Stationary Waves

What are stationary waves?

Michael Faraday discovered the stationary waves in 1831. Faraday observed these waves on the surface of a liquid present in a vibrating container. The common cause of why standing waves were formed is resonance. For example, whenever a vibrating rope is tied at one end, it will produce a standing wave. That’s why they are also called stationary waves.

Stationary waves are those waves that aren’t moving, which means that they are at rest all the time. These waves are formed when a couple of identical waves travel along the same path but in the opposite directions and interfere with each other, resulting in standing waves. The name stationary waves are also because of the disturbance, which is harmonic. It will not let the particles travel in any direction; therefore, there won’t be a transport of any kind of energy.

This article is a stationary waves study material that includes its basic definition, how their formation takes place, and its properties, to understand the concept from scratch.

Different characteristics of the stationary waves

• The disturbance, which takes place between the two points that are the starting and the reflecting point, is not confined to one particular region.
• Stationary waves tend not to move in any direction, and energy flow doesn’t occur either.
• The waves oscillate with respect to time, and their amplitude remains constant.
• An onward motion does not occur beyond a particular region from one particle to another.
• We know that the stationary wave has an incident and a reflected wave, and the total energy is twice the energy of these two.
• The medium in which the stationary waves are formed splits into several segments, and every section tends to vibrate. The particles present in a single, particular segment vibrate with the same phase.
• The pressure change in stationary waves is minimum at antinodes and maximum at nodes.
• Specific points are always at rest; these points are called nodes. There are other points where the amplitude is maximum, called antinodes.

Standing wave ratio: (SWR)

Standing Wave Ratio, or SWR of a wave, is the ratio of the amplitude at its maximum, which is the antinode, to the amplitude at its minimum, which is the node. For example, the pure stationary wave has an infinite standing wave ratio. An acceptable SWR is a non-zero value, and it says that the wave is in a condition where it is partially stationary and partially travelling.

A mathematical explanation of stationary waves

Let us consider a string of infinite length along the x-axis, which is free and can be stretched in the y-direction transversely.

In a harmonic wave travelling in the right direction, the displacement of the string along the y-direction is a function of the time t and position x.

y(x, t)−y0=Asin(2πt/T-2πxλ+ϕ)

When it travels in the left direction, the displacement along the y-direction is given by,

y(x, t)−y0=Asin(2πt/T+2πxλ+ϕ)

Here,

 y=amplitude that takes place due to the displacement of waves

 ω=angular frequency of the wave

 λ=wavelength of the wave

Examples to understand standing waves.

One of the most common uses of stationary waves is in musical instruments.

For example, in guitars, when fingers hold the strings in the form of chords, notes are created in the form of vibrations. Inside the musical instruments, stationary waves are produced through the air tube.

As kids, we have always played with ropes. Hence, understanding stationary waves with ropes is a classic example. Jump rope clearly explains a stationary wave. When the rope is shaken in sync, a regular pattern emerges in the waves. These waves oscillate in an up and down direction. The technicalities can also be understood here, the points where the rope comes to a still position that is nodes(N) and where the arc structure is formed(A), where the amplitude is maximum, that is the antinodes. In simple words, the points where the amplitude is zero are nodes and the points where the amplitude is maximum are antinodes.

Conclusion

Stationary waves, also known as standing waves, are the waves that are not moving, which means they are at rest. Two waves with the same frequency, amplitude and wavelength travelling in opposite directions are stationary. The common cause of why standing waves are formed is resonance. Due to interference, the waves moving in opposite directions have an oscillating wave fixed in space. 

The standing wave ratio of a wave is the ratio of the amplitude at its maximum, which is the antinode, to the amplitude at its minimum, which is the node. A jump rope can be considered a classic example of a standing wave.