300 British Association for the Advancement of Science. 



On the simultaneous Vibrations of a Cylindrical Tube and the Column 

 of Air contained in it. By the Rev. James Challis. 



Mr. Challis, in his report on the Analytical Theory of Hydrody- 

 namics, and elsewhere, has expressed the opinion that, to complete 

 the theory of musical vibrations in a cylindrical tube, it is necessary 

 to take into account the vibrations of the tube itself. In this com- 

 munication he states some results which he has arrived at theoreti- 

 cally, respecting the kind of influence the tube will exert on the 

 aerial column. 



It is assumed that the tube is capable of vibrating so that its par- 

 ticles move in planes perpendicular to the axis, with the same mo- 

 tion in all directions from the axis, in the same transverse section. 

 Then, if the vibrations of the tube be of very small extent, and its 

 diameter small, compared with its length, the following are the prin- 

 cipal mathematical results respecting the motion of the air, so far as 

 it is consequent upon the vibrations of the tube. 



1. The motion of the particles situated on the axis will take place 

 in the direction of the axis, and will be nearly the same as if an im- 

 pulse were originally given in this direction, and the propagation 

 were rectilinear. 



2. At all points of the same transverse section, the motion, esti- 

 mated in a direction parallel to the axis, will be nearly the same. 



3. If the tube be made to vibrate isochronously, and so as to con- 

 tain, at equal intervals along its length, nodal sections and sections 

 of maximum vibration, it will produce in the fluid vibrations of the 

 same duration, with points of quiescence and of maximum vibration 

 at intervals corresponding to vibrations of that duration in air. 



4i. But unless the nodal sections of the tube be fixed, the duration 

 of these simultaneous vibrations will not be permanent till the in- 

 tervals between the nodal sections become the same in the tube as 

 in the column of air ; and then a nodal section of the tube is nearly 

 coincident with a section of maximum vibration of the fluid. 



From these results it follows that there are certain transverse vi- 

 brations of the tube which will impress on the fluid column the same 

 kind of motion as it is known can be given to it by vibrations ex- 

 cited near one extremity of the tube when the other is open. Ma- 

 thematicians have succeeded in satisfactorily representing the cir- 

 cumstances of the motion in the latter case of disturbance, by as- 

 suming, from experiment, that the open end is a position of maxi- 

 mum vibration, or nearly so ; but hitherto no distinct cause for this 

 fact has been assigned. Mr. Challis thinks it may be shown mathe- 

 matically that the aerial vibrations, excited at the extremity of the 

 tube, and propagated along its interior, will put it into the state of 

 vibration, which, as appears from the foregoing results, will produce 

 an effect the same in kind as that observed. But to what degree the 

 phaenomenon may be attributed to this cause, can be learnt only from 

 experiment, by ascertaining whether the vibrations of the tube have 



