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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.

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    $\begingroup$ Aside from possibly the first, none of these seems like a mathematical issue. $\endgroup$
    – LSpice
    Commented Nov 27, 2020 at 3:15
  • $\begingroup$ @LSpice, I'm not sure I agree. I'm asking for help from whoever on this site might know more about the background for these calculations than I do. For instance, how the formula should vary with different boundary conditions is a mathematical question, one not addressed in the paper. Why the eigenvalues are different for different materials, and how to determine them, is a potentially mathematical question. Which units to use in a formula is certainly a mathematical question, at least in an applied sense. I could add more clarification to the question if that would help. $\endgroup$
    – Black Carrot
    Commented Nov 27, 2020 at 4:14
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    $\begingroup$ the paper you cite is not reliable; in the formula for the frequency $a$ should be the radius, not the diameter of the disc; the shape factor is not dimensionless, so a statement that says it equals 1.44 is meaningless; the formula for the frequencies only applies to a clamped unstressed disc; values of $\lambda$ for that case are in Table 3 of this paper, which seems a more reliable source --- and yes, this question is quite off-topic here. $\endgroup$
    – Carlo Beenakker
    Commented Nov 27, 2020 at 10:02
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    $\begingroup$ The name of "this paper" referenced by @CarloBeenakker: Malhaire - Comparison of two experimental methods for the mechanical characterization of thin or thick films from the study of micromachined circular diaphragms. $\endgroup$
    – LSpice
    Commented Nov 27, 2020 at 21:32
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    $\begingroup$ I would try physics.stackechange $\endgroup$
    – Carlo Beenakker
    Commented Nov 29, 2020 at 18:49

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