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The two phase drum with the deepest bass note

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Abstract

Given an open bounded connected set δ ⊂ R2 and a prescribed amount of two homogeneous materials of different density we characterize that distribution which minimizes the least eigenvalue of the associated clamped drum. We establish geometric conditions on δ under which the interface separating the two materials is an analytic Jordan curve. We bound the length of this interface and construct and test an algorithm for its calculation in the case of square δ. Our numerical results depict the dependence of this minimum least eigenvalue on the volume fractions of the two phases and suggest possible candidates for the two phase drum with the least second eigenvalue.

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Authors and Affiliations

  1. Mathematical Sciences Department, Rice University, 77251, Houston, TX

    Steven J. Cox

Authors
  1. Steven J. Cox
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Additional information

Research supported by Defense Advanced Research Projects Agency Contract F49620-87-C0065 and Air Force Office of Scientific Research Grant 89-0363.

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Cox, S.J. The two phase drum with the deepest bass note. Japan J. Indust. Appl. Math. 8, 345–355 (1991). https://doi.org/10.1007/BF03167141

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  • DOI: https://doi.org/10.1007/BF03167141

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