The wave method of Keller and Rubinow [Ann. Physics, 9 (1960), p. 24-75] is extended to the biharmonic eigenvalue problem with rectangular or circular geometry and clamped boundary conditions. First, it is noted from the clues of computer graphics that mode shapes of a clamped circular plate and those of a circular membrane look very similar to each other. This suggests that plate and membrane should have very similar vibration behavior and leads to the assumption that the covering space of a rectangular plate is still a torus. By adding several waves on the boundary, approximate eigenfrequency equations are derived. Their solutions are shown to agree remarkably with numerical solutions obtained by the Legendre-tau spectral method here and by the finite-element method elsewhere at all frequency ranges. The same idea is also applied to the circular plate and yields excellent agreement between the exact values of eigenfrequencies and the asymptotic solutions.
Siam Journal on Applied Mathematics
Chen, G.; Coleman, Matt; and Zhou, J.X., "Analysis of Vibration Eigenfrequencies of a Thin Plate by the Keller-Rubinow Wave Method .1. Clamped Boundary-Conditions with Rectangular or Circular Geometry" (1991). Mathematics Faculty Publications. 7.
G. Chen, M. P. Coleman, and J.X. Zhou. Analysis of Vibration Eigenfrequencies of a Thin Plate by the Keller-Rubinow Wave Method .1. Clamped Boundary-Conditions with Rectangular or Circular Geometry, Siam Journal on Applied Mathematics. 51(4), 967-983.