Vibration of Bars treated simply. 585 



possible values varies directly as the radius of gyration of 

 cross-section, directly as the square root of Young's modulus, 

 inversely as the square of the length o£ the bar, and inversely 

 as the square root of the density. 



(3) The frequencies of the various tones in the series for a 

 given bar and given terminal conditions vary directly as the 

 square of m. 



Satisfaction of Terminal Conditions. — Reverting now to 

 equations (6), (7), and (8), if the rod extends from a?=0 to 

 x = I, we see that they afford us four equations, two for each 

 end of the rod. To complete the solution, the constants in 

 (11) must satisfy these four equations. We may thus obtain 

 the three ratios A : B : C : D, and have left an equation which 

 must be satisfied by m. And for each value of m so found 

 the corresponding u is determined in everything save a 

 constant multiplier which must be fixed by the initial state, 

 into which matter we need not further enter. 



Free-Free Bar. — Beginning the detailed treatment of the 

 bars with one free at each end, let us rewrite equation (11) 

 in the form 



u = P(cos x' + cosh x) + Q(cos x' — cosh #') 



+ R (sin ss' + sinh x') + S(shW— sinh a?')> • ( 15 ) 



where x' denotes mx/l. As Lord Rayleigh points out, this 

 form is advantageous because the four quantities in brackets 

 repeat themselves on differentiation, and vanish with x 

 except the sum of cos and cosh, which equals 2 for x' = 0. 



Thus, for the free end at the origin, we must apply (7) to 

 (15) by differentiating the latter twice to x, put x = and 

 equate the result to zero. Then, for the second part of (7), 

 differentiate (15) thrice to x, put x = and equate to zero. 

 These operations give 



Q = and S=0 (1G) 



We have now to follow the same method for the other end, 

 where x=l, but can omit to begin with the terms involving 

 Q and S, which are already known to be zero. 



Hence we have 



w=P(cosy+cosh^') + R(sinA'' + sinh^ / ). . . (17) 



Differentiating this twice and thrice respectively and putting 

 x = l, i. e., x' = m and equating to zero, we find 



P( — cos m + cosh hi)+R( — <in m + sinh m) = 0,") 

 and X . (18) 



P(sin m + sinh m) -h R(-^ cos m-f cosh m) = 0, J 



These two equations give two expressions for the ratio P : R, 



