I would like to understand why the faster you spin, higher normal modes are excited
This is the tube i'm talking about, i think it's used as a kid's toy: https://www.youtube.com/watch?v=eCOZxzO3FvE
Thank you for reading
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You have made several assertions and asked several questions. I will address your assertions first because I think they confuse the question, and then point you toward information to explain your phenomenon in more depth.
Assertion 1: that the tube produces a pure sinusoids sound wave. This is false. The sound produced includes many harmonic overtones, not to mention the whooshing white noise. The tone produced is qualitatively similar to that of a flute or recorder, which is no coincidence, as we will discuss below.
Assertion 2: that the increasing pitch of the tone is caused by an increasing flow of air from one end of the tube to the other. (That is how I read your question. Clarify your wording if that was not your meaning.) There probably is an increasing fliw of air through the tube, but I very much doubt that it has much interaction with the tone which is produced. I hypothesize that one could tape a plastic bag over the handle end, leaving the bag free to bubble full or squish empty, and that the tone produced would be similar apart from the crinkling sounds of the bag.
And now for the general question which you were trying to ask: Why does it make that funny set of harmonic sounds?
The mechanisms at play are almost identical to those in a flute, recorder, penny whistle, beer bottle which someone is blowing across, or even that painful thumping which can occur when the windows are partially rolled down in a car. There is a resonant cavity in the form of an open ended tube, which has resonant modes at the frequencies you hear. The fluid mechanics governing which modes will be excited by a given airspeed across the hole are complex, but you should be able to find discussion of them in a text on the physics of flutes and whistles.
It appears to be like a whistle being overblown. The air going past the end causes the air inside to resonate see: Whistle Physics When you spin it fast, it is like overblowing a whistle. The air resonates in sections of the tube instead of just the end(s). For an open ended tube, the frequency doubles. If the bottom was closed, it would tripple.