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The mass therefore would be changing because I am pouring in water and testing the frequency at different volumes. Obviously as the volume increases the mass increases and the frequency decreases.

How would I exactly model this with an equation?

Perhaps using the equation for a simple harmonic oscillator: $$ \left[\text{frequency}\right] ~=~ 2 \pi \sqrt{\frac{\left[\text{stiffness constant}\right]}{\left[\text{mass}\right]}} \,.$$ But can the stiffness constant really be applied to a material like glass, since its so rigid and brittle and not like a spring?

So does anyone have any suggestions?

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  • $\begingroup$ I'm not sure at all if the simple harmonic oscillator equation would apply directly here (I'm thinking not; but I don't know enough about it's assumptions to really say); but glass definitely does have a stiffness associated with it. It's also pretty well behaved in that regard, in the sense that it's stiffness is quite linear, so it actually has a constant that doesn't vary much with applied load. $\endgroup$
    – JMac
    Mar 7, 2019 at 20:51

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The normal modes frequencies will indeed depend on the elasticity of the glass, measured by Young's modulus and shear modulus and the mass of the vibrating body. Adding water increases the mass and decreases the frequencies. The glass does has elastic properties but it is much stiffer than rubber, for example. This translates into much higher values of the elasticity modulus.

As for a formula, there is no analytic formula. The actual dependence of frequencies on the elasticity and mass will depend on the specific geometry of the glass. Even then you may find a "half'analytic " formula, including some coefficients obtained by numerically solving the differential equations for a given, simple, geometry.

All these considerations apply to the vibration of the glass body. There may be also vibrations in the air contained in the glass. But when you hot the glass you excite mostly the vibration of the body. If you blow across the top you may excite the vibrations of the air which have different laws.

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  • $\begingroup$ Do you think there is a relation between the geometric shape and the frequency. Man, im kinda sad there cannot be a formula made. Perhaps one that will just explain the relation of shape, frequency, mass. Also can the glass be thought of as a spring oscillating back and fourth. $\endgroup$ Mar 8, 2019 at 0:57
  • $\begingroup$ Yes, there is. I mean the frequency depends on shape. Not necessarily if by "relation" you mean a formula. A glass is a 3-d object so it oscillates but not just along a direction, as a simple spring. See this for example. It shows diagrams of fundamental and the n=3 mode. : mafija.fmf.uni-lj.si/seminar/files/2010_2011/… $\endgroup$
    – nasu
    Mar 8, 2019 at 13:56

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