| | Fig. 1. Traveling ocean waves.
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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.
| | Fig. 2. Standing waves on a string.
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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.
| | Fig. 3. Computer simulation of a rotating wave in a cylindrical container of water. The coloring indicates the height of the water surface at various points.
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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.
| Table 1. Properties of the three wave types
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| Traveling waves
| Resonant standing waves
| Rotating waves
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- moving constant shape
- all frequencies allowed
- in some media wave has linear momentum
- occurs in open spaces
- time and position together as the argument of a single sinusoid, e.g. cos(−ωt±κx)
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- shape oscillates with time
- only certain frequencies allowed
- wave has no momentum
- occurs around reflecting surfaces in closed spaces
- time and position as arguments in separate sinusoids, e.g. (cos ωt)×(cos κx)
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- moving constant shape
- only certain frequencies allowed
- in some media wave has angular momentum
- occurs in closed spaces of circular and cylindrical symmetry
- time and angular position together as the argument of a single sinusoid, e.g. cos(−ωt±mφ)
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