If you pull a rubber band tighter, it snaps back faster. Same idea here. Whatever the equilibrating force is that is involved with the oscillation, it’s stronger if you increase the displacement from equilibrium. That stronger force causes the wave particle velocity change to keep pace with the peak displacement change, and so the period/wavelength is unchanged.

The wavelength is determined by the amount of time between peaks or valleys in the pressure, not the amplitude of them. Said differently, the speed of sound is nearly independent of the pressure, so if the time is known, then the distance can be solved for, as simply wavelength (meters) = speed of sound (meters / second) divided by frequency (Hz or cycles / second). In other words, the wavelength depends only on the speed of sound and the frequency, but not the amplitude.

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Brilliant. I love learning things I wasn’t even looking for. Absolutely brilliant. Thank you

The analogy really ain't that good. Try with a water wave or a guitar string. If you pull a guitar string further the vibration it has gets a bigger amplitude but its mechanical resistance and border limitations won't change. This means the string will move faster or slower but it will oscillate at the same frequency.

Since the frequency is the same, the note is the same. But since amplitude varies, the sound is more intense.

Are you trying to draw a distinction here somehow between the wave dynamics in a guitar string and the dynamics in a rubber band that's pulled taut enough to support oscillation? They're the same. A rubber bands stretched taut between two supports and then plucked is exactly the same as a guitar string except that it's far more compliant. Whatever reasoning explains why a guitar string still makes the same sound even if you pluck it harder is identical to the example already given.

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