166 Dr. T. H. Havelock on tlie Instantaneous 



stretches of non-dispersive medium along which a disturbance 

 could travel with an abrupt front. 



3. Another illustration which may be treated in the same 

 way is taken from Lord Bayleigh's paper, already quoted, on 

 propagation of waves along connected systems of similar 

 bodies. Let each mass be connected to its immediate neigh- 

 bours on the two sides by an elastic rod capable of bending 

 but without inertia. The potential energy function is of the 

 form 



V= + 16(2z/ y _ 1 — y r -2 — 2/rJ 2 + i^(2//r — 1/r-l— 1/r+l) 2 



+ ±b(2y r+l -y r -y r+2 y+ (14) 



With the same initial conditions, the coordinate y unity, 

 we have for the subsequent motion — writing T 2 = a 2 /c 2 =fi/4,u, 



4r 2 y,= — #.-2 + 4# r _i— fyr + 4:y r+l — y r+2 , r>l."| 



4rVi = 47/0-7^ + 4^-7/,, k (15) 



4r 2 y = -6y + 8 yi -2y 2 .- J 



Forming a series for y r in terms of t we find this can be 

 expressed for all values of r in the form 



yr=J^:)'cos(rg--^J (16) 



Writing this in an integral form, with x for ra, we have 



y=J -4S) cos S-J) 



1 C n c 2 1 



= - } Q cos j - {.r(j> - ct sin 2 <£) J dcf> 



= -{ a cos{«(x-Yt)}dfe, . . . (17) 



.., ^ r sm 2 (i/ca) 



with V = c —r^ — -. 



tea 



Further, we have the group velocity U = csin (tea). In 

 (17) the disturbance is expressed in simple trains of wave- 

 lengths ranging from 2a to go ; in this range both V and U 

 are positive and their variation with the wave-length is 

 shown in fig. 4. The hyperbolic curves in the same diagram 

 show the limiting forms of U and Y if we wish to pass to the 

 case of a continuous beam by making a small ; the ordinary 

 theory gives V = (const.) «, hence c must become infinitely 

 large so that in the limit the product ac is finite. 



