When we imagine a wave, we usually think of a snapshot of a periodic wave with its valleys and hills. Waves propagate in space and time, so they are not static objects. They have a certain speed and movement. This is a consequence of a collective effect achieved by the displacement of all the points of a wave. Each of them moves according to the specific function defining the wave, and we can see how the wave advances. Here, we will only consider periodic waves and not wave packets to effectively compare the basic properties. To get a grasp of these concepts, see the image below.
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Fig. 1 - Periodic waves (left) and a wave packet (right)
As we can see, periodic waves fill the whole available region with amplitude, while, for wave packets, the region is finite and changes over time. Essentially, a wave packet can be pictured as a single sea wave, while a periodic wave is like a never-ending set of regularly-produced sea waves. Both move and evolve.
With this in mind, we now define the concept of stationary waves, which are waves formed by the superposition of two periodic waves with the same frequency travelling in opposite directions. By superposition, we mean that two waves are added together to form another wave. This usually leads to the concepts of destructive interference and constructive interference, which are the phenomena by which waves emerge from adding same-sign disturbances or different-sign disturbances, respectively.
Constructive interference: the phenomenon by which waves combine their amplitudes, resulting in a wave with a bigger amplitude.
Destructive interference: the phenomenon by which waves combine their amplitudes, resulting in a wave with a smaller amplitude.
See the image below for examples:
Figure 2. Constructive interference (left) and destructive interference (right). Source: Haade, Wjh31, Quibik, Wikimedia Commons (CC BY-SA 3.0).
Of course, the generation of a stationary wave can involve total constructive or total destructive interference, but we usually get a mixed interference.
On the one hand, the defining characteristic of progressive waves is that they advance in space. On the other hand, since a stationary wave is formed by the superposition of waves travelling in opposite directions, there is no movement in the direction of propagation, but almost all points displace perpendicular to it.
The main characteristics of progressive waves are:
A basic example is a wave formed in a rope where one end is loose and we quickly pull the other end up and down. The wave moves forward until it reaches the end of the rope (if we pull hard enough) and then stops. If we keep pulling up and down, this will continue to happen.
The main characteristics of stationary waves are:
We could form a stationary wave in a rope the same way we did before but with someone replicating our movements from the other end. However, a more common example involves the strings of a guitar. When we press a guitar string against the fret, we are fixing one of the ends, thus causing a reflection phenomenon: when a wave formed in the string reaches the fret, it is reflected and travels backwards. This reflection generates a wave travelling in the opposite direction, which, together with the wave travelling in the other direction, forms the stationary wave. The end of the string and the contact with the fret are the nodes of the stationary wave.
Figure 3. Different stationary waves with different amounts of nodes. In green, the wave at a certain time, and in blue, the wave after half a period. Source: CK-12 Foundation, Wikimedia Commons (CC BY-SA 3.0).
The main application of stationary waves is the generation of specific frequency sounds. The special feature in instruments is that due to interference patterns, not only a certain frequency is generated as a stationary wave, but other related frequencies corresponding to the appearance of nodes appear amplified. This is what constitutes a musical note and the mechanism behind music being harmonic. These other stationary waves are, therefore, called ‘harmonics’, and they correspond to the waves found in Figure 3 after the top one.
However, not all applications of stationary waves are restricted to music and sound waves. The functioning of a microwave, for instance, is simple: a stationary wave with a certain wavelength is generated between two walls of the microwave. An easy way to see this is to take the spinning plate out of the microwave and place red liquorice inside. After heating, some evenly-spaced parts of the liquorice will appear melted or burnt. These spots correspond to the antinodes of the wave (where a maximum transfer of energy occurred).
As the final example, we just mention the fact that many fundamental quantum systems of our world feature stationary waves. For instance, the basic distribution of an electron in a hydrogen atom is determined by a certain set of stationary wave dispositions called ‘spherical harmonics’.
A periodic wave is a wave with a repetitive pattern that extends in the whole space, while a wave packet has a non-vanishing amplitude in a certain region of space.
A stationary wave is the result of the superposition of two waves with the same frequency travelling in opposite directions. It does not have a global speed, unlike progressive waves.
Stationary waves have nodes and antinodes; every point has specific maximum and minimum amplitudes.
Stationary waves appear in day-to-day situations like microwaves or music.
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How are stationary waves formed?
They are formed by the superposition of two waves travelling in opposite directions.
What is a stationary wave?
A stationary wave is a periodic wave with no global speed of propagation and whose points have specific bounded amplitudes.
Are all points of a stationary wave in phase?
No, only points that are separated by an even number of nodes are in phase (oscillating in the same direction). On the other hand, points separated by an odd number of nodes are in opposition to phase.
Are standing waves the same as stationary waves?
Yes, they are.
Are stationary waves a type of superposition?
Yes, they are formed by a superposition of two waves travelling in opposite directions.
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