Propagation of Disturbance in a Dispersive Medium. 165 



we have 



1 f 2T cos nt -. , 1 ( " sin nt 7 /11N 

 y = — I - , - an H I - ===- an . (11) 



= " ■- f " cos ( K Yt)d K + a - r sin (jfSTt)dii ; . (12) 



where V=c sin (^Ka)./\Ka ; V r =c cosh (^a)./i/c ! a. 



In (12) each integral is equal numerically to. &Jjo[%t/t) ; 

 hence ?j is zero for t negative, J (2t/r) for £ positive, and J 

 for £ zero. 



Now we may write down a similar expression t/ r which is 

 zero for t negative and equal to J^(2^/tJ for t positive. 



da 



2r+l 



1 C% t 1 f °° f / 1\ £ ) 



y ' = J o "«»(2 T «>»*-2'tf)<'* + J | sin {(" + ^x- , ' 7r }^ 



= ~ f B oo8{*(oy-VOW*+(-l)*#-| «-«'* sin(VV7y«' (13) 



with the same notation as in (12). 



Comparing (12) and (13) we see how the central particle 

 may be regarded as a source of displacement which is sud- 

 denly created at a certain instant. Physically, the idea is 

 simpler if ij is a velocity suddenly created by an impulse, but 

 the analytical expressions are less direct. The first integrals 

 in (12) and (13) correspond to periods (>7tt) for which 

 simple wave-motion is possible, the greatest wave-velocity 

 being finite ; the second integrals arise from component periods 

 less than 7rr. Of the value of //,. at any time one half is 

 ■associated with each type of integral. 



In the ordinary analysis, (12) and (13) would be replaced 

 by twice the first integral in each case, with and cc as the 

 limits, thus assuming the possibility of representing the effect 

 of a suddenly created source of disturbance by simple wave- 

 trains of all possible wave-lengths ; the above illustration 

 suggests that the existence of two types of integral may be 

 necessary in a dispersive medium and that the apparent 

 instantaneous propagation is connected with this fact. In 

 this illustration the infinite velocity enters dynamically by the 

 neglect of the inertia of the string. If we supposed this to 

 be of uniform density we should have something analogous 

 to two interacting media, the particles being connected by 



* Sonine, Math. Ann. xvi. p. lo. 



