REFLECTION OF DIVERGING WAVES 377 



dr' 



aFnp I <p dt 

 — a \ I <p dt \ 



which reduces to 



jp = - a(p j (p dt, 



if we neglect second powers of the variables compared with unity. 

 To the same accuracy, from (14a) 



1 f dqo 

 From (11) 



^0 ^ ^oQo 

 dr' r' 



k cos (co/ — kr') + —. sin (oot — kr') 

 r 



Here k is lir over the wavelength so, if as we have assumed ri , and therefore 

 also r', is large compared with the wavelength, we may neglect the second 

 term. Then 



/ 



if = -J^ sin (co/ - kr'), 



<p dt = J' , cos {<ut — kr'), 

 2cur 



dq' _ a froQoY ..-.^ 



, , — „ . , , sin (co/ — kr') cos (w/ — kr'), 

 dr 8co \ cr 



= __i froi^«Y [cos (co/ - /^r') + cos 3(co/ - kr')]. 

 8co \ cr / 



This, when multiplied by Ar', gives the value at r' of the wave, generated 



in the interval Ar', which propagates in the negative direction of r. This is 



made up of components of frequency co and 3co. We are primarily interested, 



from the stand-point of reflection, in that of frequency co, so we shall confine 



our attention to this component, with the understanding that the other 



can be treated in exactly the same fashion. As the fundamental component 



r' 

 propagates inward to r^ it increases in amplitude m the ratio — and suffers 



a phase lag of k{r' — ri). If we call the resultant of all the reflected waves at 

 ri, qi , then the contribution to qi of the wave generated at r' is 



