Mr Sharpe, On the Reflection of Sound at a Paraboloid. 191 



For sound motion F will be of the form 



P sin {Ipat) + Q cos {'^pat), 

 where P and Q each satisfy the equation 



u + vX dv^ du^ dv duj ^ 



As in the present Paper we shall only consider stationary 

 vibration, either P or Q will always be zero. A particular solution 

 of the last equation is P = UV, where U is & function of u only, 

 and F of -y only, and 



»!?+'£+ (^''-^>'^=° (««>' 



^2 JT (ITT 



^^ +^ +(^'" +^) ^=^ (97X 



du^ du 



where A is an arbitrary constant, pa/ir is the frequency and a is 



the velocity of sound. It will be found that the condition of 



dU 

 reflection at the paraboloid LP is that -^ should = when u = l. 



In all that follows it will be found to conduce much to brevity 

 to use new independent variables, defined thus 



pu = u', pv = v', pi = r and A/p = A'; 

 (96) and (97) then become 



'''^+S7 + (»'-^'>''=« (««)• 



u'yi, + yT^-{u+A')U=Q (99), 



du^ du 



from which p has disappeared and the condition of reflection 

 becomes 



^,=0 whenw' = r (100). 



du ' ' 



We may now drop the dashes, if in all final results we re- 

 member to put 



pu for u, pv for v, pi for I, and Ajpi for A. 

 By this we come back to the original notation. 



40. We will first take a comparatively simple case. We will 

 suppose JL = 0. This case was partly treated in Arts. 10 — 17, but 

 more will be added. In this case we get F= Ju('y) and U = J^{a), 

 and the condition of reflection is that d U/du = when u = l. 



