaning a wave has, in which current and voltage are in 
phase with each other: 

i = loe~ut COS (0 =F co — 7), 
e = eQe~ut cos (<£ =F « — 7). 

In this case the flow of power is 


(4) 


P = 


= eQiQe-2ut cos2 


co - 7), 


and the average flow of power is 

p0 = avg p, 


(5) 


(6) 


Such a wave thus consists of a combination of a steady flow of 
power along the circuit, p0) and a pulsation or surge, pi, of the same 
nature as that of the standing wave (2) : 

Such a flow of power along the circuit is called a traveling wave. 
It occurs very frequently. For instance, it may be caused if by a 
lightning stroke, etc., a quantity of dielectric energy is impressed 


A • 


Fig. 39. — Starting of Impulse, or Traveling Wave. 

upon a part of the circuit, as shown by curve A in Fig. 39, or if by a 
local short circuit a quantity of magnetic energy is impressed upon 
a part of the circuit. This energy then gradually distributes over 
the circuit, as indicated by the curves B, C, etc., of Fig. 39, that is, 
moves along the circuit, and the dissipation of the stored energy 
thus occurs by a flow of power along the circuit. 


90 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

Such a flow of power must occur in a circuit containing sections 
of different dissipation constants u. For instance, if a circuit 
consists of an unloaded transformer and a transmission line, as 
indicated in Fig. 40, that is, at no load on the step-down trans- 

^> Line 


Transformer 
Line 


Fig. 40. 

former, the high-tension switches are opened at the generator 
end of the transmission line. The energy stored magnetically and 
dielectrically in line and transformer then dissipates by a transient, 
as shown in the oscillogram Fig. 41. This gives the oscillation 
of a circuit consisting of 28 miles of line and 2500-kw. 100-kv. 
step-up and step-down transformers, and is produced by discon- 
necting this circuit by low-tension switches. In the transformer, 
the duration of the transient would be very great, possibly several 
seconds, as the stored magnetic energy (L) is very large, the dis- 
sipation of power (r and g) relatively small; in the line, the tran- 
sient is of fairly short duration, as r (and g) are considerable. 
Left to themselves, the line oscillations thus would die out much 
more rapidly, by the dissipation of their stored energy, than the 
transformer oscillations. Since line and transformer are connected 
together, both must die down simultaneously by the same tran- 
sient. It then follows that power must flow during the transient 
from the transformer into the line, so as to have both die down 
together, in spite of the more rapid energy dissipation in the line. 
Thus a transient in a compound circuit, that is, a circuit comprising 
sections of different constants, must be a traveling wave, that is, 
must be accompanied by power transfer between the sections of 
the circuit.* 

A traveling wave, equation (4), would correspond to the case of 
effective power in a permanent alternating-current circuit, while 
the stationary wave of the uniform circuit corresponds to the case 
of reactive power. 

Since one of*the most important applications of the traveling 
wave is the investigation of the compound circuit, it is desirable 

* In oscillogram Fig. 41, the current wave is shown reversed with regard 
to the voltage wave for greater clearness. 


TRAVELING WAVES. 


91 


92 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 


to introduce, when dealing with traveling waves, the velocity unit 
as unit of length, that is, measure the length with the distance of 
propagation during unit time (3 X 1010 cm. with a straight con- 
ductor in air) as unit of length. This allows the use of the same 
distance unit through all sections of the circuit, and expresses the 
wave length X0 and the period T0 by the same numerical values, 

X0 = TQ = -, and makes the time angle 0 and the distance angle co 
directly comparable: 


0 = 2vft = 27T— , 

AO 

CO = 2 7T — = 2 7T/X. 
A 


(8) 


34. If power flows along the circuit, three cases may occur: 
(a) The flow of power is uniform, that is, the power remains 

constant in the direction of propagation, as indicated by A in 

Fig. 42. 


B 


B 

c' 

A' 
B' 


Fig. 42. — Energy Transfer by Traveling Wave. 

