NKTWOKKS Foi; I.MI'i;i).\\('l': FrXCTlONS 395 



and V — 3(10^) nu'ters per second. Tlie appioxiniale solution is 



where 



iirnv , . .1/2 



Pon = —J— and ff,,,. = {pon^y) 



Next we inipi'ove our estimate of the Z(M'()s by the well-known method in- 

 vol\in«>; the derivative of the func^tion, I — p'e^^'''', with i-esi)ect to /;, 

 e\aluated at pi„ . This now takes account of the actual impedance of the 

 end-plugs. The values of the zeros, so obtained, are 



Pn = - OCn + i^n , P-n = Pn = " «» " i^n 



where 



(2-15) 



^" = """ (^ + i ~ i 



where d,* is the real part of o-q,, . That is, 



8n = {mniJ-g/'2) 



where 



Trny 

 (^on — -7— • 



As an incidental matter pf interest, the above gives the Q of the cavity 

 at any resonance, namely 



Qn = ^ = d8n ^-^, (2-16) 



h 



For example, the dimensions, a = .5 cm., 6 = 1.0 cm., h = .5 cm. pro- 

 vide a (•a\ity that resonates at about 30,000 megacycles. Then the Q's at 

 the first tiu'ee I'esonances would be as follows: 



* For an\^ frequency, 5 = (a;yu(//2) '- is sometimes referred to as the "skin depth" 

 because it is the deptfi of metal at which tlie current density falls to 1/c times its 

 value at the surface of the metal. 



