162 Dr. T. H. Havelock on tlie Instantaneous 



If y r is proportional to e i(pt+Kra ^, the Telocity V of waves of 

 length 27r/fc is given by 



V- /l^ a \ sin \ Ka (K\ 



V-Vl^J-^T' * " * * (5) 



provided p 2 < -iT^jjua. If this condition is not satisfied, 

 simple wave motion is not possible. 



Instead of building up solutions from this simple type, we 

 may solve directly a particular problem and then analyse the 

 solution into simple harmonic components. 

 ~ Suppose that initially a particle (?/ ) is held displaced 

 through unit distance and then released, so that when £ = 

 we have y r and y r zero for all values of r except that ?/o=l» 



The subsequent disturbance is symmetrical with respect to 

 the particle y , and the equations of motion are 



t% = y r -i - 2y r + t/ r+1 , r >0\ 



r • ..(b) 

 ^o = %i— 2y 03 J 



where r — a/c ; c= VC^a//*). 



From these equations, with the initial conditions, we may 

 solve for y r as a series in t ; we find the general result 



oo 1 sf\2n+2r 



►-i.<- 1 )"iH5+S5rQ ■ ■ ■ w 



This can be expressed by a Bessel function, as in fact 

 follows directly from the equations (6), and we have for all 

 values of r, 



,y, = J 2 ,(^/r) (8) 



Thus the vibrations of the central particle are given by 

 J (2£/t), and those of the particles on either side by successive 

 even Bessel functions of the same argument. Using tables 

 of Bessel functions we may represent graphically the motion 

 of the particles. The curves in fig. 1 show how the dis- 

 placement of certain particles varies with the time, while in 

 fig. 2 we have the positions of the particles on one side of 

 the origin at instants given by £/t = 0, 2, 4, 6, 11. We may 

 notice how the front part of the disturbance remains a 

 crest as it travels out, becoming of greater effective wave- 

 length in the process. 



