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III. §40.] RESONANT AIR-COLUMNS. 79
pipe is produced by a note one Octave below that to which an open pipe of the same length resounds the most powerfully.
40. In order to ascertain the precise relation between the pitch of a note and the length of the corresponding air-column, we will examine the way in which resonance is produced in a column of air contained in a pipe closed at one end.
Let A, Fig. 23, be the open, and B the closed, ends of the pipe, and let us for a moment replace the contained air by an elastic spiral spring fastened at B, and of length equal to AB.
Suppose the end of the spring suddenly pushed a little way from A towards B. The coils of the spring nearest A will be squeezed together, and this condensed state of the spring will be transmitted to B and, after reflexion there, will travel back to A. In virtue of the elasticity of the spring its free end will now be caused to protrude beyond the mouth of the pipe as far as it was at first pushed into it. The coils near A being thus drawn somewhat more apart than was the case when they were in their undisturbed condition, a rarefied state of the spring will be produced at A, transmitted to Bt