CHAPTER III. STANDING WAVES. 14. If the propagation constant of the wave vanishes, h = 0, the wave becomes a stationary or standing wave, and the equa- tions of the standing wave are thus derived from the general equations (50) to (61), by substituting therein h = 0, which gives R2 = V(k2 - LCm2)2; (97) hence, if k2 > LCm2, R2 = tf- LCm2; and if /c2 < LCm2, R2 = LCm2'- tf. Therefore, two different cases exist, depending upon the rela- tive values of Ar* and LCm2, and in addition thereto the inter- mediary or critical case, in which k2 = LCm2. These three cases require separate consideration. is a circuit constant, while k is the wave length constant, that is, the higher k the shorter the wave length. A. Short waves, k2 > LCm2, (99) hence, R2 = k2 - LCm2 (100) and q = V ^ - ™\ 442 STANDING WAVES 443 or approximately, for very large k, Herefrom then follows and VLC °l= k c' mL c, -T-c, -1± - 2" k c'=---c' 2 k (102) (103) Substituting now h = 0 and (101), (103) in equations (50), (51), the two waves i', e' and i", e" coincide, and all the expo- nential terms reduce to e~ut', hence, substituting and gives , + Cv 2 + C4, i + C*', 2f + €4, (104) = t {[B^ cos (qt - kl) + BS sin (qt - kl)] - [B2 cos (qt + kl) + B2' sin (qt + kl)]} (105) and L C = k f-qBJ cos (qt-kl)-(mB, + qB,'} sin (qt-kl)] + [(mB2'-qB3) cos (qt + Jd)-(mB2 + qBJ) sin (qt + kl)]}. (106) Equations (105) and (106) represent a stationary electrical oscil- lation or standing wave on the circuit. B. Long waves, k2 < LCm2] (107) 444 hence, and TRANSIENT PHENOMENA R22 = LCm2 - k2, s = (108) (109) or approximately, for very small values of &, 1 r herefrom then follows (HO) ci = c2 = 0, and (m + s) L ~T~ (m — s) L (111) Substituting now h = Oand (109), (111) into (50) and (51), the two waves H ', e' and i", e" remain separate, having different expo- nential terms, e~(u-s» and t^"** but in each of the two waves the main wave and the reflected wave coincide, due to the vanishing of q. Substituting then = C- C and gives (112) sinkl] (113) STANDING WAVES 445 and *{[(*» + s) £/£+s< + (m - s) £2'£~8<] cos W + [(m + s) B1£+rf + (m - s) B2e~st] sin kl} s')sinA;Z] (114) + (BlS + « - B2e~st)smkl]} Equations (113) and (114) represent a gradual or exponential circuit discharge, and the distribution still is a trigonometric function of the distance, that is, ^ wave distribution, but dies out gradually with the time, without oscillation. C. Critical case, hence, o, = 0, (115) (116) and c2 = 0, raL (117) and all the main waves and their reflected waves coincide when substituting h = 0, (116), (117) in (50) and (51). Hence, writing and gives B = C, - C2 + C3 - C, 1 B' = CY 4- C2' + C,7 + C/ J i = fi-"1 {B cos kl - B' sin Id] (118) (119) 446 TRANSIENT PHENOMENA and e = y — £-M< {5' cos kl + B sin In the critical case, (119) and (120), the wave is distributed as a trigonometric function of the distance, but dies out as a simple exponential function of the time. 15. An electrical standing wave thus can have two different forms: it can be either oscillatory in time or exponential in time, that is, gradually changing. It is interesting to investigate the conditions under which these two different cases occur. The transition from gradual to oscillatory takes place at k* = m2LC; (121) for larger values of k the phenomenon is oscillatory; for smaller, exponential or gradual. Since k is the wave length constant, the wave length, at which the phenomenon ceases to be oscillatory in time and becomes a gradual dying out, is given by (63) as 27T 2, (122) m Vie In an undamped wave, that is, in a circuit of zero r and zero g, in which no energy losses occur, the speed of propagation is and if the medium has unit permeability and unit inductivity, it is the speed of light, S0 = 3 X 1010. (124) In an undamped circuit, this wave length lWo would correspond to the frequency, STANDING WAVES 447 hence, from (62), '•-,Vr.- <125> The frequency at the wave length lWo is zero, since at this wave length the phenomenon ceases to be oscillatory ; that is, due to the energy losses in the circuit, by the effective resistance r and effective conductance g, the frequency / of the wave is reduced below the value corresponding to the wave length lw, the more, the greater the wave length, until at the wave length lWo the frequency becomes zero and the phenomenon thereby non-oscillatory. This means that with increasing wave length the velocity of propagation of the phenomenon decreases, and becomes zero at wave length lWo. If m2LC = 0, k0 = 0 and lWo = oo ; that is, the standing wave is always oscillatory. If m?LC = oo, k0 = oo and lWo = 0; that