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Three types of waves: traveling waves, standing waves, and rotating waves  

Two forms of waves have long been recognized: traveling waves and standing waves. To these we add a third type, rotating waves, which present the clearest insight into rotary motion and angular momentum in waves in cases of circular and cylindrical symmetry. Traveling waves are those that we normally see in large open spaces where waves are free to propagate, such as water waves on the open ocean. They are characterized by a constant profile that moves along relative to the wave medium as shown in Fig. 1. Standing waves are characterized by nodes, points where destructive interference reduces the amplitude to zero, and by a constantly changing, oscillating wave shape in between the nodes. One situation that creates standing waves is the reflection of traveling waves off a surface as discussed in an earlier blog.
In this discussion we limit ourselves to a second situation that creates standing waves. This involves reflecting surfaces which restrict waves to a limited space, such as the inside of an organ pipe or on a violin string. These are confined waves, bouncing around inside the limited space. Such standing waves inside a closed space are a form of resonance and are limited to certain wavelengths and frequencies. We call these resonant standing waves. The allowed frequencies are the resonant frequencies of the system. The limitation to select frequencies makes them useful in musical instruments and allows musicians to select to play particular musical notes. An example of standing waves is shown in Fig. 2.
In closed resonators of circular symmetry we see a third category of waves, called rotating waves. These waves have some properties of traveling waves and some properties of standing waves. Rotating waves have a constant profile like traveling waves, but are restricted to discrete wavelengths and frequencies like resonant standing waves. Rotating waves offer the clearest insight into waves inside a cylindrical resonator. Rotating waves see application in the quantum mechanical description of atoms. In Fig. 3, we show an example of a rotating wave. Table 1 lists properties of traveling, resonant standing waves and rotating waves.

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