Standing Waves

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A "standing wave" may sound like a contradiction in terms, as waves are fundamentally defined by the transport of energy through space. However, a standing wave (or stationary wave) is actually a manifestation of wave interference and resonance.

When a continuous progressive wave encounters a boundary, it reflects back into the medium. The superposition of the incident wave and the reflected wave—moving in opposite directions with identical frequency and amplitude—creates a resultant wave pattern that appears trapped in place.

Key characteristics of standing waves include:

  • Nodes: Points of permanent zero displacement resulting from continuous destructive interference (governed by boundary conditions).
  • Antinodes: Points of maximum oscillation amplitude resulting from continuous constructive interference.
  • Energy Confinement: Unlike traveling waves, standing waves store kinetic and potential energy within fixed spatial regions rather than propagating energy along the medium.

This principle governs phenomena across classical and modern physics, from acoustic resonance in musical instruments and microwave cavities to standing electron probability waves in quantum mechanics.

Interactive
Let's first explore why a standing wave is produced​
  1. Set the type to pulse, have damping to zero, and fixed end
  2. Now send down a single pulse - You will note that it reflects and switches side, or changes phase
  3. If you send another pulse whilst the previous one is returning, they will interfere with each other.

This is key to standing waves, but this only occurs at set frequencies, Let's see this in action.
  1. Set the system to oscillate, damping zero and fixed end
  2. If you press play you will see that you might get a standing wave when the reflected wave interferes with the coming wave.
  3. But the frequency (1.5 Hz) is not quite right. It may behave as a standing wave for a short time, but eventually it will become chaotic.
  4. See if you can alter the frequency slightly to make a standing wave. Press restart when you do this

Hopefully you will see that you will get a large amplitude (its in resonance) but the position of these amplitude does to move to the left or right (Hance its standing wave)

.
The places where the waves are at maximum displacement, or amplitudes are called the antinodes, and this is where are getting constructive interference
The places where there is no motion are called nodes and are the places where there is destructive interference .​
Picture
​Theory
In the first in a series of videos I discuss the principles of a transverse standing wave using a guitar and a string to demonstrate and explain.

Experiment

Now go to this online interactive lab on standing waves

Sample Problem
We are now ready to try a sample problem
Below is a sample problem with a video that explain how to solve it. It is suggested you try the problem beforehand, as this actually aids understanding, even if you are unsure if you are correct.
Picture
 
Interactive
Here is another Standing wave in string interactive developed in Geogebra by Tom Walsh. Again, you will notice only specific frequencies will result in a standing wave
 

Standing waves can also form in longitudinal sound waves, where air itself serves as the oscillating medium. Many wind and brass instruments rely on the resonance of standing waves inside air columns to produce pitch.

A closed pipe (a tube sealed at one end and open at the other) creates specific acoustic boundary conditions:

  • Closed End: Air molecules are constrained from moving, forming a displacement node (pressure antinode).
  • Open End: Air molecules oscillate freely, forming a displacement antinode (pressure node).

Because a node must exist at the closed end and an antinode at the open end, closed pipes only support odd harmonics (n = 1, 3, 5, ...), given by the frequency relation fn = n v / (4L).

Sample Problem
Picture
 
Interactive
In this animation you can explore both the standing wave in longitudinal form as well as the transverse form used in representations
​​Standing wave in pipe interactive developed in Geogebra by Tom Walsh

Standing Waves in 2 dimensions - Chladni plates

In the previous examples we were looking at standing waves in one dimension. However standing waves can occur in two and three dimensions. Chladni plates provide a fascinating visual representation of standing waves in 2 dimensions​, this video will discuss the principles of how it works.
 
Applications
As the above videos demonstrate, there are applications for standing waves, most obviously the production and use of musical instruments
There are other applications
  • Microwave ovens use microwave standing waves to heat your food. This principle was used to measure the speed of light and is further discussed here.
  • Louis de Broglie used the concepts of standing waves to explain why electrons have a set energy of discrete orbits. This is discussed further here
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