Devn 



The characteristic equation of (II- 5 . 2) reads 



2 

 A (a , o- ) A + B (a , o- ) A + C (a,cr ) = 



where . . > A . 2 , .2 , T (o)2 



A ( a, o- ) = 4 a <r (m+mzz ) + a N 



zz 



B (a, cr)^N Co) 

 zz 



2 3 2 



pg S(o) (2a- 4 a + a ) +2acr (m+m ) 



5 ZZ 



7Th 2 2h 2 2 



'] 



C (a, «r)4 ar 2 ^ + 



2 3 2 



PgS(o)(a- 4 a + a ) -air (m+m )|« 



37?h 8h 2 z z 



Pg s( 



°>( 



1- 8a + 3a 



37Th. 



8 h. 



i 



a (m+m ) 



ZZ 



For a given frequency and a given value of a , the 

 quantity a H (i <r , a) defines a point in the Nyquist plane . When a 

 varies , this point describes a curve called an equifrequency curve . 

 The solutions of equation (II- 6 . 2) are obtained as the intersections 

 of this equifrequency curve and a circle centered at the origin and 

 of radius equal to the numerator of expression (II- 6. 4) times J 

 (This construction is but the geometric interpretation of equation 

 II-6.4) .These solutions are stable if the real parts of the roots of 

 the characteristic equation are negative , they are unstable otherwise. 



When the wave frequency is small , equation ( II - 6 . 2) has 

 only one solution (Figure 18) . For a slightly larger frequency , 

 equation (II- 6. 2) has two stable solutions and one unstable solution 

 (figure 19) . Only the smallest solution has been obtained experimen- 

 tally , because during the experiments we did not attempt to see if 

 a second stable motion was possible . When the frequency ^s 0. 182 

 Hz there are only two solutions both stable (Figure 20) . 

 Figure 21 clearly shows the jump phenomenon . Beyond £ ■=• 0. 182 , 

 there is only one stable solution (Figure 22) . 



From the quantitative point of view , Figures 18 to 22 

 lead to solutions slightly different from the experimental results . 

 In particular ; the jump phenomenon occurs for f = 0. 182 Hz instead 

 of f = 0. 190 Hz . It is believed that this discrepancy is due to the 



1044 



