Standing wave (physics)

A standing wave is a wave that oscillates in place without actually traveling anywhere. Also known as a stationary wave, a standing wave occurs when a wave interferes with itself after reflecting from the boundaries of the system in which it is contained. For this to happen, the frequency of the wave, or the number of waves that pass a given point in a certain amount of time, must be just right. When the frequency is correct, the interference of the incident wave and the reflected wave occur in such a way that there are specific points along the medium that look as though they are standing still. The production of standing waves also depends on the presence of the right type of wavelength. Some easily observable examples of standing waves include plucked or bowed musical instrument strings, sloshing water in a bathtub, and the vibrating columns of air that produce sound in pipe organs.

Background

To understand standing waves, it is necessary to understand how normal waves work. The simplest example of how a normal wave works can be seen in a typical mechanical wave. Created by a vibrating object, a mechanical wave moves through a medium like air or water and transports energy as it does so. The movement of a wave through a medium is tied directly to interactions that occur between its constituent particles. In response to vibration, a given particle will exert a pushing or pulling force on the nearest neighboring particle. This results in the displacement of the neighboring particle, which subsequently exerts the same force on its own neighboring particle. All of this particle interaction ultimately leads to the wave's movement through the medium. As this movement occurs, a crest can be observed moving from particle to particle. Between each crest, there is also a trough. These crests and troughs make up the overall sine wave pattern that can be seen moving through the medium. Waves that move in this manner are called traveling waves. A traveling wave will continue moving until it encounters either another wave or the boundary of the medium. As a result, traveling waves are most likely to be seen when a wave is not confined to a specific space along a medium. Ocean waves are a common type of traveling wave.

The appearance of a traveling wave can be altered under the right circumstances when a wave is introduced in a confined medium. For example, if two people hold the ends of an elastic jump rope just a few feet apart, a wave can travel for only a short distance before it reaches the end of the rope. Once the wave reaches the end of the rope, it will reflect and automatically start traveling back in the opposite direction. These reflected waves will eventually encounter the incident waves being generated at the other end of the rope. When this happens, the reflected waves interfere with the incident waves, and a new wave shape is formed in the medium. In most cases, the new wave shape does not conform to the typical crest-trough appearance of a normal sine wave. Rather, the new wave shape is usually irregular and non-repeating.

Overview

Under the right conditions, it is possible for a wave traveling through a confined part of a medium to have a regular wave pattern. If the wave in question is vibrated at just the right frequency, the resulting wave pattern will take the form of a traditional sine wave despite the interference caused by reflection. In such instances, the interference between the incident and reflected wave occurs in a very specific way that makes it look as though certain points along the medium are standing still. These points are called nodes. At the same time, there are also other points along the medium called antinodes that experience displacement changes over time in a regular pattern. These antinodes continuously move from a positive displacement to a negative displacement at regular intervals. This movement ultimately yields a regular, repeating wave pattern that can be easily observed—a wave pattern that is known as a standing wave pattern.

The key to the formation of a standing wave is the frequency at which energy is fed into the system, or the driving frequency. Specifically, the driving frequency must match up with the system's natural frequency. The phenomenon that occurs when this happens is called resonance. Resonance is marked by a dramatic increase in the amplitude of the resulting vibrations. In the type of systems that are capable of producing standing waves, there are many possible natural frequencies and, therefore, many potential standing waves. This array of possible standing waves is known to as the system's harmonics. The simplest of these is the fundamental harmonic or the first harmonic. Subsequent harmonics are referred to as the second harmonic, the third harmonic, and so on. Together, all of the harmonics beyond the fundamental are also called overtones.

Another important element in the formation of standing waves is wavelength. Wavelength is the measure of distance between the crest and the trough of a wave. In one-dimensional systems, there are three types of wavelengths that can produce standing waves: wavelengths with two fixed ends, wavelengths with two free ends, and wavelengths with one fixed end and one free end.

In mediums that have two fixed ends, the nodes will be located at either end of the wave. A wave that has one antinode in the middle creates what is known as half a wavelength. Adding an additional node in the middle leads to the creation of a whole wavelength. As further nodes are added, more wavelengths are created.

In mediums that have two free ends, antinodes will be found at either end of the wave. If a node is added in the middle, half a wavelength is created. With the addition of another antinode in the center, a full length is created.

In mediums that have a fixed end and a free end, the fixed end forms a node and the free end forms an antinode. If both a node and an antinode are added, the result will be three-quarters of a wavelength.

Bibliography

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