124 DYNAMICAL THEORY OF SOUND 



form into simple-harmonic functions of x. Each of these would 

 travel with its own velocity, so -that the resultant wave-profile 

 would continually alter. For this reason it would be hopeless 

 to look for a general solution of (7), or even of the modified 

 form (13) below, of the same simple character that we met 

 with in the theory of strings ( 23), and again in that of the 

 longitudinal vibrations of rods. 



A further remark is that when we substitute from (10) 

 in (7), the second term is of the order &V as compared with 

 the first. When the wave-length is large compared with the 

 dimensions of the cross-section this is a very small quantity, 

 and the term in question, which arose through taking account 

 of the rotatory inertia of the elements of the bar, viz. in 

 equation (2), may be neglected. It is easy to see, and it may 

 be verified a posteriori, that the same simplification is legitimate 

 in discussing the vibrations of a finite bar, at all events so long 

 as the distance between successive nodes is large compared 

 with K. We accordingly take the equation 





 P 



as the basis of our subsequent work, together with the formulae 



M = E**%{, FW-E^. ...(14) 



3# a doc W 



46. Free-Free Bar. 



To ascertain the normal modes of a finite bar we assume as 

 usual that 77 varies as cos (nt + e). The equation (13) of the 

 preceding section then reduces to 



where m* = ri i p/K 2 E. ........................ (2) 



It is to be noted that m is of the nature of the reciprocal of 

 a line. The solution of (1) is 



77 = A cosh mx + B sinh mx -f- C cos mx + D sin mx, (3) 

 the time-factor being for the present omitted. The three ratios 

 A:B:C:D, and the admissible values of m, and thence of n\ 

 are fixed by the four terminal conditions, viz. two for each end. 



