Why do two tubes with different lengths produce different frequencies?

Question

I've learned that it has to do with harmonic frequencies and the relationship between length and wavelength in the equation $L = n(\frac{\lambda}2)$, but my question is why?

If you were to blow air into two tubes that differ only in length, the result would be two different frequencies. But this applies even if you are blowing air in exactly the same way for both tubes (equal velocity, quantity, pressure, etc.). This should mean that the motion of the particles in each tube should be the same, and although the wave speed is the same, frequencies and wavelengths are different. How does changing the location of the tube's bottom cause this when the particles at the top have no direct interactions with it?

This question also applies to the idea of changing the frequency of a vibrating string by changing the location of one end (holding it down with your finger).

Answer 1

A wave disturbance, be it either in a stretched string or a tube with one closed end, requires time to travel from one end to the other. Taking your example- a tube- we see that the longer the tube, the longer it would take for a wave disturbance at the open end to make it to the closed end, reflect off it, and return to the open end again.

In simple terms (and ignoring things like phase shifts during reflection, etc. for now) if the disturbance source is cyclic, the system will then resonate when the round-trip transit time of a pressure wave matches the repetition frequency of the source. What this means, then, is that the longer tubes will resonate at lower frequencies, and vice versa.

Much has been written on this topic in the arena of organ pipe physics, where you can find detailed treatments of things like phase shifts at the ends of both open and closed tubes; including these effects then allows the resonant frequencies of tubes to be mathematically solved for as functions of their length.

Let us know if you want more detail than this, so that one of us here can steer you to it.