CHAPTER IV. TRAVELING WAVES. 20. As seen in Chapter III, especially in electric power cir- cuits, overhead or underground, the longest existing standing wave has a wave length which is so small compared with the critical wave length — where the frequency becomes zero — that the effect of the damping constant on the frequency and the wave length is negligible. The same obviously applies also to traveling waves, generally to a still greater extent, since the lengths of traveling waves are commonly only a small part of the length of the circuit. Usually, therefore, in the discussion of traveling waves, the effect of the damping constants on the fre- quency constant q and the wave length constant k can be neglected, that is, frequency and wave length assumed as inde- pendent of the energy loss in the circuit. Usually, therefore, the equations (74) and (75) can be applied in dealing with the traveling wave. In these equations the distance traveled by the wave per second is used as unit length by the substitution /I = [C3 cos q(t- X)+ C3' sin q (t - /I)] - <•-* ('+A) [C4 cos g (t + ;) + C/ sin q (t + ^)] } (141) and — v/I-- [Cj cos g (^ — ^) 4- C/ sin q (t — X)] [C2 cos q (t + X) + C27 sin g 0 + ^)] [C8 cos g ft - A) 4- C87 sin g (* - A)] [(74 cos g (^ + X) + (7/ sin g (t + A)] } , (142) or cos q(t - /I)] + X)] (143) and [A2 cos q (t - [A3 cos g (t + [A 4 cos g (t — A/ sin g (* A/ sin q (t A± sin q (t where and ^ - o-Z, a- = vie. (144) (145) In these equations (141) to (144) the sign of ^ may be reversed, which merely means counting the distance in opposite direction. TRAVELING WAVES 459 This gives the following equations: i = £-ut {£+s(f A) [Bl cos q (t — X) + .B/ sin q (t — X)] 4- £~s(<~A) [53 cos g (£ — .A) + jBs7 sin q (t — X)] and e - •-«"-" i cos « - + sn « T + e+s(t+» [B2 cos g (t + X) + 5/ sin q (t + X)] + £-s«-*) [J58 cos g (t - X) + Bz' sin q (t - /I)] . + e-*f(<+x) [54 cos g (< + ^)+ 547 sin q (t + ^)]}, (147) or [A2 cos g (t + X) + Aa7 sin q (t + X)] [A3 cos ?(«->!)+ ^3r sin q (t - X)] [^I4cosg (« + ;)+ A/sin q (t + ;)]} (148) and [A3 cos ?(* + ;)+ A/ sin q (t + X)] [A3 cos g (« - X) + A3' sin g (t - X)] [A4 cos q(t + X)+ AJ sin q(t+ X)]}. (149) In these equations (141) to (149) the values A, B, C, etc., are integration constants, which are determined by the terminal conditions of the problem. The terms with (t - X) may be considered as the main wave, the terms with (t + X) as the reflected wave, or inversely, depend- ing on the direction of propagation of the wave. 21. As the traveling wave, equations (141) to (149), consists of a main wave with variable (t - X) and a reflected wave of the same character but moving in opposite direction, thus with the variable (t + X), these waves may be studied separately, and afterwards the effect of their combination investigated. 460 TRANSIENT PHENOMENA Thus, considering at first one of the waves only, that with the variable (t - X), from equations 148 and 149 we have [Alcosq(t - X)+ A/singft - X)] + £-•«-*> [A3 cos q (t - X) + Az' sin q (t - X)}} e~ut { (Aie+8('-x) + A3£-S('-A)) cos q(t- X) + (A1's+a('-A) + A8'e — ('-A)) sin £ (t - X)} (150) and that is, in a single traveling wave current and voltage are in phase with each other, and proportional to each other with an effective impedance (152) This proportionality between e and i and coincidence of phase obviously no longer exist in the combination of main waves and reflected waves, since in reflection the current reverses with the reversal of the direction of propagation, while the voltage remains in the same direction, as seen by (148) and (149). In equation (150) the time t appears only in the term (t — X) except in the factor e~ut, while the distance X appears only in the term (t — X). Substituting therefore hence f .