Sound – understanding standing waves

A standing wave is the combination of two waves that are moving in opposite directions. Standing waves are typically formed in situations where a wave is bouncing back and forth in an environment that produces constructive interference. To put this more simply, a standing wave occurs when the length of the wave matches the length of the environment.

To understand why standing waves behave the way they do, consider what happens when you send a single pulse (0.5 of a wave) down a rope that is fixed at the end.

In the image below, the pulse travels down the rope (figures A and B). In figure C, the pulse encounters the fixed end. The pulse is reflected off the end (figure D) and returns on the opposite side of the rope (figure E). When it returns to where it started, it is reflected again, and it continues until it runs out of energy.

Continuous standing waves

A continuous standing wave is produced by moving the end of the rope up and down to produce a wave that is exactly the right length to ‘fit’ on the rope. In the image below, the (red dot) end of the rope is moved up and down to produce a wave that is exactly 0.5 times the length of the rope (figure 2). In figure 4, the wave has begun to be reflected back from the fixed end. Eventually, the reflected wave will make it back to where it started (red dot) and be reflected again (figure 7). Because the wave is reflected back and forth, it takes very little energy input to keep the standing wave going (figure 8). Energy input is indicated by the arrows on the left side of each figure.

When both ends of the wave are fixed (not free to move back and forth), a standing wave can occur when the medium (rope in this case) is exactly 0.5, 1.0, 1.5, 2.0, 2.5 etc. the length of the wave.

Waves are also reflected when the ends are not fixed. It is difficult to see this in the rope example, but in many situations, standing waves can occur when one end is fixed and the other free. This occurs when the medium is exactly 0.25, 0.75, 1.25, 1.75 etc. the length of the wave.

Hearing sound waves

When you blow across the top of a bottle, the sound you hear is the result of a standing sound wave. Sound waves are longitudinal or compression waves. If you could see the standing sound wave, you would see alternating areas of high and low pressure. Since it is difficult to see (and draw) compression waves, we usually represent them using sine waves.

The air at the open end (top) of the bottle is exposed to the atmosphere so it is at constant pressure. The air at the closed end of the bottle varies in pressure. In this example, 2.25 waves are shown in the bottle. The density of dots in the left bottle corresponds to the solid transverse wave drawn at the left, and the dots in the other bottle correspond to the dashed transverse wave.

Nodes and antinodes in a standing wave

The areas in a standing wave that are constant and do not change are called nodes. The areas that fluctuate the most are called antinodes. It is interesting that, in a standing wave, the nodes do not move even though waves are constantly passing through them in both directions.