228 BELL SYSTEM TECHNICAL JOURNAL 



employed for phase and group velocity . The abscissae are values of to 

 and the ordinates are values of phase in radians. A portion of the function, 

 6, in the neighborhood of coi is shown. The distance, OB, is di . The slope 

 of the tangent, CA, to the curve at A \?, d\. The distance, CB, is wi^i. 

 Consequently, OC, or the intercept of this tangent on the phase axis, is 

 di — w\d[ . If, as shown in the figure, the absolute value of this intercept 

 is greater than it, we may transform (6) to a form in which the angle is 

 less than tt, by the substitution 



<p = 01 — ioidi + 2mr, (7) 



where n is an integer and 



1^1 < TT. 



In Fig. 2, n is 3, and (f is the distance DC. (6) then becomes 

 /(/) = 2bM exp ( — ai) — — cos (ojiT — (p), 



OT 



and cp is the ordinary phase lag of the sinusoid, relative to an origin of time 

 given by the instant of maximum envelope. 



We may choose as the instant at which the disturbance occurs, not Te , 

 at which the envelope is a maximum, but Ta , at which the instantaneous 

 value of the function has its maximum absolute value. Since 5 is small 

 compared with wi , this will occur very nearly at the smallest absolute value 

 of T for which cos(a)iT — ^) is ±1. This will occur for 



and for 



T = —, when -^ < ^ < "^j 



CO] A • L 



when —iz<<p<—- or ^ < (p < tt. 



From (4), (3) and (7), 



where k is an integer such that 



-^ < ^ = 01 - Wl^l + ^TT < '^. 



The significance of this can be seen from Fig. 3. Here, in addition to the 6 

 curve of Fig. 2, there are plotted a series of curves whose ordinates differ 



^Lamb, "Hydrodynamics," Cambridge U. Press 1916, p. 371. 



