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124 VIBRATION-FORMS FOR AN OPEN PIPE. [V. § 60.
gives us the fundamental-tone [§ 43]. The reasoning here adopted evidently applies equally well to cases in which the air-column is subdivided to any assigned extent. It follows, therefore, that the notes obtainable from a stopped pipe are all odd partial-tones belonging to one and the same clang.
60. The case of the open pipe shall next be investigated. Here, as in the previous case, the centre of a ventral segment must coincide with the end of the pipe at which the direct pulses enter. The considerations alleged on pp. 120 and 121 indicate that the same thing must also hold good at the opposite orifice1.
Referring once more to Fig. 35 (l), we obtain all the possible modes of vibration which satisfy both the above conditions by placing one end of the pipe midway between A and 1, and the other successively half way between 1 and 2, 2 and 3, 3 and 4, and so on. The first four of the cases thus obtained, for a tube of constant length, are shown in the next figure, which is drawn on precisely the same plan as Fig. 39.
In each case, the two half-segments at the ends
1 I here assume, for the sake of simplicity, what is not rigorously true, viz. that reflexion at an open end of a pipe is as complete as at a stopped end. It is however approximately true for the pipes employed for musical purposes, whose transverse dimension is small compared to their length.