One can make a sound by blowing into an empty bottle. The sound is a product of a resonance phenomenon called the Helmholtz resonance. We can determine the resonance frequency by noting the shape and dimension of this cavity, the Helmholtz resonator, where V denotes the volume of the body, S denotes the cross-sectional area of the neck, and l denotes the length of the neck:
The resonance frequency of the Helmholtz resonance is determined by the following formula:
$$f = \frac{ c }{ 2 \pi }\sqrt{ \frac{S}{Vl} }$$
Let’s think about why resonance occurs. The body of the cavity is filled with air, the volume of which is V, while the neck contains air with a mass of m. When one blows into the open end of the neck, the air in the neck is pushed down into the body, compressing the air inside. When the air in the body cavity is compressed, the restoring force expands the air, and the mass of air in the neck is pushed upwards. Because of the inertia of the mass, the mass passes through the equilibrium. However, again the restoring force of the air in the body compresses the air. And again, the mass of air in the neck is pushed upwards, and this cycle repeats again and again. This process can be viewed as a simple harmonic motion, producing a resonance.
- Stevens, K. N., Acoustic Phonetics, MIT Press, Cambridge, MA, 1998.