The frequency of vibration of particles in a medium in a sound wave is equal to the frequency of the source of the sound due to the process of wave propagation and resonance.
When a sound is produced, it generates pressure variations that travel through the medium as a series of compressions and rarefactions, forming a sound wave. The source of the sound, such as a vibrating object or a speaker diaphragm, creates these compressions and rarefactions at a specific frequency.
As the sound wave travels through the medium, it interacts with the particles of the medium. The particles themselves do not travel with the wave but instead vibrate back and forth about their equilibrium positions as the wave passes through them. The frequency at which these particles vibrate corresponds to the frequency of the source of the sound.
This phenomenon is known as resonance. When the frequency of the sound wave matches the natural frequency of vibration of the particles in the medium, resonance occurs, leading to a stronger interaction and more significant particle vibrations. In other words, the particles respond most effectively when the frequency of their vibrations matches the frequency of the sound wave.
For example, if a tuning fork with a frequency of 440 Hz (which produces the musical note A) is struck, it sets the surrounding air particles into motion, vibrating at the same frequency of 440 Hz. This creates a sound wave that propagates through the air, and the particles in the air continue to vibrate at 440 Hz as the wave travels.
Therefore, the frequency of vibration of particles in the medium aligns with the frequency of the source of the sound wave due to the phenomenon of resonance.