134 BELL SYSTEM TECHNICAL JOURNAL 



In Eq. (5.18) /3 is a parameter (with the dimensions of area) characterizing 

 the ampUtude of the surface roughness, and J; is the unit tensor in two di- 

 mensions. Expressing (5.16) in terms of the Fourier space-amplitudes hk 

 and pk of //i and p respectively, we have 



-ikhk = pk, (5.19) 



giving finally 



\lik\r =kPkPt ' 'k/k' (5.20) 



Expressing (5.18) in terms of Fourier space-ampHtudes we get 



(e) 



Pktt' = l3A,'Ukk', (5.21) 



which, when inserted into (5.20) gives the following desired result: 



-(e) ^^_i ^_2 



|//*p = I3A2 k~\ (5.22) 



-(e) 



Now replacing 1/^ \^ by|/fe |- in Eq. (5.14) and substituting the ex- 

 pression (5.22) with the use of Eq. (5.15), we obtain 



Sic.) = ? • ^ -^crR' E . _/ ^,,, (5.23) 



If the frequencies of interest are larger than a certain characteristic frequency 

 coo = 47r2 D/L^ where L is the smaller of Zi and Zo , the summation in 

 (5.23) may be replaced by an integration giving finally 



k dk 



Si.) = ^D/rs":WM- I -,_^^,^, 



(5.24) 



This result is in agreement with experiment in most respects. The dependence 

 on frequency is, of course, that experimentally observed by all investigators. 

 The non-dependence on the voltage applied across the contact is implied 

 by the basic assumptions common to all of the mechanisms considered here, 

 and is in approximate agreement with the results of Christensen and Pearson 

 (see Eq. (1.2)). For our result to agree with the results of CP as regards the 

 dependence on the average resistance-^ R, the factor c-R^ A^ must be pro- 

 portional to 7^2+'' where b '~ 1.25. These authors also imply that some of 



''^It must be remembered that the resistance Ti in the CP formula is the total contact 

 resistance equal to sum of the gap resistance and the spreading resistance, whereas the R 

 in our theory evidently should be considered the gaj) resistance. For the purposes of com- 

 parison we make the ad hoc assumption that the gap resistance is proportional to the total 

 contact resistance. 



