Plate vibration solutions require proper edge boundary conditions, unresolved until Kirchhoff in 1850.

Evidence

The contest was reopened in 1813. This time, Germain was awarded an honorable mention—again as the only entrant. The contest was reopened for a final time in 1816, and, for the third time, she was the only entrant. This time, the judges awarded her the prize; she had found the right equation. However, the all-important boundary conditions were wrong. A string, for example, can be fixed at an end or freely oscillate up-and-down (tension can be kept by having a frictionless vertical track in which it slides). Chladni plates have edges free to vibrate up and down, but for thick plates specifying these conditions is a very subtle problem. The subtleties were not mopped up until the work of Gustav Kirchoff (1824–1887) in 1850. — Eric J. Heller, Why You Hear What You Hear- An…, 1393

Categorization

• XREF: Connects to the standing question of why plates/surfaces radiate or cancel sound - the boundary-condition subtlety resurfaces in instrument acoustics and loudspeaker design.
• DEFINE: Clarifies what ‘boundary conditions’ mean physically for a vibrating plate versus a string - fixed versus free edges.