Published December 9, 2025
| Version v1
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The Spectral Content of a Sagging String
Authors/Creators
- 1. Heritage Xperiential Learning School
Description
This paper develops a precise correspondence between the static deformation of an
interval and the spectral structure of the Dirichlet Laplacian. The torsion function generated by
a uniform load decomposes into local quadratic profiles that coincide with the nodal intervals of
each eigenfunction, while the reciprocal of each local profile identifies the unique extremum of
the corresponding vibrational mode. These local descriptions are unified by a global spectral
expansion that expresses the torsion function as a uniformly convergent superposition of all odd
eigenfunctions. The results provide a complete one dimensional framework in which equilibrium
geometry, nodal structure, and vibrational behaviour are fully aligned.
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The Spectral Content of a Sagging String.pdf
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