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Today as I was filling up a bottle of water, and I could hear the distinctive change in pitch that signals that the bottle is near filling up, I started wondering what equation determines the frequency of the sound.

More specifically, I suspect this is fundamentally a problem in vibration dynamics. I suspect that there is some type of equation that changes the natural frequency of the cavity but I cannot put my finger on it.

Does someone know how to derive this equation? I suspect it will involve a lot more variables than the the natural frequency of the harmonic oscillator $\sqrt{\frac{k}{m}}$, or the pendulum frequency $\sqrt{\frac{g}{l}}$.

This is not for a project, just plain old curiosity, so I would be very happy for pointers to books or links.

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Most acoustic vibrations of air more or less confined to a given geometry can be explained by two basic models. One is the one-dimensional treatment of the air in a tube, and the other is the lumped-parameter treatment of the air within a volume that has a small hole in its side. The former is called a quarter-wave or half-wave tube, depending on the nature of the vibration, and the latter is called a Helmholtz resonator.

When we fill a bottle with water, further investigation is required in order to determine which of the above models apply. Key to that determination is the frequency range of the sound we hear. The two models will predict very different frequency ranges.

The proper tube model in this case is the quarter-wave tube model simply because the liquid in the bottle provides a closed end and the mouth of the bottle the other end. The natural frequency of vibration of such an air column has a wavelength four times the distance from the edge of the mouth to the water. If say the water surface is a couple inches from the edge, the wavelength will be about 8 inches, giving a vibration frequency of about 1500 Hz, which our ears can readily detect.

The natural vibration frequency of a Helmholtz resonator is given by f = (c/2/pi)(A/L/V)^0.5, where c is the speed of sound, pi = 3.14, A the area of the opening, L the approximate length of the air "slug" that vibrates in the opening, and V the volume. Let's say the mouth diameter is 0.5 inch, the length of the air slug about twice the mouth diameter, and the volume at a certain moment of filling is roughly (piD^2/4)*L', where D is the bottle diameter and L' the length of the air space in the bottle. If we say that D = 3 inch and L' = 4 inch, we get f = 164 Hz, which is also well within the range of hearing.

Many of us can get a feel for the Helmholtz frequency when we blow edge-on across the mouth of an empty bottle, and the 164 Hz is very reasonable. The quarter-wave frequency is much higher, at 1500 Hz. In my view, during the process of filling an empty bottle, the frequencies start quite low, due to Helmholtz resonance, and as the liquid level rises, the Helmholtz formula shows that the Helmholtz frequency will also rise (the air volume decreases). At some point, however, the Helmholtz model breaks down, because we no longer have a well defined volume and a relatively small opening. Thus, the quarter wave model then becomes more applicable, producing a higher range of frequencies.

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The frequency of the sound you're hearing is related to the changing resonant frequency of your bottle based on the frequency/velocity relationship of sound waves v=f*λ

Velocity (v) is the speed of sound in air which is nearly constant, typically 343 m/s.

The length of the air column in your bottle is the wavelength (λ) of the resonant frequency. So as the wavelength decreases (bottle fills up), the frequency of the sound produced increases.

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  • $\begingroup$ I am trying to wrap it around my head this concept. I was half expecting that the fluid viscosity or density would have played a role (e.g. I would expect oil to have slightly different wooshing sound). However, if I understand correctly what you are saying is that, the empty part of the bottle (its length in particular) acts as a cavity? And essentially I can think of the sound frequency filling up the available length as a standing wave of sorts? $\endgroup$
    – NMech
    Nov 26 '20 at 13:56

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