Peoples Physics Concepts

11.7. Standing Waves http://www.ck12.org

11.7 Standing Waves

• Describe how standing waves form through constructive interference.


• Solve standing wave problems where either both ends are constrained or neither of the ends are constrained.

Students will learn how standing waves form through constructive interference and how to solve standing wave

problems where either both ends are constrained or neither of the ends are constrained.

Key Equations

Standing waves for a string restricted at both ends or unrestricted at both ends

fn= 2 nvL ; n is an integer

Guidance

• A typical standing wave is shown below. This is the motion of a simple jump-rope. Nodesare the places
  where the rope doesn’t move at all;antinodesoccur where the motion is greatest.

FIGURE 11.1

For this wave, the wavelength is. Since , the frequency of oscillation is.

• Higher harmonicscan also form. Note that each end, where the rope is attached, must always be a node.
  Below is an example of a rope in a 5thharmonic standing wave.

FIGURE 11.2

In general, the frequency of oscillation is , where n is the number of

antinodes. The thick, dotted lines represent the waveenvelope: these

are the upper and lower limits to the motion of the string.

• Importantly, each of the above standing wave examples can also apply to sound waves in a closed tube,
  electromagnetic waves in a wire or fiber optic cable, and so on. In other words, the standing wave examples
  can apply toanykind of wave, as long as nodes are forced at both ends by whatever is containing/reflecting
  the wave back on itself.


• Resonance is a phenomenon that occurs when something that has a natural frequency of vibration (pendulum,
  guitar, glass, etc.) is shaken or pushed at a frequency that is equal to its natural frequency of vibration. The
  most dramatic example is the collapse of the Tacoma Narrows bridge due to wind causing vibrations at the
  bridge’s natural frequency. The result is the dramatic collapse of a very large suspension bridge.