I would like to reproduce the results of Manas - The music of gold: Can gold counterfeited coins be detected by ear?, but it skips a lot of steps, and the mathematics behind it is a bit advanced for me (Bessel functions etc.). Summarizing the paper, it shows how to calculate the dominant frequencies of the sound of a ringing metal coin for given dimensions and metal composition, using the formula $f=\frac{h}{2 \pi a^2}\sqrt\frac{E}{12 \rho (1-v^2)}\lambda^2$, with tables for values of each of the physical variables for various metals.

There are a few particular places where I've gotten stuck.

• The "shape factor" $\frac{h}{2 \pi a^2}$ is calculated as 1.44 given values $h=\text{1 mm}$ and $a=\text{21 mm}$ (page 6), but I got the value 0.00036107861.
• The dimensions in these factors do not seem to cancel out, and it doesn't say what is the correct unit to measure in.
• There is a distinction made between a free-standing disc, a clamped disc, and a disc supported on a column, but it is not clear to me how to modify the formula in each case.
• The format of the formula suggests that the values of $\lambda$ are independent of the physical variables, but then there is a table on page 8 where for instance, $\lambda_{2,0}$ has values ranging from 2.21 to 2.33 for different metals.

Edit: To make my question more concrete, I would like to be able to calculate the frequencies for, say, a silver coin with thickness 2mm and diameter 30mm, and determine which mode of testing to use in order to reproduce that spectrum with a real coin. I am not able to get the same result as the paper using their formula, and I don't know what mistake or omission to fix in order to fill that gap.