(l) Second -pin-end frequency: tan q?, = tanh q2,, q2. = 2.25tt = T.OT 

 (within 0.01). Here s = c = 0.707, C = 588, a^^= 0, 'a.^^ = - 1.00, 

 a = 2.00, and A = - 1.00 (within O.Ol). 



(m) At ql = 2.490-n - 7.823: a = 31.6, a = - 32.7, 8l^2 ^ ^3.7, and 

 A = - 1.05. 



(n) Second free-free frequency: cC = 1; qi£ = 2.500tt = 7.o'-l 



(within O.OOl). Here a , a „, and a are all infinite. In '--&' wicn 



Case (g), finite amplitudes may occur only if neither F nor G vanisr.c-o 



and F/G = b /b,^ = 1.001 q, or if F = G = 0. 

 11 12 



(o) Third built-in frequency: cC = -1; hence q£ = 2.50025Tr = 7.855- 

 Here C = 1290 and a = a = - 6U5, a = 6i+5 , and A = (exactly). 



To find qH and a-,p, let cC = -1; but C ~ S, hence c = - 



1 



- 0.000775. Also c = - e, the slight angle beyond 2.5^^ or 7.85398, there- 

 fore 6 = 0.000775 = 0.0002i+7Tr and ql = 7-85398 + 0.000775 = 7-855 or 



2.50025Tr. 7 = - i tan q£ - - ( ^'^^J^^^ • ' - ) = _ i (_i290) = 6U5- 

 12 2 2 '^ 2 



Alternatively, a = ; r =— sS = -S = — sinh 7-855 because cC = -1, 



12 1-cC 2 2 2 



I = 1 (e'^-855 _e-'^-855) ^i. .7-855 ^ 



12 2 2 



^12 " ^^^" 



(p) At ql = 2.51-!^ = 7.885: C = 1322, I^^ = - 32.1, a = 31-2, 

 a = - 30.2, and A = - 0.95- 



Ab 



ove qX, = 3, at least, curves representing a-, -i , a p, and a lie 



close to those defined by a = tan ql -1, a-.^ = - "tan ql , and 



17 



