Napoleon saw Chladni demonstrate in Paris in 1808 and the Academy of Science set a prize for a mathematical description of the elasticity of metal plates capable of predicting the figures. It was hard enough that Germain was the only person who entered.
Her first two submissions were rejected for mathematical errors; the third won the prize in 1816. Getting there meant inventing a way to describe how a material bends and deforms under stress, which is why her name is attached to plate theory rather than to a footnote about sand.
Hackaday tells the biography well: teaching herself from her father's books during the revolution, submitting coursework to the École Polytechnique under the name Monsieur Le Blanc because it did not admit women, and corresponding with Gauss under the same pseudonym. It is a maker-audience piece rather than a scholarly one, but it gets the dates and the sequence right.