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BELL SYSTEM TECHNICAL JOURNAL 



Agi 





~j- cos (co/ + A-ri - 2kr')Ar'. 



This is to be integrated from Vi to the farthest point from which a reflected 

 wave has reached ri at the instant / under consideration. This point is at 

 K^i + cl). So 



3 ^i(ri+ct) 



'■ = -87; 



Here the integrand is a function of r' and / and the upper hmit of integration 

 is also a function of /. We therefore make use of the relation* 



'Wo A f i- cos (co/ + )^ri - 2k/) dr'. 



c / Jr, r- 



d_ 

 da 



f'fix, a) dx = £ (J^fix, a)) dx + f(b, a) ^ - f(a, a) 



da 

 da 



Putting / for a, r' for .v we have 



dqi _ a /roQo 

 IF ~ Sr, \c 



/ 



i(ri+et) 



2c 



^, sin (o)/ + kri — 2kr') 



r- CO (ri + ct)-_ 



which, upon integration becomes, 



dt 8ri \ c / \ri 



■ sin (co/ + kr,) - [C/(w/ + kt\) - Ci{2krx)\ 



cos (co/ + kr^) 



Ic 



) 



co(/-i + cif 

 Since q\ is zero when / is - , its value at / will be found by integrating from 



to /, so 



--^;^^-^-^-'''^ "■ 



2kr^ 



• CO / .V/(co/ + kri) sin (co/ + kr^) dt + Si{2krx) 



• [cos (co/ + A';-]) — cos 2/.';'i| — co / C'/(co/ + kr^) cos (co/ + ^rj rf/ 



''n/<- 



+ Ci{2kr^) lsin(co/ + ^r,) - sin 2kr^\ 



which reduces to 



'* Byerly, Integral Calculus, second edition j). W. 



