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Note on the Use of Fourier's Series in the Problem of the Transverse Vibrations of Strings

Published online by Cambridge University Press:  20 January 2009

H. S. Carslaw
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Fourier's Series were first applied to the solution of the Differential Equations which occur in the Theory of the Conduction of Heat and that of the Vibrations of Stretched Strings, namely,

In the first, when the flow of heat is linear and the initial temperature is given in the range 0 < x < π by f(x), the solution can be put in the form

In this case there is no difficulty with regard to the differentiation of the infinite series, term by term, as the factor causes the series which we thus obtain to remain uniformly convergent with respect to x and t in the intervals concerned.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 20 , February 1901 , pp. 23 - 28
Copyright
Copyright © Edinburgh Mathematical Society 1901

References

* Lindemann,

Die Schwingungsformen gezupfter und gestrichener Saiten.

Berichte über die Verhandlungen der naturforschender Gesellschaft zur Freiburg. B. VII. 1879.

This paper is translated in the Philosophical Magazine for 1880, under the title “On the Forms of the Vibrations of Twitched and Stroked Strings.”

† Harnack,

Über die mit Ecken behafteten Schwingungen gespannter Saiten.

Math. Annalen, XXIX.

‡ Morera,

Sul Problema delle Corde Vibrante.

Atti della Academia delle Scienze di Torino. T. XXIII.

* See the discussion in Burkhardt's Entwickelungen nach oscillirenden Functionen, pp. 10–24.

† Untersuchungen über die mit dem Fortbestehen linearer partieller Differential Gleichungen verträglichen Unstetigkeiten.

Annali di Matematica. T. VIII.