344 BELL SYSTEM TECHNICAL JOURNAL 



pose that the formula will give a fair approximation to the facts in 

 this case also, but it should not be expecled to be accurate. 



APPENDIX C 



The deduction of the curves gi\en in Fig. 2 from ihc cur\cs given 

 in Fig. 1 requires some explanation. Looked at casually, it would 

 seem as if an isolated dot would not possess any frequency character- 

 istics whatsoever. Nevertheless, if a voltage, such as any of those 

 represented in Fig. 1, is applied to a network capable of being thrown 

 into oscillation, the network will respond to the voltage by oscillating. 

 Suppose, for simplicity, that the network consists of an inductance, 

 a capacity and a very small resistance in series, the response of the 

 network to the application of any of the voltages illustrated is that 

 it oscillates at constant frequency and gradually decreasing ampli- 

 tude. Further, the response varies when the natural period of the 

 circuit is varied. 



There are two ways of looking at this iilu'iiiinu'iion. We ina\' say, 

 on the one hand, that the oscillations of the fretjucncy in question 

 are manufactured by the network out of the \oltage applied and that 

 the frequency does not exist in the original voltage. On the other 

 hand, we may say that the original \oltage contains components 

 at or near the resonant frequency and that the circuit responds to 

 these components, because it offers them a small impedance, while 

 it does not respond to other components because it offers them a large 

 impedance. Hither of these views is permissible, but it is con\eiiient 

 for the purposes of this paper to use the nomenclature of the second 

 view and to consider the applied voltages to be made up of an in- 

 definitely large number of frequencies. The problem of determining 

 the response of oscillating networks is then sohcd by deducing the 

 frequency characteristic or the response characteristic of the im- 

 pressed voltage. This characteristic may be determined 1)\' means 

 of the Fourier integral, whose computation is described in any stand- 

 ard textbook on the subject. The following is intended to outline 

 the considerations, from a i)h\sical staii<l|)(iiiU, which lead to estab- 

 lishing this integral. 



To deduce the freciuency characteristic of an isolated dot, it is 

 simplest to start with a long series of dots which are uniformly spaced. 

 If such a series of dots is considered to extend indefinitely, it is possible 

 to analyze the resultant wave into a Fourier series by well known 

 methods. Now, suppose that such a Fourier series has been ob- 

 tained for a given spacing of the dots. The next step is to increase 