<* V-M; that is, counting the time differently at any point X, and counting it at every point of the circuit from the same point in the phase of the wave from which the time t is counted at the starting point of the wave, X = 0, or, in other words, shifting the starting point of the counting of time with the distance X, and substituting in (150), we have TRAVELING WAVES 461 e = (A j cos qtt + A / sin g^) £ stl (A 3 cos £ '' * (Aj COS g-"1 (A 3 cos A 3' sin A / sin A 3 sin (153) The latter form of the equation is best suited to represent the variation of the wave, at a fixed point ^ in space, as function of the local time tL. Thus the wave is the product of a term £-uA which decreases with increasing distance ^, and a term e0 - e~utl {e+stl (A1 cos qtt + A/ sin q$ + e~stl (A3 cos qtt + A/ sin qtt) } (154) which latter term is independent of the distance, but merely a function of the time tt when counting the time at any point of the line from the moment of the passage of the same phase of the wave. Since the coefficient in the exponent of the distance decrement £~uA contains only the circuit constant, but does not contain s and q or the other integration constants, resubstituting from equations (71) to (68), x = ai = i VLO, we have uX = u \/LC I where I is measured in any desired length. 462 TRANSIENT PHENOMENA Therefore the attenuation constant of a traveling wave is }, '(155) and hence the distance decrement of the wave, depends upon the circuit constants r] L, g, C only, but does not depend upon the wave length, frequency, voltage, or current; hence, all traveling waves in the same circuit die out at the same rate, regardless of their frequency and therefore of their wave shape, or, in other words, a complex traveling wave retains its wave shape when traversing a circuit, and merely decreases in amplitude by the distance decrement e~w\ The wave attenua- tion thus is a constant of the circuit. The above statement obviously applies only for waves of con- stant velocity, that is, such waves in which q is large compared with s, u, and m, and therefore does not strictly apply to ex- tremely long waves, as discussed in 13. 22. By changing the line constants, as by inserting inductance L in such a manner as to give the effect of uniform distribution (loading the line), the attenuation of the wave can be reduced, that is, the wave caused to travel a greater distance / with the same decrease of amplitude. As function of the inductance L, the attenuation constant (155) is a minimum for — °=o- dL hence, rO - gL = 0, or (156) and if the conductance g = 0 we have L = ; hence, in a per- fectly insulated circuit, or rather a circuit having no energy losses depending on the voltage, the attenuation decreases with increase of the inductance, that is, by "loading the line," and the more inductance is inserted the better the telephonic transmission. TRAVELING WAVES 463 In a leaky telephone line increase of inductance decreases the attenuation, and thus improves the telephonic transmission, up to the value of inductance, L = -, (157) 9 and beyond this value inductance is harmful by again increasing the attenuation. For instance, if a long-distance telephone circuit has the following constants per mile: r = 1.31 ohms, L = 1.84 X 10~3 henry, g = 1.0 X 1Q-" mho, and C = 0.0172 X 10~6 farad, the attenuation of a traveling wave or impulse is u0 = 0.00217; hence, for a distance or length of line of 1Q = 2000 miles, e-nA = £-4.34 = 0.0129; that is, the wave is reduced to 1.29 per cent of its original value. The best value of inductance, according to (157), is L = -C = 0.0225 henry, i/ and in this case the attenuation constant becomes u, = 0.00114, and thus e-«A = £-2.24 = 0.1055, or 10.55 per cent of the original value of the wave; which means that in this telephone circuit, by adding an additional inductance of 22.5 - 1.84 = 20.7 mh. per mile, the intensity of the arriving wave is increased from 1.29 per cent to 10.55 per cent, or more than eight times. If, however, in wet weather the leakage increases to the value g = 5 x 10~8, we have in the unloaded line UQ = 0.00282 and e~^ = 0.0035, while in the loaded line we have u0 = 0.00341 and e~u°l = 0.0011, 464 TRANSIENT PHENOMENA and while with the unloaded line the arriving wave is still 0.35 per cent of the outgoing wave, in the loaded line it is only 0.11 per cent; that is, in this case, loading the line with inductance has badly spoiled telephonic communication, increasing the decay