Vibrational modes of circular free plates under tension

Abstract

The vibrational frequencies of a plate under tension are given by the eigenvalues ω of the equation Δ² u − τΔu = ωu. This article determines the eigenfunctions and eigenvalues of this bi-Laplace problem on the ball under natural (free) boundary conditions. In particular, the fundamental modes – the eigenfunctions of the lowest nonzero eigenvalue – are identified and found to have simple angular dependence.

Keywords:

• free plate
• bi-Laplace
• Bessel functions

AMS Subject Classifications::

• Primary 35J40
• Secondary 35P15
• 33C10

Acknowledgements

I am grateful to the University of Illinois Department of Mathematics and the Research Board for support during my graduate studies, and the National Science Foundation for graduate student support under grants DMS-0140481 (Laugesen) and DMS-0803120 (Hundertmark) and DMS 99-83160 (VIGRE), and the University of Illinois Department of Mathematics for travel support to attend the 2007 Sectional meeting of the AMS in New York. I would also like to thank the Mathematisches Forschungsinstitut Oberwolfach for travel support to attend the workshop on Low Eigenvalues of Laplace and Schrödinger Operators in 2009. Finally, I would like to thank my advisor Richard Laugesen for his support and guidance throughout my time as his student and for his assistance with refining this article.