is, the standing wave is always non-oscillatory, or gradually dying out. In the former case, m?LC = 0, or oscillatory phenomenon, substituting for m2, we have and r _L g c' or rC — gL = 0 (distortionless circuit). In the latter case, m2LC = oo , or non-oscillatory or exponen tial standing wave, we have r \ — — Q\ — =00 448 TRANSIENT PHENOMENA and since neither r, g, L, nor C can be equal infinity it fol- lows that either L = 0 or C = 0. Therefore, the standing wave in a circuit is always oscillatory, regardless of its wave length, if rC - gL = 0, (126) or - = §J (127) that is, the ratio of the energy coefficients is equal to the ratio of the reactive coefficients of the circuit. The standing wave can never be oscillatory, but is always exponential, or gradually dying out, if either the inductance L or the capacity 0 vanishes ; that is, the circuit contains no capacity or contains no inductance. In all other cases the standing wave is oscillatory for waves shorter than the critical value L = -— , where 0 V - 9 V §} > (128) and is exponential or gradual for standing waves longer than the critical wave length lWo; or for k < ko the standing wave is exponential, for k > ka it is oscillatory.0 The value kQ = m VLC thus takes a similar part in the theory of standing waves as the value r02 = 4 L0C0 in the condenser discharge through an inductive circuit; that is, it separates the exponential or gradual from trigonometric or oscillatory conditions. The difference is that the condenser discharge through an inductive circuit is gradual, or oscillatory, depending on the circuit constants, while in a general circuit, with the same circuit constants, usually gradual as well as oscillatory standing waves exist, the former with greater wave length, or m VLC > k, (129) the latter with shorter wave length, or m VLC < k. (130) STANDING WAVES 449 An idea of the quantity k0, and therewith the wave length lu,o, at which the frequency of the standing wave becomes zero, or the wave non-oscillatory, and of the frequency /0, which, in an undamped circuit, will correspond to this critical wave length lWo, can best be derived by considering some representative numerical examples. As such may be considered: (1) A high-power high-potential overhead transmission line. (2) A high-potential underground power cable. (3) A submarine telegraph cable. (4) A long-distance overhead telephone circuit. (1) High-power high-potential overhead transmission line. 16. Assume energy to be transmitted 120 miles, at 40,000 volts between line and ground, by a three-phase system with grounded neutral. The line consists of copper conductors, wire No. 00 B. and S. gage, with 5 feet between conductors. Choosing the mile as unit length, r = 0.41 ohm per mile. The inductance of a conductor is given by = I (2 loge lf 10~9, in henrys, (131) where I = the length of conductor, in cm.; lr = the radius of conductor; ld = the distance from return conductor, and /* = the permeability of conductor material. For copper, fi = 1. As one mile equals 1.61 X 105 cm., substituting this, and reducing the natural logarithm to the common logarithm, by the factor 2.3026, gives L = f 0.7415 log ^ + 0.0805\ in mh. per mile. (132) , For lr = 0.1825 inch and ld = 60 inches, L = 1.95 mh. per mile. The capacity of a conductor is given by C = I I - *—=\ 109, in farads, (133) 450 TRANSIENT PHENOMENA where S0 = 3 X 1010 = the speed of light, and d = the allow- ance for capacity of insulation, tie wires, supports, etc., assumed as 5 per cent. Substituting £0, and reducing to one mile and common loga- rithm, gives mf.; (134) logf lr hence, in this instance, C = 0.0162 mf. Estimating the loss in the static field of the line as 400 watts per mile of conductor gives an effective conductance, which gives the line constants per mile as r = 0.41 ohm; L = 1.95X10-3 henry; g = 0.25 X 10~6 mho, and C = 0.0162 X lO"6 farad. Herefrom then follows :>-i.S-.S-'* a- = VLC = V31.6 X 10~6 = 5.62 X lO"6, &0 = ra\/57 = 545 X 10~6; hence, the critical wave length is ^o =: IT = 11»500 miles, /c0 and in an undamped circuit this wave length would correspond to the frequency of oscillation, - m /0= ^~ = 15.7 cycles per See. STANDING WAVES 451 Since the shortest wave at which the phenomenon ceases to be oscillatory is 11,500 miles in length, and the longest wave which can originate in the circuit is four times the length of the circuit, or 480 miles, it follows that whatever waves may originate in this circuit are by necessity oscillatory, and non-oscillatory currents or voltages can exist in this circuit only when impressed upon it by some outside