of the wave more than threefold. A loaded telephone line, therefore, is much more sensitive to changes of leakage g, that is, to meteorological conditions, than an unloaded line. 23. The equation of the traveling wave (153), e+* (Al cos qtt + A/ sin gtt) e~sti(A3 cos gti + A3' sin qtt) } , e = e-t* e-«* { + can be reduced to the form e = e-^jj^fi-^ (fi+*i - e-*i) Smqth + Ef-*** (e+^ - e-'S cos qtk } , (158) where k = ^ - 7i = * ~ * - ?i 1 and L (159) By substituting (159) in (158); expanding, and equating (158) with (153), we get. the identities (160) Eje~8yi cos qy^ — E2e~sy* sin qy2 = Av E^'^1 sin qy{ + E2£~sy* cosqy2 = A/, E^+Syi cos qy^ — E2e+Sy* sin qy2 = - A3, ^iS4"^1 sin gyt + #2e+SY» cos gy2 = -A/, and these four equations determine the four constants Ev E2, Any traveling wave can be resolved into, and considered as consisting of, a combination of two waves: the traveling sine wave. £-*V) sin qt^ (161) and the traveling cosine wave, e2 = E2e~^ £-«'«. (e+^ - e-*4) cos ^. (162) TRAVELING WAVES 465 Since q is a large quantity compared with u and s, the two component traveling waves, (161) and (162), differ appreciably from each other in appearance only for very small values of th that is, near tti = 0 and tlt = 0. The traveling sine wave rises in the first half cycle very slightly, while the traveling cosine wave rises rapidly; that is, the tangent of the angle which the wave de makes with the horizontal, or— , equals 0 with the sine wave and has a definite value with the cosine wave. All traveling waves in an electric circuit can be resolved into constituent elements, traveling sine waves and traveling cosine waves, and the general traveling wave consists of four component waves, a sine wave, its reflected wave, a cosine wave and its reflected wave. The elements of the traveling wave, the traveling sine wave ev and the traveling cosine wave e2 contain four constants: the intensity constant, E\ the attenuation constant, u, and u0 respectively; the frequency constant, q, and the constant, s. The wave starts from zero, builds up to a maximum, and then gradually dies out to zero at infinite time. The absolute term of the wave, that is, the term which repre- sents the values between which the wave oscillates, is +stl - e~s^. (163) The term e0 may be called the amplitude of the wave. It is a maximum for the value of tl} given by < which gives _ (u _ 8) ,-<*-•>> + (U + S) hence, u — s and 1 u + s 2s u — s 466 TRANSIENT PHENOMENA and substituting this value into the equation of the absolute term of the wave, (163), gives u 9 Q hi 4- - e~stl), (168) we have el = e0 sin qlt 1 and (169) e2 = e0 cos qtt. J If ^ = 0, e0 = 0; that is, tt is the time counted from the beginning of the wave. » It is tt = t - X - r, or, if we change the zero point of distance, that is, count the distance X from that point of the line at which the wave starts at time t = 0, or, in other words, count time t and distance X from the origin of the wave, tt = t - ;, and the traveling wave thus may be represented by the amplitude, the sine wave, el = Ee~ut (s+stl — s~stl) sin qti = eQ sin qti\ the cosine wave, e2 = Es~wt (s+stl — e~stl) cos qti = e0 cos qtt', (170) and ti = t — X can be considered as the distance, counting backwards from the wave front, or the temporary distance; that is, distance counted with the point X, which the wave has just reached, as zero point, and in opposite direction to X. Equation (170) represents the distribution of the wave along the line at the moment t. As seen, the wave maintains its shape, but progresses along the line, and at the same time dies out, by the time decrement 468 TRANSIENT PHENOMENA Resubstituting, ti = t - X, the equation of the amplitude of the wave is (171) As function of the distance X, the amplitude of the traveling wave, (171), is a maximum for dX which gives X = 0; that is, the amplitude of the traveling wave is a maximum at all times at its origin, and from there decreases with the distance. This obviously applies only to the single wave, but not to a combination of several waves, as a complex traveling wave. - For X = 0, and as function of the