[Physics] Why would a decreased length mean a higher frequency of sound


This is in terms of the Water Bottle Lab: adding water (effectively decreasing length) to a bottle and finding its fundamental frequency at different lengths/amounts of water.

I understand what happens, but why exactly does less length mean higher frequency?

Best Answer

The sound wave in the bottle reflects off the water surface (with the reflected wave changing sign - so there is a displacement node there) and again at the mouth of the bottle (without changing sign - so there is a displacement antinode there).

Now resonance occurs when the frequency with which you stimulate the oscillations in the bottle "fit" - that is, when the wavelength $\lambda$ (which is a function of frequency, since sound velocity is independent of frequency: $\lambda = \frac{c}{f}$) is such that $(2n+1)\lambda/4=\ell$.

As $\ell$ gets smaller, $\lambda$ gets smaller and the resonant frequency gets higher.