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I've been thinking that 'modes of vibration' refers to the frequencies which produce standing waves in a system. But now I'm confused, because ChatGPT gave me the below response for my question.

Question: Are there modes of vibration that does not produce a standing wave?

Yes, there are modes of vibration that do not produce standing waves. These modes typically arise in systems with different boundary conditions or configurations that do not allow for the formation of standing wave patterns.

One example is an open tube or an open string. In an open tube, such as an open-ended pipe or a flute, or in an open string, such as a guitar string that is free at one end, the modes of vibration do not result in standing waves. Instead, they produce traveling waves that propagate in both directions along the length of the system. The waves reflect at the open end and interfere with each other to create complex wave patterns.

In these systems, the modes of vibration are typically referred to as "harmonics" rather than "harmonic modes" because they are not associated with standing wave patterns. Each harmonic corresponds to a specific frequency at which the system can vibrate, but the resulting wave is not stationary and does not exhibit nodes and antinodes like a standing wave.

So, while standing waves are characteristic of certain systems with fixed boundaries, other systems with open boundaries or different configurations can exhibit modes of vibration that give rise to traveling waves rather than standing waves.

What does modes of vibration actually mean?

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  • $\begingroup$ In simple terms, it gives the ratio of the length of the vibrating column to the wavelength of waves associated. $\endgroup$
    – Fire
    Commented Jun 22, 2023 at 11:43

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A system here is a collection of atoms held together elastically, as if connected by springs. It isn't perfectly rigid. Examples are a spring and mass, or a guitar. Or air which is held together by pressure.

The system is vibrating if every atom follows some oscillatory path. They move back and forth without ever getting too far from their rest position. Vibration is bigger than thermal motion, so we will ignore it.

Vibration occurs when a force is applied to one part of the system. The end of the spring is bumped or moved up had down. A disturbance spreads out and sets other parts of the system vibrating. This traveling disturbance is a wave.

Sometimes the wave spreads out and sets the whole system vibrating. A mode of vibration is motion where each atom oscillates at the same frequency so the whole system returns to the same state over and over.

An example is the fundamental note of a guitar string. The wave bounces back and forth between the fixed ends. Each harmonic is also a mode. Each has a different frequency. The real motion is a sum of modes.

An organ pipe is similar, but one end is open. Waves bounce back and forth between the ends.

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  • $\begingroup$ "A mode of vibration is motion where each atom oscillates at the same frequency so the whole system returns to the same state over and over." - I hope this means that all the atoms must get back to their equilibrium position at the same time thus bringing the entire system back to its initial state. And if I'm right, this is possible only in the case of standing waves. Doesn't that mean each modes of vibration refer to a standing wave? $\endgroup$ Commented Jun 23, 2023 at 7:26
  • $\begingroup$ It is common, but not always true. For example, see this old film of a damped wave machine. AT&T Archives: Similiarities of Wave Behavior. Go to 13:30 $\endgroup$
    – mmesser314
    Commented Jun 23, 2023 at 14:41

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