time t this amplitude is a maximum, according to equations (163) to (165), at and is 1 u + s 2s u — s 2s /u + (172) At any other point X of the circuit, the amplitude therefore is a maximum, according to equation (164), at the time and is tm - em = E 2*-* /^y Vw2 - s2 w - s/ (173) TRAVELING WAVES 469 25. As an example may be considered a traveling wave having the constants u = 115, s = 45, q = 2620, and E = 100, hence, where tt = t — L In Fig. 99 is shown the amplitude eQ as function of the dis- tance X, for the different values of time, t = 2, 4, 8, 12, 16, 20, 24, and 32 X 10~3, v 16, 2^ Distin 4 6 8 10 12 14 16 18 20 22 24 Fig. 99. Spread of amplitude of electric traveling wave. with the maximum amplitude em, in dotted line, as envelope of the curve of eQ. - As seen, the amplitude of the wave gradually rises, and at the same time spreads over the line, reaching the maximum at the starting point A = 0 at the time t0 = 9.2 X 10~3 sec., and then decreases again while continuing to spread over the line, until it gradually dies out. It is interesting to note that the distribution curves of the amplitude are nearly straight lines, but also that in the present instance even in the longest power transmission line the wave has reached the end of the line, and reflection occurs before the maximum of the curve is reached. The unit of length A is the distance traveled by the wave per second, or 188,000 miles, and during the rise of the wave, at the origin, from its start to the maximum, or 9.2 X 10~3 sec., the wave thus has traveled 1760 miles, and the reflected wave would have returned to the origin before the maximum of the wave is reached, providing the cir- cuit is shorter than 880 miles. 470 TRANSIENT PHENOMENA 2 4 6 8 10 12 14 16 18 20 22 21 34 36 38 40 42 44 46 48 Fig. 100. Passage of traveling wave at a given point of a transmission line U.5 °*(qti 1.0 + 45°) 2 5.2621 2.0 2.5 \ \ SOX 10'a Sec. Fig. 101. Beginning of electric traveling waves. TRAVELING WAVES 471 With s = 1 it would be t0 = 8.7 X 10"3 sec., or nearly the same, and with s = 0.01 it would be t0 = 3.75 X 10~3 sec., or, in other words, the rapidity of the rise of the wave increases very little with a very great decrease of s. Fig. 100 shows the passage of the traveling wave, el = e0 sin qtb across a point X of the line, with the local time tt as abscissas and the instantaneous values of el as ordinates. The values are given for ^ = 0, where tt = t] for any other point of the line X the wave shape is the same, but all the ordinates reduced by the factor £~115* in the proportion as shown in the dotted curve in Fig. 99. Fig. 101 shows the beginning of the passage of the traveling wave across a point X = 0 of the line, that is, the starting of a wave, or its first one and one-half cycles, for the trigonometric functions differing successively by 45 degrees, that is, e = e sin t cos qtt = e0 sin ( qtt + ^ = 6 cos + = e sin The first curve of Fig. 101 therefore is the beginning of Fig. 100. In waves traveling over a water surface shapes like Fig. 101 can be observed. For the purpose of illustration, however, in Figs. 100 and 101 the oscillations are shown far longer than they usually occur; the value q = 2620 corresponds to a frequency / = 418 cycles, while traveling waves of frequencies of 100 to 10,000 times as high are more common. Fig. 102 shows the beginning of a wave having ten times the attenuation of that of Fig. 101, that is, a wave of such rapid decay that only a few half waves are appreciable, for values of the phase differing by 30 degrees. 