## Thin oscillating bending bar with free ends

Solution of
u´´´´ = freq² u with
boundary condition u´´´(1)=u´´´(-1)=
u´´(1)=u´´(-1)=0. Here
freq is proportional to the frequency of oscillation.

## Thin oscillating disc with free boundary

Three views of the same disc at a given moment. Black is below the neutral
plane, white is above it.

Solution of Δ Δ u = freq² u
with as boundary conditions that Δ u and its gradient
vanish on the edge of the disc. Here Δ is the Laplacian.

## Thin oscillating square plate with free boundary

Picture for square plate obtained from solutions for the thin bar by the
method of Wheatstone.

Many more pictures.

Wilberd van der Kallen