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



axis of r, where r 2 stands for y 2 + z 2 . If P be any point within 

 the reflector, whose coordinates are x, y, z, draw through it two 

 confocal and coaxial paraboloids PU, PV. I shall put 20U=u, 

 20 V = v, whence it is evident that u and v define the position of 

 P as well as x, y, z. 



I shall not treat the sound from the source and the reflected 

 sound separately, but shall consider them as both included under 

 one velocity-potential F, and if need be, afterwards discriminate 

 between the primary and reflected sound, when it is possible to do 

 so. F satisfies the equation 



d 2 F d 2 F <PF_l <PF 



dx 2 dy 2 dz 2 a 2 dt 2 



We may regard F here as a function of u and v, where 

 u = p — x, v = p + x, and p 2 — x 2 + y 2 + z 2 . 



For all points within the reflector v will vary from to oo , and 

 u will vary from to I. u = Q is the equation to the line Ox, v = 

 is the equation to the line OL. 



For points on the axis to the right of 0, v = 2a? and u = 0. 



For points on the axis to the left of 



u = — 2x and v = 0. 



(3) when transformed becomes 



4 / d 2 F d?F_ dF dF\ _ 1 <PF 

 u + v \ dv 2 du 2 dv du ) a 2 dt 2 



For sound motion F will be of the form 



P sin mat + Q cos mat (5). 



P and Q will each satisfy the equation 



4 / d\P u dPP dP dP\, ap = (6) 



u + v\ dv 2 du 2 dv du) 



In nearly all that follows it will be found very convenient 

 to put 



p = h m ( 7 )- 



A particular solution of (6) is P = UV where U is a function of u 

 only and V of v only, and 



.£+£+(l*-V-0 (8),. 



ud £ +d I + w +A > u = < 9 >' 



where A is an arbitrary constant. 



