Stationary Waves

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.

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.

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.

The differences between stationary and progressive waves

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.

Progressive waves

The main characteristics of progressive waves are:

• Global amplitude: this means that all points eventually have a certain allowed amplitude value. In the end, if the wave propagates, it reaches all points in space, and this propagation, at some point in time, will reach every value between the minimum and the maximum amplitude.
• Inexistence of nodes: there are no nodes, that is, points that do not vibrate at any time.
• Points in phase: all points have a relative phase (state of oscillation) between 0º and 360º. It is 0º when the points are a wavelength apart.
• Energy transmission: energy is transferred in the direction of propagation.

• Wave speed: there is a global wave speed determined by its propagation. All points of the same wave have this same speed.

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.

Stationary waves

The main characteristics of stationary waves are:

• Local amplitude: depending on the amount of superposition, each point has certain specific amplitude maximum and minimum values.
• Presence of nodes: there are nodes, points where the state of vibration is null and constant in time. The points that oscillate continuously and reach the maximum possible amplitude are called antinodes.
• Points in phase: points between two nodes oscillate in phase, i.e., they simultaneously oscillate in the same direction (with different amplitudes). Points on either side of a node oscillate with opposite phases, i.e., they simultaneously oscillate in opposite directions (with different amplitudes).
• Energy transmission: there is no energy transmission in the direction of propagation of the original waves since the transmission of the two waves travelling in opposite directions neutralises them.
• Wave speed: there is no net global wave speed since both waves travelling in opposite directions neutralise each other. We find that the speed of each point is specific and transversal to the direction of propagation of the original waves.

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).

Applications of stationary waves

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’.