Shape of a drum, a constructive approach

  • Authors:

  • P. N. Shivakumar;Yan Wu;Yang Zhang

  • Affiliations:

  • The University of Manitoba, Department of Mathematics, Winnipeg, Manitoba, Canada;The University of Manitoba, Department of Mathematics, Winnipeg, Manitoba, Canada;The University of Manitoba, Department of Mathematics, Winnipeg, Manitoba, Canada

  • Venue:
  • WSEAS Transactions on Mathematics
  • Year:
  • 2011

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Abstract

For the classical question, "Can you hear the shape of the drum?", the answer is known to be "yes" for certain convex planar regions with analytic boundaries. The answer is also known to be "no" for some polygons with reentrant corners. A large number of mathematicians over four decades have contributed to the topic from various approaches, theoretical and numerical. In this article, we develop a constructive analytic approach to indicate how a preknowledge of the eigenvalues lead to the determination of the parameters of the boundary. This approach is applied to a general boundary and in particular to a circle, an ellipse, and a square. In the case of a square, we obtain an insight into why the analytical procedure does not, as expected, yield an answer. For the Mathieu equation with a parameter, we demonstrate the determination of the parameter when the eigenvalues are known.