I Standing Wave Fundamental Frequency and Particle Vibration

For a wave A sin ( kx - ωt) and a wave A sin ( kx + ωt) travelling opposite to each other, on evaluating by applying superposition principle , the resultant displacement function is 2A sin ( kx ) cos (ωt) . For different Node Anti-node configurations we calculate natural frequencies of the standing waves in the medium/string. Is this totally different from the frequency of the individual particles in the medium which execute an SHM with a fixed amplitude, calculated out of ω . What is the physical difference of frequency that maybe calculated from ω and the natural frequencies ?
 

sophiecentaur

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. Is this totally different from the frequency of the individual particles in the medium which execute an SHM
It's the geometry / dimensions of the whole oscillating system that determines the resonance and not the individual particles. The particles are not associated with any particular frequency - they will have a whole range of speeds and collision rates and can be changing direction, individually in a random way due to collisions. They cannot 'know' about the oscillations - they can only move in the direction that the instantaneous pressure is pushing them in bulk. But the temperature in a gas (aka average Energy of particles) can affect the bulk properties of the medium (wave speed).
 
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The particles themselves don't have any frequency associated. So you cannot say that the standing wave frequency is different from the frequency of the particles. A particle attached to a spring (like in the simple harmonic oscillator) oscillates with a specific frequency. The same particle attached to a different spring oscillates with a different "natural frequency". So the frequency is associated with the particle in a specific elastic potential.

If you have multiple particles interacting by elastic forces there are multiple natural (or normal modes) frequencies associated with the system, like in the waves on a string. So there is nothing to compare. The isolated "particles" don't have frequencies.
 

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