(b) The flow of power is decreasing in the direction of propaga- 
tion, as illustrated by B in Fig. 42. 

(c) The flow of power is increasing in the direction of propaga- 
tion, as illustrated by C in Fig. 42. 

Obviously, in all three cases the flow of power decreases, due to 
the energy dissipation by r and g, that is, by the decrement e~ut. 
Thus, in case (a) the flow of power along the circuit decreases at 


TRAVELING WAVES. 93 

the rate e~ut, corresponding to the dissipation of the stored energy 
by e-"', as indicated by A ' in Fig. 42; while in the case (6) the 
power flow decreases faster, in case (c) slower, than corresponds 
to the energy dissipation, and is illustrated by B' and C' in Fig. 42. 
(a) If the flow of power is constant in the direction of propa- 
gation, the equation would be 

i = io<rut cos (0 — o> — 7), 

e = e^~ut cos (0 - co - 7), (9) 


In this case there must be a continuous power supply at the 
one end, and power abstraction at the other end, of the circuit 
or circuit section in which the flow of power is constant. This 
could occur approximately only in special cases, as in a circuit 
section of medium rate of power dissipation, u, connected between 
a section of low- and a section of high-power dissipation. For 
instance, if as illustrated in Fig. 43 we have a transmission line 

Line 


Transformer LoadCT 

Line ^-) 

Fig. 43. — Compound Circuit. 

connecting the step-up transformer with a load on the step-down 
end, and the step-up transformer is disconnected from the gener- 
ating system, leaving the system of step-up transformer, line, and 
load to die down together in a stationary oscillation of a compound 
circuit, the rate of power dissipation in the transformer then 
is much lower, and that in the load may be greater, than the 
average rate of power dissipation of the system, and the trans- 
former will supply power to the rest of the oscillating system, the 
load receive power. If then the rate of power dissipation of the 
line u should happen to be exactly the average, w0, of the entire 
system, power would flow from the transformer over the line into 
the load, but in the line the flow of power would be uniform, as 
the line neither receives energy from nor gives off energy to the 
rest of the system, but its stored energy corresponds to its rate 
of power dissipation. 


94 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

(b) If the flow of power decreases along the line, every line 
element receives more power at one end than it gives off at the 
other end. That is, energy is supplied to the line elements by 
the flow of power, and the stored energy of the line element thus 
decreases at a slower rate than corresponds to its power dissipation 
by r and g. Or, in other words, a part of the power dissipated in 
the line element is supplied by the flow of power along the line, 
and only a part supplied by the stored energy. 

Since the current and voltage would decrease by the term e~w<, 
if the line element had only its own stored energy available, when 
receiving energy from the power flow the decrease of current and 
voltage would be slower, that is, by a term 


hence the exponential decrement u is decreased to (u — s), and s 
then is the exponential coefficient corresponding to the energy 
supply by the flow of power. 

Thus, while u is called the dissipation constant of the circuit, s 
may be called the power-transfer constant of the circuit. 

Inversely, however, in its propagation along the circuit, X, such 
a traveling wave must decrease in intensity more rapidly than 
corresponds to its power dissipation, by the same factor by which 
it increased the energy supply of the line elements over which it 
passed. That is, as function of the distance, the factor e~ sX must 
enter.* In other words, such a traveling wave, in passing along 
the line, leaves energy behind in the line elements, at the rate 
e + st, and therefore decreases faster in the direction of progress 
by e~ sX. That is, it scatters a part of its energy along its path 
of travel, and thus dies down more rapidly with the distance of 
travel. 

Thus, in a traveling wave of decreasing power flow, the time 
decrement is changed to e~(u~s^, and the distance decrement e+sX 
added, and the equation of a traveling wave of decreasing power 
flow thus is 

--- ( 

( 

* Due to the use of the velocity unit of length X, distance and time are 
given the same units, ^ = X0; and the time decrement, e+*<, and the distance 
decrement, e~sX, give the same coefficient s in the exponent. Otherwise, the 
velocity of propagation would enter as factor in the exponent. 


