IN CYLINDRICAL TUBES. 237 



The substance forming the stop being known, so that we might 

 regard the vibrations produced in it under given circumstances de- 

 terminable, the relation between the functions xj^ and J" would be 

 known, and the function y would be the only unknown one in the 

 above functional equation, from which, any particular form being 

 assigned to (p, that of y must be determined. The arbitrary quantity 

 which will be involved in the solution of this equation, must be 

 determined by the original value of the function jf. 



6. We have here supposed the tube to be stopped, but the 

 equation (B) will still be true for the open tube, \|/ {«/-(/ + c)}, de- 

 noting always the velocity of the 'extreme section at the time f. 



Equation (2) gives us 



F{at + l)=-f{at-l) + y\,{at-{l+c)}, 

 and writing at + x, for at + l, 



F{at + x)= -f{at-{2l-x)} + >// {at -{2l + c -x)}'. 



Hence, 



v = f{at-x)-f{at-{2l-x)} +^ {at-{2l + c-x)\-\ 



as = f{at-x)+f{at-{2l-x)}-yl^{at-{2l + c-x)}] 



The form of J" being determined by equation J?, these last equations 

 will give the complete solution of the problem. 



7. Before we proceed to consider particular cases, we will exhibit 

 these equations (C) under another form, which will be useful in 

 deducing some general inferences as to the nature of the motion in 

 the tube. 



Let T denote a period of time, from the commencement of the 

 motion at A, less than that which is necessary for the pulse to 

 travel twice the length of the tube ; consequently at will be less 

 than 21. 



Equation (B) gives us 



/(ar + 9.1)=/ {ar) - v// («T - c) + ^ {aT + 2l), 

 Vol. V. Part II. Hh 



•(C). 



