Plane Waves of Sound. 21 



Substituting for \^ and f,,+i and equating coefficients of 

 cosnt and of smnt we get 



Qsin;;zSs+i + C/cosw^s+i = Q-f isin/z/s-^+i + Cs4-i'cos;;/5r^,-|-i . . . (iii) 

 and a similar equation for the V>'s. 



We have also another surface condition, viz.: that of 

 pressure, which gives us for all values of the time 



V dxjx, = 4 V.T,-^ i/^^s+i = 



from which we obtain 



Fg^CgCOsmZg+i - CJs\n?nZgj^i} = Vg+ilCg+iCOsmZt^i - 

 C;+isin;;/^,+i} . . . (iv.) 



and a similar equation for the B's. 



Hence from these two equations (iii.) and (iv.) we can 

 determine Cs+i] C/+i in terms of G; C/ and so in terms of 

 C • C ' 



Now the constants Q+i, C/+i must be expressible as 

 some function of the compartment to which they belong 

 and hence must be some function of the density of that 

 compartment. Hence since the difference of density 

 between any two compartments is extremely small we may 

 write 



^D< 



+1 



Where aD,+i is the small change of density, its square and 

 higher powers being negligible. 



If we substitute these values in equations (iii.) and (iv.) 

 and neglect small quantities we obtain the equations 



dC dC „ 



^fdC dO . \ dVf^ -,, . \ ^ 



r[-^co?,mz - -^j^smmz j + -^^( Ccosws - Usiwnz j = U 



or since -j:^ occurs in each term, these may be written 



