CONTEMPORARY ADVANCES IN PHYSICS 667 



wave, with the ratio of the constants w/m for its speed of propagation u: 



ic = n/m = -^JtJ^. (108) 



This result justifies the title wave-equation for the dififerential equation 

 (103), and the meaning speed of propagation for its coefficient -ylT/p. 

 The reader will scarcely have failed to notice, however, that the 

 result was obtained only by prescribing very sharply defined physical 

 conditions. The string was supposed infinitely long; it was supposed 

 distorted into the form of a sine-wave; the transverse speeds of its 

 successive particles at the instant t = were preassigned as rigorously 

 as their positions. Were we to alter this last specification, we should 

 arrive at very different results. If for instance we should make the 

 assumption that at / = the string is distorted into a sine-wave and 

 is stationary, the equations (106) would not be adequate to describe 

 what happens. We should then be forced to have recourse to a 

 more general solution of the dififerential equation: 



y — C sin (nt + mx) + D sin (nt — nix) (109) 



and to adjust the constants C and D so as to conform to the newly 

 prescribed initial conditions, which are: 



y = A sin mx, y = at / = 0. (110) 



The adjustment is attained by making C = D = hA, whereupon we 

 get: 



y = A sin nt cos mx, (111) 



an equation which describes not a wave advancing perpetually along 

 the string, but a stationary oscillation with nodes and loops of vibra- 

 tion, like those which a violin-string properly bowed exhibits, those in 

 the air-column of Kundt's tube which the hillocks of dust reveal. 

 One would hardly detect by instinct in this stationary wave-pattern 

 the superposition of two oppositely gliding wave-trains each traveling 

 with the speed u = n/m = VT/p. Yet the one is always equivalent 

 to the other, and in the equation (111), the coefficients n and m are 

 linked to one another through the wa\'e-speed characterizing the string, 

 and the equation may be written 



y = A sin //;;;/ cos ;;/.v, u = si'/ p. (11^) 



Although the tension and the density of the string thus determine 

 n when m is preassigned (or vice versa), nothing so far brought upon 

 the scene compels any limitations upon the coefficient m. The infi- 