TRAVELING WAVES. 95 

the average power then is 


Po = avg e, 


-s)t e-2s\ — L° €-2ute+2s(t-\) ^ 

2 


Both forms of the expressions of i, e, and po of equations (11) 
and (12) are of use. The first form shows that the wave de- 
creases slower with the time t, but decreases with the distance X. 
The second form shows that the distance X enters the equation 
only in the form t — X and 4> — co respectively, and that thus for 
a constant value of t — \ the decrement is e~2ut} that is, in the 
direction of propagation the energy dies out by the power dissi- 
pation constant u. 

Equations (10) to (12) apply to the case, when the direction 
of propagation, that is, of wave travel, is toward increasing X. 
For a wave traveling in opposite direction, the sign of X and thus 
of co is reversed. 

(c) If the flow of power increases along the line, more power 
leaves every line element than enters it; that is, the line element 
is drained of its stored energy by the passage of the wave, and thus 
the transient dies down with the time at a greater rate than corre- 
sponds to the power dissipation by r and g. That is, not all the 
stored energy of the line elements supplies the power which is 
being dissipated in the line element, but a part of the energy 
leaves the line element in increasing the power which flows along 
the line. The rate of dissipation thus is increased, and instead 
of u, (u + s) enters the equation. That is, the exponential time 
decrement is 

e~ <" + •)', (13) 

but inversely, along the line X the power flow increases, that is, 
the intensity of the wave increases, by the same factor e+sX, or 
rather, the wave decreases along the line at a slower rate than 
corresponds to the power dissipation. 
The equations then become: 

-u<-s^-x)COs(0-co-7), ) 
6-s(t-x) cos <> — a — * 


and the average power is 


96 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

that is, the power decreases with the time at a greater, but with 
the distance at a slower, rate than corresponds to the power 
dissipation. 

For a wave moving in opposite direction, again the sign of X 
and thus of co would be reversed. 

35. In the equations (10) to (15), the power-transfer constant 
s is assumed as positive. In general, it is more convenient to 
assume that s may be positive or negative; positive for an increas- 
ing, negative for a decreasing, flow of power. The equations (13) 
to (15) then apply also to the case (6) of decreasing power flow, 
but in the latter case s is negative. They also apply to the case 
(a) for s = 0. 

The equation of current, voltage, and power of a traveling wave 
then can be combined in one expression: 


i = '^ _ _ ^ 

Q = ^£~VW-r*Vc=csAr»rka i ^3^,., — /v } ==/?„£— at £—s u-r A; r»r\c f .^^ /.i — o/ 1 I V§W 


where the upper sign applies to a wave traveling in the direction 
toward rising values of X, the lower sign to a wave traveling in 
opposite direction, toward decreasing X. Usually, waves of both 
directions of travel exist simultaneously (and in proportions de- 
pending on the terminal conditions of the oscillating system, as 
the values of i and e at its ends, etc.). 

s = 0 corresponds to a traveling wave of constant power flow 
(case (a)). 

s > 0 corresponds to a traveling wave of increasing power flow, 
that is, a wave which drains the circuit over which it travels of 
some of its stored energy, and thereby increases the time rate of 
dying out (case (c)). 

s < 0 corresponds to a traveling wave of decreasing power flow, 
that is, a wave which supplies energy to the circuit over which it 
travels, and thereby decreases the time rate of dying out of the 
transient. 

If s is negative, for a transient wave, it always must be 


since, if — s > u, u -\- s would be negative, and e~(u + s}t would 
increase with the time; that is, the intensity of the transient would 


TRAVELING WAVES. 97 

increase with the time, which in general is not possible, as the 
transient must decrease with the time, by the power dissipation 
in r and g. 