source, and then are of such great wave length that the circuit is only an insignificant fraction of the wave, and great differences of voltage and current of non-oscillatory nature cannot exist. Since the difference in length between the shortest non- oscillatory wave and the longest wave which can originate in the circuit is so very great, it follows that in high-potential long- distance transmission circuits all phenomena- which may result in considerable potential differences and differences of current throughout the circuit are oscillatory in nature, and the solution case (A) is the one the study of which is of the greatest importance in long-distance transmissions. With a length of circuit of 120 miles, the longest standing wave which can originate in the circuit has the wave length lw = 480 miles, and herefrom follows k = - = 0.0134 and ft 0.01342 LC 31.6 X 10-12 hence, in the expression of q in equation (101), = V 5.7 X 106 - 0.00941 X 10e, Ar5 m2 is negligible compared with — ; that is, 452 TRANSIENT PHENOMENA or / = — = 380 cycles per sec. 2 7T Hence, even for the longest standing wave which may origi- nate in this transmission line, q = 2380 is such a large quantity compared with m =* 97 that m can be neglected compared with q, and for shorter waves, the overtones of the fundamental wave, this is still more the case; that is, in equations (105) and (106) the terms with m may be dropped. In equation (106) ~ thus fc become common factors, and since from equation (135) by substituting m = 0 and (136) in (105) and (106) we get the general equations of standing waves in long-distance transmission lines, thus : i = e-* {[Bl cos (qt - kl) + £/ sin (qt - kl)] - [B2 cos (qt + kl) +. £2' sin (qt + kl)]}, (137) = - \J p£~ut{ [#i cos (qt - kl) + £/ sin (qt - kl)] + [£2cos (qt + kl) + £2'sin (qt + kl)]}, (138) or e = e-«rf{ [Al cos (qt + kl) + A/ sin (qt + kl)] + [A2 cos (qt - kl) + A2' sin (qt - kl)]}, (139) fc i = y ^ £~ui{[A, cos (qt + kl) + A/ sin (qt + kl)] - [A2 cos (qt - kl) + A2' sin (qt - kl)]}, (140) where 2> A' • - V 5/, etc. (2) High-potential underground power cable. 17. Choose as example an underground power cable of 20 miles length, transmitting energy at 7000 volts between con- STANDING WAVES 453 ductor and ground or cable armor, that is, a three-phase three- conductor 12,000-volt cable. Assume the conductor as stranded and of a section equiva- lent to No. 00 B. and S. G. Calculating the constants in the same manner, except that the expression for the capacity, equation (119), multiplies with the dielectric constant or specific capacity of the cable insula- tion, and that f ig verv small, about three or less; or taking the ^r values of the circuit constants from tests of the cable, we get values of the magnitude, per mile of single conductor, r = 0.41 ohm; L = 0.4 X 10~3 henry; g = 10~6 mho, corresponding to a power factor of the cable-charging current, at 25 cycles, of 1 per cent; C = .6 X 10~6 farad. Herefrom the following values are obtained : u = 513, m = 512, * -- VLC - 15.5 X 10~6, k0 = m VLC = 7.95 X 10~3, and the critical wave length is lWo = 790 miles, and the frequency of an undamped oscillation, corresponding to lWo, is /0 = 81.5 cycles per second. As seen, in an underground high-potential cable the critical wave length is very much shorter than in the overhead long- distance transmission line. At the same time, however, the length of an underground cable circuit is very much shorter than that of a long-distance transmission line, so that the critical wave length still is very large compared with the greatest wave length of an oscillation originating in the cable, at least ten times as great. Which means that the discussion of the possible phe- nomena in any overhead line, under (1), applies also to the under- ground high-potential cable circuit. In the present example the longest standing wave which may originate in the cable has the wave length lw = 80 miles, which gives k = 0.0785 and -4= - 5070, VLC 454 TRANSIENT PHENOMENA or about ten times as large as m, so that m can still be neglected in equation (87), and we have = 5070, VLC or / = 810 cycles per second, and the general equations of the phenomenon in long-distance transmission lines, (123) to (125), also apply as the general equa- tions of standing waves in high-potential underground cable circuits. (3) Submarine telegraph cable. 18. Choosing the following values: length of cable, single stranded-conductor, ground return, = 4000 miles; constants per mile of conductor: r = 3 ohms, L = 10~3 henry, g = 10~6mho? and C = 0.1 X 10~6 farad, we get u = 1500; m = 1500;