CONTEMPORARY ADVANCES IN PHYSICS 665 



ohall encounter functions not so widely known as the simple sine and 

 cosine which suffice for the case of the stretched string. The little- 

 known example of the ball of fluid, with its three dimensions and four 

 variables, will repeat these lessons, and will serve as the final stepping- 

 stone to the wave-motions imagined by de Broglie and by Schroedinger. 

 To proceed to these, it will suffice to imagine strings and fluids not 

 uniform like those of the simple theory of vibrating systems and sound, 

 but varying from point to point in a curious and artificial way. 



Example of the Stretched String 

 Imagine a stretched string, infinitely long, extended along the 

 X-axis of a system of coordinates. Designate the tension in the string 

 by T, the (linear) density of the string by p. To derive the differential 



equation governing the motion, conceive the 



string as a succession of short straight seg- y<-rr^^^Z\.^-*^9 _ 



ments (Figure 1). Each segment exerts upon yV' l'^^ 

 its neighbors a force, which is the tension in l\~ 

 the string. When the string lies straight \ 

 along the axis of x, each segment lies in equi- i 



librium between the equal and opposite forces | 



which its neighbors exert upon it. When how- ""diT^ 

 ever the string is drawn sidewise (remaining, pjg ^ 



we shall suppose, in the xj'-plane) the neigh- 

 bors of each segment are oblique to it and to one another, the forces 

 which they exert upon it have components along the v-direction. 

 These components are in general unequal, and their algebraic sum 

 is a force urging the segment along the 3'-direction. Denote by dx 

 the length of such a segment, by y its lateral displacement, by 6 the 

 angle between it and the axis of x; so that dyjdx = tan 6, and pdx 

 stands for the mass of the segment. The resultant force upon the 

 segment is given by: 



F = r[sin {d + dd) - sin 6'] = T[tan (6 + dd) - tan d'] 



= T-d{tan dldx)dx = T{d''y/dx'-)dx (101) 



to the degree of approximation to which the difference between sin d 

 and tan 6 may be neglected.^ 



Equating this to the product of mass by acceleration, we obtain: 



pd^y/df" = T{d^y/dx^) (102) 



* This is the degree of approximation all but universal in the theory of vibrating 

 systems and sound. The conclusions from this theory are therefore strictly valid 

 only in the limit of infinitesimal displacements or distortions. 