Standing waves and traveling waves, in which the coefficient 
in the exponent of the time exponential is positive, that is, the 
wave increases with the time, may, however, occur in electric cir- 
cuits in which the wave is supplied with energy from some outside 
source, as by a generating system flexibly connected (electrically) 
through an arc. Such waves then are "cumulative oscillations." 
They may either increase in intensity indefinitely, that is, up to 
destruction of the circuit insulation, or limit themselves by the 
power dissipation increasing with the increasing intensity of the 
oscillation, until it becomes equal to the power supply. Such 
oscillations, which frequently are most destructive ones, are met in 
electric systems as "arcing grounds," "grounded phase," etc. 
They are frequently called "undamped oscillations," and as such 
find a use in wireless telegraphy and telephony. Thus far, the 
only source of cumulative oscillation seems to be an energy supply 
over an arc, especially an unstable arc. In the self-limiting cumu- 
lative oscillation, the so-called damped oscillation, the transient 
becomes a permanent phenomenon. Our theoretical knowledge of 
the cumulative oscillations thus far is rather limited, however. 

An oscillogram of a "grounded phase " on a 154-mile three- 
phase line, at 82 kilovolts, is given in Figs. 44 and 45. Fig. 44 
shows current and voltage at the moment of formation of the 
ground; Fig. 45 the same one minute later, when the ground was 
fully developed. 

An oscillogram of a cumulative oscillation in a 2500-kw. 100,000- 
volt power transformer (60-cycle system) is given in Fig. 46. It 
is caused by switching off 28 miles of line by high-tension switches, 
at 88 kilovolts. As seen, the oscillation rapidly increases in in- 
tensity, until it stops by the arc extinguishing, or by the destruc- 
tion of the transformer. 

Of special interest is the limiting case, 

— s = u; 

in this case, u + s = 0, and the exponential function of time 
vanishes, and current and voltage become 

i = i0e±sX cos (0 =F co — 7), 


e = e0e±sX cos (0 T co - 7), v 


98 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 


8 


TRAVELING WAVES. 


99 


100 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

that is, are not transient, but permanent or alternating currents 
and voltages. 

Writing the two waves in (18) separately gives 


cos (0 - co - 70 - i'0'e-sX 


e = e0e+sx cos (0 - co - 


and these are the equations of the alternating-current transmission 
line, and reduce, by the substitution of the complex quantity for 
the function of the time angle <f>, to the standard form given in 
"Transient Phenomena," Section III. 

36. Obviously, traveling waves and standing waves may occur 
simultaneously in the same circuit, and usually do so, just as in 
alternating-current circuits effective and reactive waves occur 
simultaneously. In an alternating-current circuit, that is, in 
permanent condition, the wave of effective power (current in 
phase with the voltage) and 'the wave of reactive power (current 
in quadrature with the voltage) are combined into a single wave, 
in which the current is displaced from the voltage by more than 0 
but less than 90 degrees. This cannot be done with transient 
waves. The transient wave of effective power, that is, the travel- 
ing wave, 

i = iQ€- ut €- s (t ±\) cos (^ =p w _ T)? 

e = eQ€~ ut e~ s (i ±X) cos (0 =F co — 7), 

cannot be combined with the transient wave of reactive power, 
that is, the stationary wave, 

i = io'e-ut cos (0 T co - 7'), 
e = e0'e-ut sin (<£ =F co - 7'), 

to form a transient wave, in which current and voltage differ in 
phase by more than 0 but less than 90 degrees, since the traveling 
wave contains the factor e-s«TX), resulting from its progression 
along the circuit, while the stationary wave does not contain this 
factor, as it does not progress. 

This makes the theory of transient waves more complex than 
that of alternating waves. 

Thus traveling waves and standing waves can be combined only 
locally, that is, the resultant gives a wave in which the phase angle 
between current and voltage changes with the distance X and with 
the time t. 


TRAVELING WAVES. 


101 


When traveling waves and stationary waves occur simultane- 
ously, very often the traveling wave precedes the stationary 
wave. 

