# What is the effect of liquid density on the pitch of a singing wine glass?

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?

• Here are a couple studies to get you started: newt.phys.unsw.edu.au/music/people/publications/… (page 5) and tuhsphysics.ttsd.k12.or.us/Research/IB12/AlbeKastGard/index.htm – pentane Apr 5 '16 at 13:00
• It would be great if you could explain why this happens though. Also if you could explain the formula (8) and essentially what he is talking about. It seems to complex for a person in grade 11 such as me to understand – Wilrs Apr 5 '16 at 13:54
• You may be in grade 11 but you identified a key formula from the paper! Formula 8 essentially relates the frequency of a wine glass with water in it $\omega$ to a constant, the empty glass frequency $\omega_0$, divided by the mass of liquid in the glass $\alpha h^n$ (where $\alpha$ is the density and $h^n$ is the volume of the liquid). If you've studied the simple harmonic oscillator (mass on a spring) you know that the frequency is the square root of a constant over the mass. That's the model behind equation 8. – pentane Apr 5 '16 at 14:28
• h is the distance of water from the top of the glass (correct?), but what is the n? – Wilrs Apr 6 '16 at 9:21
• – Qmechanic Dec 7 '16 at 11:59

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.