Mr. R. H. M. Bosanquct on the Theory of Sound. 27 



The electrical nietbod entirely puts aside all consideration 

 of reflexion ; and between it^ as exhibited by Helmholtz and 

 his followers, and the Bernoulli theory there exists a want of 

 connexion. It will be my endeavour to extend the Bernoulli 

 theory in the direction above indicated, so as to arrive (in the 

 present note by the application of rigorous analysis to the case 

 of svmmetrical diveroence, and in the followino; note with the 

 assistance of the experimental study of the .unsymmetrical 

 motions concerned) at a more clear idea of the whole thing 

 than was afforded by the electrical analogies. 



Preliminary Illustration. — Reflexion hy Change of Section. 



If a stream of sound traverse a cylinder whose section 

 changes from S^, to Si, a reflected stream will arise at the 

 chano-e of section — the chano-e of section beino- small, and the 

 wave-length great compared with the dimensions of the sec- 

 tion ; so that the form of the plane-waves is substantially un- 

 disturbed, and the expansion may be considered as taking 

 place completely in the plane of the section. 

 Let A be the amplitude of the incident stream, 

 b of the transmitted stream, 

 a of the reflected stream. 

 Let MSqA^ be the energy per second of the incident stream, 

 MSi(5>-^ of the transmitted stream, 

 MS^a" of the reflected stream. 

 By Prop. II. of the last note, the incident and reflected 

 streams in S^ do not interfere with each other. 



Let the axis of the cylinder be the axis of .i', and its inter- 

 section with the plane of change of section the origin. Then 

 the wave-systems may be represented by 



y^ — A sin ^■( vt — .r) + a sin k(yt + x)^ 

 yi=^b sin k(vt — x\ 



We suppose that the whole reflexion takes place at the change 

 of section, so that no change of phase comes into the reflected 

 system except it, which is determined by the sign of a. This 

 is legitimate as long as the change of section is small. Then 



-p = —kK cos k(vt — x) + ka cos k(vt + x), 

 ax ^ - /; 



^ = -bkcosk(vi-x). 

 ax ^ 



And these must be equal at the origin (,i' = 0); for the pressure 

 must be the same at the common surface ; 



.-. A-a = b. 



