[Physics] Relationship between volume of water and resonant frequency


I am currently researching the relationship between the volume of water in an object, and the frequency at which resonance occurs.

I have conducted an experiment in which I added volumes of water, and I found that with an increase in the volume of water, the resonant frequency decreases. This is what I had expected, as a result of increased damping by the water, but I have yet to find an explanation for this in terms of a formula.
Could someone here help? I have so far found that in an underdamped system


However, I am not sure if this is the correct formula to apply in this case as I have also found others.

Best Answer

The resonant frequency has less to do with damping, and more with the geometry of your setup (which is hard to determine from your description). More water = 2 things: higher mass, and greater volume. Depending on the role of the water in your resonator, either of these things would give a lower resonant frequency.

If the water mass is the critical factor (and there is some "Constant Spring" giving the restoring force in your resonator) then the frequency will scale with the inverse square root of the mass $$f\propto\sqrt{\frac{1}{m}}$$

If the resonance is inside the liquid, and the vessel is approximately cylindrical, then resonant frequency will be inversely proportional to the dimension of the vessel (the length of the water column), $$f\propto\frac{1}{m}$$

But without actual knowledge of your setup (what are the candles doing???) it's hard to give a definite answer.