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



Suppose a source of sound at S spread equally over the section 

 Ss of the tube. Let OS = 01 = c and let OP = x, where P may 

 be on the right or left of Ss. 



To find the motion. 



Fig. 8. 



Let 1} 4>. 2 , 3 , 4 be velocity-potentials, and let 

 X = cos m (at — x + c), 



2 = cos to (at + x — c), 



3 = cos to (at — x — c), 



4 = cos m (at + x + c). 



In the part OS 2 and 3 exist and the vibration is stationary. 

 In the part Sx 0j and 3 exist and the vibration is progressive. 

 The velocity vanishes at and is discontinuous at S. 



Next suppose (Fig. 9) SOx a conical tube with vertex at and 



Fig. 9. 



open at x. Suppose a source of sound at S spread equally over 

 the spherical section Ss of the tube. Let OS = c. 

 Let 02, 2 , 03 be velocity-potentials, and let 



0! = - cos m(r — at — c), 

 2 = - cos m (r + at — c), 



03 = cos to (r — at + c). 



Then in OSs 0. and 3 exist and the vibration is stationary ; 

 in Ssx 0! and 3 exist and the vibration is progressive. 

 The velocity vanishes at and is discontinuous at S. 



From both cases we see that, whether the source be near 

 or far off, there will always be near a region of stationary 

 vibration. 