The phenomenon may start with a traveling wave or impulse, 
and this, by reflection at the ends of the circuit, and combination 
of the reflected waves and the main waves, gradually changes to a 
stationary wave. In this case, the traveling wave has the same 
frequency as the stationary wave resulting from it. In Fig. 47 is 
shown the reproduction of an oscillogram of the formation of a 
stationary oscillation in a transmission line by the repeated re- 


i, 


Fig. 47. — CD11168. — Reproduction of an Oscillogram of Stationary Line 
Oscillation by Reflection of Impulse from Ends of Line. 

flection from the ends of the line of the single impulse caused by 
short circuiting the energized line at one end. In the beginning of 
a stationary oscillation of a compound circuit, that is, a circuit com- 
prising sections of different constants, traveling waves frequently 
occur, by which the energy stored magnetically or dielectrically in 
the different circuit sections adjusts itself to the proportion cor- 
responding to the stationary oscillation of the complete circuit. 
Such traveling waves then are local, and therefore of much higher 
frequency than the final oscillation of the complete circuit, and 
thus die out at a faster rate. Occasionally they are shown by the 
oscillograph as high-frequency oscillations intervening between 


102 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

the alternating waves before the beginning of the transient and 
the low-frequency stationary oscillation of the complete circuit. 
Such oscillograms are given in Figs. 48 to 49. 

Fig. 48A gives the oscillation of the compound circuit consisting 
of 28 miles of line and the high-tension winding of the 2500-kw. 
step-up transformer, caused by switching off, by low-tension 
switches, from a substation at the end of a 153-mile three-phase 
transmission line, at 88 kilovolts. 


Fig. 4SA. — CD10002. — Oscillogram of High-frequency Oscillation Preced- 
ing Low-frequency Oscillation of Compound Circuit of 28 Miles of 
100,000-volt Line and Step-up Transformer; Low-tension Switching. 

Fig. 48# gives the oscillation of the compound circuit consisting 
of 154 miles of three-phase line and 10,000-kw. step-down trans- 
former, when switching this line, by high-tension switches, off the 
end of another 154 miles of three-phase line, at 107 kilovolts. 
The voltage at the end of the supply line is given as ei, at the 
beginning of the oscillating circuit as e2. 

Fig. 49 shows the oscillations and traveling waves appearing 
in a compound circuit consisting of 154 miles of three-phase line 
and 10,000-kw. step-down transformer, by switching it on and 
off the generating system, by high-tension switches, at 89 kilo- 
volts. 

Frequently traveling waves are of such high frequency — 
reaching into the millions of cycles — that the oscillograph does 
not record them, and their existence and approximate magnitude 
are determined by inserting a very small inductance into the 


TRAVELING WAVES. 


103 


104 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

circuit and measuring the voltage across the inductance by spark 
gap. These traveling waves of very high frequency are extremely 
local, often extending over a few hundred feet only. 

An approximate estimate of the effective frequency of these very 
high frequency local traveling waves can often be made from their 
striking distance_across a small inductance, by means of the 

relation -^ = V/ 7^ = z0, discussed in Lecture VI. 
lo * Co 

For instance, in the 100,000- volt transmission line of Fig. 48A, 
the closing of the high-tension oil switch produces a high-frequency 
oscillation which at a point near its origin, that is, near the switch, 
jumps a spark gap of 3.3 cm. length, corresponding to ei = 35,000 
volts, across the terminals of a small inductance consisting of 34 
turns of 1.3 cm. copper rod, of 15 cm. mean diameter and 80 cm. 
length. The inductance of this coil is calculated as approximately 
13 microhenrys. The line constants are, L = 0.323 henry, C = 

2.2 X 10~6 farad; hence z0 = y 5 = Vo.1465 X 103 = 383 ohms. 

The sudden change of voltage at the line terminals, produced 

i on nno 
by closing the switch, is - -~— = 57,700 volts effective, or a 

V_3 

maximum of e0 = 57,700 X V2 = 81,500 volts, and thus gives 
a maximum transient current in the impulse, of i0 = — = 212 

amperes. iQ = 212 amperes maximum, traversing the inductance 
of 13 microhenrys, thus give the voltage, recorded by the spark 
gap, of e\ = 35,000. If then / = frequency of impulse, it is 

e\ = 2-jrfLiQ. 

