REFLECTION OF DIVERGING WAVES 379 



,; = -JL. {'^^ (cos („/ - kr,) - ^, + 2kr, 



• [-[Siicot + kr,) - Si{2kn)] cos (co/ + /trO 

 - [CiW + /feri) - C/(2/^ri)J sin (co/ + )^ri) 



+ Si{2c^l + 2/&r,) - ^'i(4/feri)] j . 



The first term represents the value at Vi of an outwardly moving wave in 

 phase quadrature with the main wave. The second is a transient, the value 

 of which is equal and opposite to that of the first term at the instant that 

 the main wave passes ri . The first two terms in the inner bracket are waves 

 which propagate inward and so are to be regarded as reflections of the 

 main wave. The last two terms represent a velocity which is zero when the 

 main wave passes fi , and subsequently oscillates about and approaches 



- — Si(4^ri). Physically it appears to result from the particular form chosen 



for the main wave, which starts abruptly as a sine wave. The time integral 

 of the impressed force, and so the applied momentum, has a component in 

 one direction. Presumably if the main wave built up gradually these terms 

 would be absent. 



Returning to the reflected waves, their amplitudes are zero when the 

 main wave passes Vi , after which they become finite. Si{x) and Ci(x) os- 



cfllate about and approach - and zero respectively as .v approaches infinity. 



Hence, as / increases indefinitely, the amplitudes of the reflected waves 



approach - — Si(2kri) and Ci{2kr^. For the assumed large values of 2kri 



these quantities are small compared with unity. When multiplied by 2kri 



1 



their variation is very slow. Hence the amplitudes vary roughly as ^ , 



and approach zero as the main wave at ri approaches an ideal plane one. 



However, the significant fact is not that the reflected waves are small 

 but that they are of finite magnitude. Because of this the main wave will 

 not behave exactly as we assumed above, but will decrease slightly more 

 rapidly with increasing radius. This should increase the reflection slightly, 

 for the existence of the reflected wave is dependent on the decrease in am- 

 plitude with distance when the radius of curvature is finite. 



To describe exactly what happens when the generator begins sending out 

 waves from a central point would be hopelessly complicated, but we may 

 form a general picture. In the early stages where the curvature is consider- 

 able, the reflected waves would be quite large and the main wave would be 



