There are all sorts of resonances around us, in the world, in our culture, and in our technology. A tidal resonance causes the 55 foot tides in the Bay of Fundy. Mechanical and acoustical resonances and their control are at the center of practically every musical instrument that ever existed. Even our voices and speech are based on controlling the resonances in our throat and mouth. Technology is also a heavy user of resonance. All clocks, radios, televisions, and gps navigating systems use electronic resonators at their very core. Doctors use magnetic resonance imaging or MRI to sense the resonances in atomic nuclei to map the insides of their patients. In spite of the great diversity of resonators, they all share many common properties. In this blog, we will delve into their various aspects. It is hoped that this will serve both the students and professionals who would like to understand more about resonators. I hope all will enjoy the animations.

For a list of all topics discussed, scroll down to the very bottom of the blog, or click here.

Origins of Newton's laws of motion

Non-mathematical introduction to relativity

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

History of mechanical clocks with animations
Understanding a mechanical clock with animations
includes pendulum, balance wheel, and quartz clocks

Water waves, Fourier analysis



Friday, March 11, 2016

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

All postings by author Postings in this section next: Mathematical expressions for one dimensional traveling wave

Three types of waves: traveling waves, standing waves, and rotating waves
Fig. 1. Traveling ocean 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.

Fig. 2. Standing waves on a string.

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.

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
Traveling waves Resonant standing waves Rotating waves
  • 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)
  • 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)
  • 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±)

Wave topics covered in this and subsequent postings

Three types of waves (above).

One dimension:
Mathematical expressions for one dimensional traveling waves (next posting)
Standing waves in one dimension
Waves in a ring resonator - standing and rotating waves
Two dimensions:
Traveling and standing waves in two and three dimensions - Cartesian coordinates
Standing waves and rotating waves in two dimensional circular resonators
Three dimensions:
Standing waves and rotating waves in three dimensional cylindrical resonators
Spherical harmonics
The Lagrangian approach to simple waves - several common waves that lack momentum

All postings by author Postings in this section next: Mathematical expressions for one dimensional traveling wave