i8 = 



1/2 



(B7) 



H 



The result simplifies to the taut cable equation in the limit s/l == when mgl << Hin equation (B4). 



In that case X^ approaches zero and equation (B6) reduces to 



lim 



S//-0 



tanf 



(B8) 



and 



(j3/)^= (2k- Dtt, k= 1,2,3... (B9) 



^"^ii^'" 



odd. (BIO) 



A graphical solution to equation (B9) is presented in Fig. B2 for several values of the parameter 



X^. The arrows indicate the values of ^r which correspond to the natural frequencies of a taut string. 



For small X^ the symmetric mode frequencies approach those of a taut string. However, as X^ increases 

 the first symmetric mode frequency increases toward the first antisymmetric frequency. They coincide 

 for X^ = 4ir^ and thereafter the first symmetric mode frequency is greater than the first antisymmetric 

 mode frequency. At still larger values of X^ these frequency crossovers occur at the higher symmetric 

 modes. 



As an example consider X^ = 36it^. The antisymmetric mode values of — are given by --— , n 



= 2,4,6, etc. as before while the first four symmetric mode solutions are indicated by the encircled 

 intersections in Fig. 32. The lowest two symmetric mode frequencies have crossed over and lie above 

 the 'owesl two antisymmetric frequencies. The frequencies of the third symmetric and antisymmetric 

 modes are equal (crossover is occuring) while the fourth symmetric mode frequency is quite close to 

 the rt = 7 frequency of a string. For the modes higher than « = 7 the natural frequencies are essen- 

 tially those of the taut string. The catenary effects progress into the higher modes as X^ increases, but 

 for finite X^ some unaffected modes remain. Returning to the first symmetric mode frequency of the 



125 



