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

$\omega&space;_{d}=\sqrt{1-\zeta&space;^{2}}\omega&space;_{n}$

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

• What exactly are you asking for? Apr 13, 2017 at 13:09
• Thank you for your reply. I am looking for a formula to back up and explain this found relationship, as without a correct formula I have no basis for a correct analysis of my findings. Apr 13, 2017 at 13:45
• Not clear. What kind of "object"? Please provide more details of your experiment. Apr 20, 2017 at 22:33

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), $$f\propto\frac{1}{m}$$