164 



Dr. T. H. Havelock on tl\e Instantaneous 



We have also the group velocity \] = d(fcY)/d/c = c cos (^ica). 

 In the range for k from to ir/a, both U and V are positive 

 and vary between and c ; fig. 3 shows the curves for U and 

 V as functions of the wave-length X, and their limiting- 

 positions for a zero. 



When a is diminished indefinitely we approach as a limit 

 the case of a uniform string with V and U equal to c. For 

 an initial displacement / at the origin, we have y given by (9) 

 multiplied by / ; as a approaches zero / must become infinite 

 in order that we may obtain the divergent integral (3) which 

 is used to represent the effect of a concentrated initial 

 disturbance. 



Fiv. 



In the integral (9) the solution (8) is expressed in terms 

 of simple harmonic components of wave-lengths ranging from 

 2a to co . The effect begins without delay at each point, and 

 the solutions give equal values for equal positive and negative 

 values of t. Suppose we regard the central particle (?/ ) as a 

 source of displacement. Then if i/ is zero for all negative 

 values of t, and equal to f(t) for positive values, we can write 



/<Q 



= ^J>jo /( 



o))cos n (w— /) dco, 



(10) 



with 2/ =i/(0+) for* = 0. 



If we put f(t) equal to J (2t/r) we must cut out of the 

 integration with respect to n the region in the vicinity of 

 n — ltjz ; in this way, or writing down the result directly, 



