to a Travelling Disturbance. 387 



given by (2), if we replace x by ct — x, and multiply by St. 

 Integrating from £ = to £ = co , we obtain 



^Vo Uo Jo 



JO 



. . . (3) 

 In the absence of the factor e~& the result would in some 

 cases be indeterminate, the physical reason being that we 

 can superpose a system of free waves having the prescribed 

 velocity c, if such are possible. 



The integration with respect to t gives 



i r°° <f>(ky k *dk ir 4>(k)e ik *dk 



V ~27rJ fJ,-i((T-kc) + 2tt) IJU-i{G+kc) * l ' 



The coefficient yu, being supposed small, the most important 

 part of the result will be due to values of k in the first 

 integral which make 



a=kc, (5) 



approximately, if such exist. Writing, then, k=/e + k\ 

 where k is a root of (5) and k' is assumed to be small, we 

 have 



o— kc=(~-c\k' = (TJ-c)k ! , . . . (6) 



nearly, where U is the group-velocity corresponding to the 

 wave-velocity c. The important part of (4) is therefore 



v^4>(«y"f ^JJl c)k , -, ••• (7) 



since the extension of the range of integration to k'=±co 

 makes no serious difference. 



The integral comes under a known form. If a be positive, 

 we have 



f°° e imx dm = f2we-* t , [>>0] \ 

 J^a + im^ I ; [«<0] I * ' " " W 

 whilst 



f" f!!!^__ r o , [*>o].-> 



J_,o a-z'm ~ I 2w«« . [><0] j * ' " W 

 Hence if U<c, we have 



?7 



6— U 



according as #J0 ; whilst if JJ>c, 



e-^ c ~ u \ or 0, ... (10) 



,7 = 0, or ^j^ e **W-°\ .... (11) 

 in the respective cases. } ,-. 9 



