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In the singing wine glass experiment what's the effect on the frequency produced if the effect of liquids with different densities was tested? Everything else would be kept constant (same glass, height etc.)
I believe that the frequency would decrease as density increases, but is there a true and pure scientific reason behind it?
Mechanical oscillators have elasticity, which causes them to "spring back" after being disturbed from their rest position and inertia, which allows the oscillation to cycle due to continual overshoot of the rest position. For a mass $m$ attached to a spring with stiffness $k$, the frequency of oscillation $\omega$ is given by:
$$\omega =\ \sqrt\frac{elasticity}{inertia} =\ \sqrt\frac{k}{m}$$
A singing wine glass is a mechanical oscillator just like the mass on the spring and so its frequency is also determined by an equation that balances elasticity and inertia. A 2006 paper in the Journal of the Acoustical Society of America$^1$ derived (and experimentally verified) the following equation for the fundamental frequency of oscillation $\omega$ of a wine glass filled with liquid:
$$\omega =\ \sqrt\frac{\omega_{_0}^2}{1 + \alpha h^n}$$
where
$\omega$ = fundamental frequency of the wine glass with the liquid in it
$\omega_{_0}$ = fundamental frequency of the empty wine glass
$\alpha$ = a constant proportional to the liquid density, glass shape, and wall thickness of the glass
$h$ = height of liquid in the glass
$n$ = a constant that depends on the shape of the glass
Looking at the denominator first, $h^n$ is basically the volume of liquid and $\alpha$ contains the density of the liquid so that term measures of the mass or inertia added by the liquid. The $\omega_{_0}^2$ in the numerator acts as a measure of the intrinsic elasticity of the glass. Everything else equal, more mass in the glass means a lower fundamental frequency. For the same volume of liquid, the denser liquid has more mass thus lower frequency.
Below are the results of a study$^2$ consistent with this conclusion. They put various mixtures of corn syrup and water (corn syrup has a higher density than water) in a wine glass:
$^1$ "Vibrational modes of partly filled wine glasses" Jundt et al. 2006. Journal of the Acoustical Society of America.
$^2$ "An Examination of the Relationship between the Percentage of Water to Corn Syrup and Resonant Frequency" Gardner et al. 2012. Tigard-Tualatin High School.
