670 BELL SYSTEM TECHNICAL JOURNAL 



A plot of this equation as a family of curves for various values of (p is shown 

 in Fig. 6. It is noted that, in general, this group of curves resembles those 

 of Fig. 4 of the triangular whip. 

 The steady state is 



dko^ojo r 1 . 1 . 



X = ~2 i ~ sm coo t — -sin co/ 



CO — a*o [_a.'o <o 



+ - sm coo (/ — - ) — ~sm co (/ — - ) 



COo COo O) Wo J 



(17) 



and, in dimensionless quantities or expressed as a ratio of the pulse dimen- 

 sions, we obtain 



A(p 

 da = ^ _ ^ 2 ^^^ '^^ ^^^ ^'^^('^ - !)• (18) 



From (18) it follows that the maximum ampHtude of the steady state is 



. 4:(p C03 TTif , ^ 



A plot of this curve is shown in Fig. 7. 



Shifted Cosine Whip. 



The shifted cosine whip produces results of a similar nature. We have 

 seen that the transform equation for this whip is (1.05) 



Using equations (3) and (5) and transferring to dimensionless quantities, in 

 which 



X 27r/coo CO / COo/ 



- = 0, = (^ = —^ J = — = — 



d 1 COo ZTT/COo 27r 



we obtain 



^ ^ 2 ( 2 — I') \^" ^'^'^^ ~ "^^ ^^^ ^'^'^ 



(21) 

 - cos 27rvj(r - \)u{t - 1) + (^' co3 27r(r - 1)w(t - 1) j. 



Since we are interested only in the transient displacement, (21) becomes 



. ^ (1 - C03 iTTipr) - <p^{l - C03 Ittt) . 



2(1 - ^2) V^^j 



A family of curves showing 8 for various values of <p is shown in Fig. 8. 



