1168 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



As the frequency is increased there will he a range in which the terms 

 in parentheses in equation (455) are small compared to unity, so that 

 the square root may be expanded by the binomial theorem. This gives 



a = ^ ^ + -^^^ , (459) 



/3 = icoV^e . (4()p) 



If the line is of finite total thickness a and the frequency is so low or 

 the laminae are so thin that the first term on the right side of (459) 

 is large compared to the second, we have approximately 



P IT 



2\/iJL/e ga' 



(4G1) 



This is the frequency-independent attenuation constant that we found 

 in Section X, equation (407), for the pth mode in a plane Clogston 2 

 with infinitesimally thin laminae. We shall call the range over which the 

 attenuation is essentially flat the "low-frequency" range. On the other 

 hand, if the laminae are of finite thickness the second term on the right 

 side of (459) ultimately becomes dominant, and the attenuation constant 

 is then given approximately by 



o^'ulgll _ Tr'nlgtlf 



24:\/fI/e ()\/M/e 



(462) 



This is also the attenuation constant of a plane wave in an unbounded 

 laminated medium (except at very high frequencies), as may be seen 

 by letting the stack thickness a tend to infinity in equation (459). By 

 "high frequencies" we shall mean the frequency range in which the 

 attenuation constant is approximately proportional to / . 



Finally at very high frequencies when | | » 1, w^e have from (451), 



q^2/@, (463) 



and so from (443), 



7 = ?'co\//i2e2 



1 + 



2dn, 



(1 - 0)m20J 



(464) 



Expanding by the binomial theorem and substituting for from (452), 

 we get after a little rearrangement, 



