Stationary Waves

Stationary Waves

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If you think a, you might picture a snapshot of it hills and valleys. But waves aren't just still objects - they move through space and. This happens because all the points of the wave move together, which creates a collective effect. Each point moves in a specific way, and that's how the wave moves forward. We're only talking about periodic waves here, not wave packets. To really understand this, take a look at the picture below. And speaking of pictures, this brings us to stationary waves.

Periodic waves (left) and a wave packet (right)
Periodic waves (left) and a wave packet (right)

Let's take a closer look at periodic waves and wave packets. Periodic waves fill up all available space with amplitude, while wave packets are limited in space and change over time. Think of a wave packet as a single sea wave, while a periodic wave is like a never-ending set of regularly-produced sea waves. Both types of waves move and evolve. Now, let's talk about stationary waves. These are waves that form when two periodic waves with the same frequency travel in opposite directions and superimpose, or combine together. This leads to constructive interference, where waves combine to create a larger amplitude, and destructive interference, where waves combine to create a smaller amplitude.

See the image below for examples:

Constructive interference (left) and destructive interference (right)
Constructive interference (left) and destructive interference (right)

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

Let's talk about progressive waves and their main characteristics. These waves have a global amplitude, which means that eventually, all points will have the same maximum and minimum amplitude. Unlike stationary waves, there are no nodes in progressive waves - all points vibrate at some point in time. And all points have a relative phase between 0º and 360º. Energy is transferred in the direction of propagation, and all points of the same wave have the same speed. A classic example of a progressive wave is a rope wave - imagine one end of a rope is loose and you quickly pull the other end up and down. The wave moves forward until it reaches the end of the rope and stops, but if you keep pulling up and down, the wave will keep moving forward.

Stationary waves

Let's dive into the main characteristics of stationary waves. These waves have a local amplitude, which means that each point has specific maximum and minimum amplitude values depending on the amount of superposition. Unlike progressive waves, stationary waves have 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 between two nodes oscillate in phase, while points on either side of a node oscillate with opposite phases. 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. Therefore, there is no net global wave speed either. We can create a stationary wave in a rope by replicating movements from the other end. A common example involves the strings of a guitar, where pressing a string against the fret creates a reflection phenomenon that generates a wave travelling in the opposite direction. This wave, together with the wave travelling in the other direction, forms the stationary wave, with the end of the string and the contact with the fret acting as nodes.

Different stationary waves with different amounts of nodes

Applications of stationary waves

The main application of stationary waves is generating specific frequency sounds, which is why they play a crucial role in music and instruments. Due to interference patterns, not only a certain frequency but also related frequencies are generated as harmonics, thus creating a harmonic musical note. However, stationary waves are not limited to music and sound waves. They also play a fundamental role in the functioning of a microwave, where a stationary wave with a certain wavelength is generated between two walls. Evenly spaced burnt or melted spots in heated red liquorice correspond to the antinodes of the wave, where maximum energy transfer occurs. Stationary waves also play a crucial role in fundamental quantum systems, such as the distribution of an electron in a hydrogen atom, which is determined by a set of stationary wave dispositions called 'spherical harmonics'. In summary, a stationary wave is the result of the superposition of two waves with the same frequency travelling in opposite directions, and it has nodes and antinodes, with each point having specific maximum and minimum amplitudes. Stationary waves have various day-to-day applications, such as in music, microwaves, and quantum systems.

Stationary Waves

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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