STRINGS 63 



as is seen at once by considering two consecutive positions of the 

 wave-form. Thus if in Fig. 24 the curves A, B represent the 

 positions at the instants t, t + St, we have PQ = c&t, RP = ySt, 

 RP/PQ = y', whence the former 

 of the relations (6). The same 

 thing follows of course from 

 differentiation of (2). Con- 

 versely, it is easily seen from 

 (5), or otherwise, that if the 

 initial conditions be adjusted so that either of the relations (6) 

 is everywhere satisfied, a single progressive wave will result. 



When the string is started with initial displacement, but 

 no initial velocity, the formula (5) reduces to 



y = i((a-c*)+(0 + cO} 00 



The two component wave-forms resemble the initial profile, but 

 are of half the height at corresponding points. It is easily seen 

 without analysis that this hypothesis satisfies the condition of 

 zero initial velocity. 



It appears from (6) that in any case of a single progressive 

 wave the expressions (4) and (6) of 22 for the kinetic and 

 potential energies are equal. Lord Rayleigh has pointed out 

 that this very general characteristic of wave motion may be 

 inferred otherwise as follows. Imagine the wave as resulting 

 from an initial condition in which the string was at rest, and 

 the energy E therefore all potential, in the manner just 

 explained. The two derived waves have half the amplitude (at 

 corresponding points) of the original form, and the potential 

 energy of each is therefore J E. Since the total energy of each 

 wave must be ^ E, it follows that the kinetic energy of each 

 must be \E. 



In mathematical investigations it is not unusual to find the 

 effect of dissipation represented by^jt he hypothesis that each 

 element of the string is resisted by a force proportional to its 

 velocity, so that the differential equation takes the form 



dt - 



As regards the theory of stringed instruments this particular 



