and J)iittr}lmfion of Sound. 290 



tliftVrences imlicated by tlio flame. It will he remarked that 

 T(i carry out this experiment ii[)on a ])ractical scale will mean 

 a very large structure, the linear magnification being that 

 {6 times) required to pass from an 8-inch wave-length to one 

 (say) of 48 inches. 



Work done hy Detached Sources. 



In the case of a single source the pressures to be overcome 

 are proportional to the magnitude of the source, and thus the 

 work done is proportional to the square of the magnitude of 

 the source, as is indeed otherwise evident. If, as is usually 

 the case in practice, the object be to emit sound in one (hori- 

 zontal) plane only, an economy may be effected by distributing 

 the source. If sources all in one phase be distributed along 

 a vertical line, the effect is the same at distant points in the 

 horizontal plane as if they were all concentrated in one point, 

 but the icork required to be done may be much less, the 

 saving corresponding to the fact that in directions other than 

 horizontal the sound is now diminished. We will begin by 

 considering two unit sources in the same phase. 



If ^, y\r be the potentials of these sources at a point whose 

 distances are r, r', we have, as in (1), 



, co^ k (at— r) , corik (at— 7^') 

 *= - -iTTr ■ ' ^= W" 



Thus, when r = 0, 



4:7rr- ~ =cos kat — kr sin kat. 



(10) 



dr 



d(b , dylr ka sin kat ka . , . , ^. 

 dt dt i-rrr ^ttD ^ " 



if D denote the distance between the sources. The work 

 done by the source at /' = is accordingly proportional to 



^^^' • • (^^) 



-and an equal w^ork is done by the source at r' = 0. If D be 

 infinitely great, the sources act independently, and thus the 

 scale of measurement in (11) is such that unity represents the 

 work done by each source when isolated. If D = 0, the work 

 done by each source is doubled, and the two sources become 

 equivalent to one of doubled magnitude. 



If D be equal to ^X, or to any multiple thereof, sin Z;D = 0, 

 -and we see from (11) that the work done by each source is 

 unaffected by the presence of the other. This conclusion 

 may be. generalized. If any number (n) of equal sources 



