18 BELL SYSTEM TECHNICAL JOURNAL 



complex quantities, which had begun early in the nineteenth century 

 among mathematicians, was popularized among engineers by Kennelly 

 and Steinmetz. Then the proportionality relation V = ZI, which had 

 previously been true only if V and / were interpreted as amplitudes, 

 acquired a more general significance, for it was found that this relation 

 could express the phase relationship as well, provided Z was given a 

 suitable complex value. 



An important generalization came when the close similarity of the 

 laws connecting V and / in an electric circuit to those governing force 

 and velocity in mechanical systems suggested that the ratio "force/ 

 velocity" be called a "mechanical impedance." This usage is now 

 well nigh universal. 



The next step was a short one : it amounted to extending the term 

 to include also the ratio "force per unit area/flow per unit area" ; that 

 is, "pressure/flux." This usage is well known in such fields as acous- 

 tics, but it has not penetrated as far into the electrical field as con- 

 venience seems to warrant. 



If we read these remarks with a view to appraising the direction in 

 which future growth might be expected, we are immediately impressed 

 by the strong trend toward interpreting the ratio "force/velocity" in 

 an ever widening sense. It is my purpose in the present paper to 

 indicate some further extensions which I have found to be useful. 

 They are founded upon five basic ideas. The first is to recognize 

 and use whenever possible analogies between dynamical fields in which 

 the impedance concept is common and others (heat, for instance) in 

 which it is not. The second is the idea of extending the F// relation 

 from circuits to radiation fields, in much the same way that the 

 "force/velocity" concept has been made to embrace "pressure/flux" 

 in hydrodynamics. The third is, to regard the impedance as an attri- 

 bute of the field as well as of the body or the medium which supports 

 the field, so that the impedance to a plane wave is not the same as 

 the impedance to a cylindrical wave, even when both are propagated 

 in infinite "free space." The fourth basic idea is that of assigning 

 direction to the impedances of fields. This does'not mean, however, 

 that the impedances are vectors; in fact, they are not, since they fail 

 to obey the laws of addition and the laws of transformation peculiar 

 to vectors. And finally the fifth is a generalization of the idea of a 

 one-dimensional transmission line or simply a transmission line. While 



apparent resistance of a circuit containing resistance and self-induction only." 

 Frederick Bedell and Albert C. Crehore, "Derivation and Discussion of the General 

 Solution of the Current Flowing in a Circuit Containing Resistance, Self-induction 

 and Capacity, With Any Impressed Electromotive Force," Journal A. I. E. E., Vol. 

 IX, 1892, pp. 303-374, see p. 340. 



