I would like to reproduce the results of [Manas - The music of gold: Can gold counterfeited coins be detected by ear?](https://www.researchgate.net/publication/276297204_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.