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Non-Linear Differential Equation having a Periodic Coefficient

Published online by Cambridge University Press:  03 November 2016

N. W. McLachlan
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Consider a thin thread (preferably nylon, which is strong, has low loss, and is resilient) attached to an electromagnetically-driven reed or tuning fork at one end, the other passing over a pulley, and tensioned by a weight W Suppose the free length of the thread is about 1 metre, and that the reed vibrates transversely to the thread. By adjusting W to get the correct tension, the first overtone of the thread may be obtained, there being a node at the middle. The frequency of the thread is the same as that of the reed. If the latter is rotated through 90° (the length of the thread baing unaltered), vibration occurs along the thread, thereby causing its tension to vary periodically The main frequency of tho thread is now half that of the reed, so a subharmonic of order two occurs, there being no node at the middle. This was demonstrated by F. Melde ninety years ago, using a tuning fork excited by a violin bow.

Type
Research Article
Information
The Mathematical Gazette , Volume 35 , Issue 311 , February 1951 , pp. 32 - 36
Copyright
Copyright © The Mathematical Association 1951

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References

Page 32 of note * A true node cannot occur in a dissipative system owing to energy loss. It is a position of least amplitude.

Page 32 of note † Poggendorff’s Annalen, 109, 193, 1860.

Page 32 of note ‡ The reed frequency is 100 c.p.s., since the magnetic attraction is independent of the direction of the current through the magnet winding.

Page 32 of note § McLachlan, , Theory and Application of Mathieu Functions, p. 289,Google Scholar which will be designated by M.F