Geiger counter detects ionizing radiation, which is radiation that can sufficient energy to detach the electron from an atom it hit. The way that works is to exploit that phenomenon.
The sensor tube is filled with an inert gas at low pressure so that the radiation can ionize. It also has a cathode and anode with a high voltage over them. The negatively charged free electrons will accelerate toward the positive anode because of electrostatic attraction. The positively charged ionized gas atom will move to a negatively charged charge.
When this happens the electron that is accelerated will hit other gas atom and ionizes them to. This is an avalanche effect, a form of amplification that makes it possible to get a detectable current. The principle is called https://en.wikipedia.org/wiki/Townsend_discharge
The ionized atoms will get electrons from the cathode and the electrons that hit the cathode will be absorbed by it. This is fundamentally the same as a current pulse that flows through the tube. The detector circuit counts the number of electric pulses.
The sound is this electric signal used to drive a speaker, It is a way to get auditory feedback of how much radiation it detects without needing to look at it all the time. This is typically something portable Geiger counter used because typical usage is to move them around so you can detect something radioactive and for a rough estimate listening to the sound is enough.
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