Or' '=2^' ; .' . Y 

35,000 


27rX 13 X 10-6 X212 
= 2,000,000 cycles. 

37. A common form of traveling wave is the discharge of a 
local accumulation of stored energy, as produced for instance by 
a direct or induced lightning stroke, or by the local disturbance 
caused by a change of circuit conditions, as by switching, the 
blowing of fuses, etc. 


TRAVELING WAVES. 105 

Such simple traveling waves frequently are called "impulses." 
When such an impulse passes along the line, at any point of 
the line, the wave energy is zero up to the moment where the 
wave front of the impulse arrives. The energy then rises, more 
or less rapidly, depending on the steepness of the wave front, 
reaches a maximum, and then decreases again, about as shown in 
Fig. 50. The impulse thus is the combination of two waves, 


Fig. 50. — Traveling Wave. 


one, which decreases very rapidly, e~(u + s}i} and thus preponder- 
ates in the beginning of the phenomenon, and one, which decreases 
slowly, e-(u~s)t. Hence it may be expressed in the form: 

a2e-2^-s)^e-2sX, (20) 


where the value of the power-transfer constant s determines the 
" steepness of wave front." 

Figs. 51 to 53 show oscillograms of the propagation of such an 
impulse over an (artificial) transmission line of 130 miles,* of the 
constants : 

r = 93.6 ohms, 

L = 0.3944 henrys, 

C = 1.135 microfarads,— 

thus of surge impedance ZQ = y ~ = 590 ohms. 

The impulse is produced by a transformer charge, f 

Its duration, as measured from the oscillograms, is TQ = 0.0036 

second. 

In Fig. 51, the end of the transmission line was connected to a 

noninductive resistance equal to the surge impedance, so as to 

* For description of the line see "Design, Construction, and Test of an Arti- 
ficial Transmission Line," by J. H. Cunningham, Proceedings A.I.E.E., January, 
1911. 

t In the manner as described in "Disruptive Strength of Air and Oil with 
Transient Voltages," by J. L. R. Hayden and C. P. Steinmetz, Transactions 
A.I.E.E., 1910, page 1125. The magnetic energy of the transformer is, however, 
larger, about 4 joules, and the transformer contained an air gap, to give constant 
inductance. 


106 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 


Fig. 51. — CD11145. — Reproduction of Oscillogram of Propagation of 
Impulse Over Transmission Line; no Reflection. Voltage, 


Fig. 52. — CD 11 152. — Reproduction of Oscillogram of Propagation of Im- 
pulse Over Transmission Line; Reflection from Open End of Line. Voltage. 


TRAVELING WAVES. 


107 


give no reflection. The upper curve shows the voltage of the 
impulse at the beginning, the middle curve in the middle, and the 
lower curve at the end of the line. 

Fig. 52 gives the same three voltages, with the line open at the 
end. This oscillogram shows the repeated reflections of the vol- 
tage impulse from the ends of the line, — the open end and the 
transformer inductance at the beginning. It also shows the in- 
crease of voltage by reflection. 


Fig. 53. — CD11153. — Reproduction of Oscillogram of Propagation of Im- 
pulse Over Transmission Line; Reflection from Open End of Line. 
Current. 


Fig. 53 gives the current impulses at the beginning and the mid- 
dle of the line, corresponding to the voltage impulses in Fig. 52. 
This oscillogram shows the reversals of current by reflection, and 
the formation of a stationary oscillation by the successive reflec- 
tions of the traveling wave from the ends of the line.