26. A specially interesting traveling wave is the wave in which s = u, (174) 472 TRANSIENT PHENOMENA since in this wave the time decrement of the first main wave and its reflected wave vanishes, e-«--* = 1; (175) that is, the first main wave and its reflected wave are not tran- sient but permanent or alternating waves, and the equations of 12 16 \ oanfqti ^edooaqtj + 60 h ±± + 60 20 Time t =,1150 \ LiM X Fig. 102. Passage of a traveling wave at a given point of a line. the first main wave give the equations of the alternating-current circuit with distributed r, L, g, C, which thus appear as a special case of a traveling wave. Since in this case the frequency, and therewith the value of q, are low and comparable with u and s, the approximations made TRAVELING WAVES 473 in the previous discussion of the traveling wave are not per- missible, but the general equations (50) and (51) have to be used. Substituting therefore in (50) and (51), s = u, gives i = [£~hl {Cl cos (qt - kl) + C/ sin (qt - kl) } - e+hl \C2 cos (qt + kl) + C/ sin (qt + kl)}] - r2ut [£~hl {C4 cos (qt + kl) + C4' sin (qt + kl) } - +hl and - e {C3 cos (qt - kl) + <73' sin (qt - kl)}] (176) - (c/C, + c/7/) sin (qt - kl) } l { (c/CY- ctC2) cos (qt + kl) hl { (c2'C4 - C2C4) cos (qt + kl) + £ c, - c23 cos $ - ; -(C/C3 + c2(73Osin(^-^)}]. (177) In these equations of current i and e.m.f. e the first term represents the usual equations of the distribution of alternating current and voltage in a long-distance transmission line, and can by the substitution of complex quantities be reduced to a form given in Section III. The second term is a transient term of the same frequency; that is, in a long-distance transmission line or other circuit of distributed r, L, gt C, when carrying alternating current under an alternating impressed e.m.f., at a change of circuit conditions, a transient term of fundamental frequency may appear which has the time decrement, that is, dies out at the rate In this decrement the factor 474 TRANSIENT PHENOMENA is the usual decrement of a circuit of resistance r and inductance Lj while the other factor, may be attributed to the conductance and capacity of the circuit, and the total decrement is the product, A further discussion of the equations (176) and (177) and the meaning of their transient term requires the consideration of the terminal conditions of the circuit. 27. The alternating components of (176) and (177), io = s-ja{C1 cos (qt - kl) + C/ sin (qt - kl) } - e+hl{C2 cos (qt + kl) + C/ sin (qt + kl) } (178) and /> . * fit \ ( f+ '§* * /> (i \ r*r\cy ( nl 1/*7\ /*/» '/^ ^l_ /> /• f\ 01 T~I /'/v/ 7^7\ (. t/fl — c i I O- vy .| ' O^vy -. / l^\_/o \tyt/ IV v J V^i ^ i ~1~ ^•|^-/ -i / olll I Ul> — fi/L ) I (179) are reduced to their usual form- in complex quantities by resolv- ing the trigonometric function into functions of single angles, qt and kl, then dropping cos qt, and replacing sin qt by the imagi- nary unit j. This gives i0 = s~hl { (Cl cos kl — Of sin kl) cos qt' + (C/ cos kl + Cj sin A;Z) sin ^} — e+hl { ((72 cos kl + C27 sin A;Q cos qt + (C2 cos kl — C2 sin A^Z) sin qt } ; hence, in complex expression, / = e~hl { (C1 + yc/) cos &Z — (C/ — yCJ sin A;Z} - e+w { (Ca + yCa7) cos kl + (C/ - yCa) sin &£}, (180) and in the same manner, + [c/ (c, + yc/) + Cl (C/- ycj] sin w} - [c/ (Ca + yc/) + c\ (C/ - j(722)] sin JW } . (181) TRAVELING WAVES 475 However, from equation (52), qk + h (m + s) T _ j , k (m + s) — qh since q = 2 and lr 1/r and we have s 4- m = — and c/ = xk + rh h2 + k2 = h2 + k2 rk — 2 TT/L/Z, rk — xh h2 + k2 = h2 + k2 ' where x = 2 TT/L = reactance per unit length. From equation (54), R2 = V(s2 + q2 - m2)2 + 4 — ; — 7~9 = T T- v^y^y *7 * , 1 jv»^ A * _JI A»* Aj /i /^ IV \ /v l(j ~\~ A/ ft — Jfi/ However, T- k2) - 2jhk = V(rg - xb) - jV(zy)2 - (rg - xb}2 = V (rg - xb) - j V (r2 + x2) (g2 + b2) - (rg - (rg - xb) - j Vr^b2 + x2g2 + 2 rgxb V(rg - xb) - j (rb + xy) = V(r - jx) (jgf - y&); or A - y/b = VZF (194) TRAVELING WAVES 477 substituted in (193) gives and (195) substituted in (192) gives E= VZ {B^+hl (coskl-jsmkl) -B2e~hl (cos JW-f /sin kl)}, (196) where Bl and B2 are the complex imaginary integration constants. Writing h = a and k = f), B, = Dl and B2 = -D2 the equations (191) and (196) become identical with the equa- tions of the long-distance transmission line derived in Section III, equations (22) of paragraph 8. It is interesting to note that here the general equations of alternating-current long-distance transmission appear as a special case of the equations of the traveling wave, and indeed can be considered as a section of a traveling wave, in which the accelera- tion constant s equals the exponential decrement u.