IN CYLINDRICAL TUBES. 241 



other terms in the general vahie of v, shew how the general waves 

 in which we have 



If 



v, = /,(af-x), and v, = Jl{at-(2l-x)}, 



are formed by the superposition of successive waves, as the time 

 increases. If the velocity becomes by this superposition so large, that 

 it can no longer be considered extremely small as compared with 

 the velocity of propagation (a), our analysis will be no longer ap- 

 plicable ; but if V never exceed a certain value, the motion will 

 become regular, and follow the law which our investigations indicate. 

 Let us consider in what cases we may expect these effects to be 

 produced. 



9. We have at present imposed no restrictions on the forms of 

 the functions denoted by cp, f and \//, except that their greatest 

 values shall be small compared with a. In order however that the 

 undulations may be sonorous, <p, and consequently y and \f/, must 

 denote periodical functions, so that the values of (p {z), f (2), and ^ (ss), 

 will recur as often as % is increased by a certain quantity. We will 

 also iinpose an additional limitation upon them, to which, in all 

 practical cases they will probably be subject very nearly, as will 

 certainly be the case in the experiments to which I shall hereafter 

 more immediately refer. Supposing then their values to recur, when 

 s becomes %-Vm\, {m any whole number), we will also suppose them 



to recur with different signs when z becomes x±m' -; {m! being 



any odd number). 



10. First suppose the greatest value of \//, small as compared with 

 that of y or 0, as must be the case in a closed tube. In the above 

 expression for v, it will be observed that the quantity represented by 

 % increases as we proceed from one term to the next, in a vertical 

 line by 2/. 



Suppose then 



%l = m' . -, or l = m' - 



2 4 



