Stationary waves (4.4.4)

Waves are classified as either progressive or standing. Progressive waves transfer energy from one point to another, which is visible as a series of moving peaks. In contrast, standing waves do not transfer energy; their formation is a unique phenomenon that occurs when waves are spatially confined.

A stationary (standing) wave forms from the superposition (overlap) of two progressive waves. They can be formed using either transverse or longitudinal waves.

For this to occur, the progressive waves must satisfy three conditions:

1. Must travel in opposite directions
2. Have the same frequency
3. Have the same amplitude

The diagrams below show two progressive waves in red and blue travelling in opposite directions. The green line is the stationary wave formed as they superpose.

Stationary transverse waves can be formed by microwaves using the experimental arrangement shown below.

1. A microwave transmitter continuously fires microwaves of a constant frequency at a metal plate.
2. The microwaves reflect off the metal plate.
3. The microwaves travelling in either direction superpose to form a stationary microwave.

The intensity of the stationary wave is measured at different points by the microwave detector.

It is important to note that microwaves are used because their wavelengths are in the range to meaning the distances between points of maximum intensity (antinodes) and minimum intensity (nodes) of the stationary wave can be measured easily using a classroom ruler.

Stationary transverse waves can be formed on a stretched string with the experimental setup below.

1. A string is attached to a vibration generator and held taught by hanging masses over a pulley at the other end.
2. The vibration generator produces progressive waves travelling right that reflect from the movable bridge.
3. The progressive waves superpose to form a stationary wave.

It is important to note that changing the position of the movable bridge alters the effective length of the vibrating section of the string. This changes the allowed resonant frequencies and therefore the stationary-wave pattern that forms, i.e. which harmonics can be produced.

Stationary waves can be formed in air columns, such as those found in a pipe organ.

Sound waves are longitudinal waves, consisting of compressions and rarefactions of air particles.

Longitudinal waves superpose in the same way as transverse waves to form stationary waves.

1. In an organ pipe, air is blown in through the labium, creating a sound wave.
2. The sound wave reflects at the opposite end of the pipe.
3. The sound waves travelling in opposite directions superpose to form a stationary wave.

A stationary wave can be visualised by adding the amplitudes of two progressive waves as they move through each other.

Consider two progressive waves of frequency:

where is the time period.

1. At time the progressive waves perfectly overlap; they form a stationary wave with the largest maximum amplitude due to constructive interference.
2. A time later, the progressive waves mirror each other and completely cancel due to total destructive interference.
3. After a time later, the progressive waves overlap again, and the largest stationary wave is formed but shifted so it mirrors the initial stationary wave.
4. Finally, after a time the progressive waves cancel again.

Although progressive waves share several similarities with stationary waves, there are four key areas where they differ.

1. Energy transfer
2. Measurement of wavelength
3. Amplitude of each point
4. Phase difference between points

A progressive wave transfers energy in the direction of wave propagation, whereas a stationary wave has a net energy transfer of zero.

The wavelength of a progressive wave is the distance between two adjacent points oscillating in phase. For example, the wavelength of a progressive wave is equal to the distance between two peaks.

The wavelength of a stationary wave refers to the wavelengths of the progressive waves that form it.

A stationary wave is shown in the diagram above. The distance between two nodes (points of zero amplitude) is always equal to half a wavelength.

The distance between antinodes (points of maximum amplitude) is also equal to half a wavelength.

A complete wavelength is therefore twice the distance between two nodes (points of zero amplitude) or between two antinodes (points of greatest amplitude).

Any particles in a progressive wave oscillate with the same amplitude, although different particles reach their maximum displacement at different times.

The amplitude of a stationary wave varies along the wave:

• The maximum amplitude points are antinodes, and the zero amplitude points are the nodes.
• Points in between the antinodes and nodes have intermediate amplitudes decreasing from the maximum to zero.

Phase difference can be measured in degrees or radians and is often expressed as a multiple of

For a progressive wave, the phase difference between two points depends on their separation along the wave. Because the wave is travelling, any two arbitrary points on the wave generally oscillate with different phases, and the phase difference can take any value between zero and

Stationary waves behave differently because they are formed by the superposition of two identical waves travelling in opposite directions.

All particles in the same segment (between two adjacent nodes) reach their maxima, minima, and zero displacement at the same time. Therefore, they always oscillate in phase:

Whereas, two points on adjacent segments, on opposite sides of a node, reach their maxima and minima at opposite times. So, oscillate in antiphase:

When one side is at maximum positive displacement, the other side is at maximum negative displacement.

Since the progressive waves that form stationary waves have the same frequency, at some points they will be in antiphase and always cancel each other out.At other points, the progressive waves will always be in phase and superpose to the greatest amplitude.

• Nodes are points of zero amplitude on a stationary wave.
• Antinodes are points of maximum amplitude on a stationary wave.

The amplitude of oscillation of each point on the wave varies between zero at the nodes to the maximum amplitude at the antinodes.

• Any two points between adjacent nodes oscillate in phase with each other.
• Any two points on opposite sides of a node oscillate in antiphase with each other.

Stationary waves can only form on a stretched string if a whole number of half-wavelengths fits between the two endpoints.

This means only waves with certain wavelengths can form a stationary wave on a stretched string. Both endpoints are held in place, so they must be nodes.

• The longest wavelength of a stationary wave that can be formed by a stretched string is where is the length of the unstretched string, and has one antinode in between the ends.
• The second longest wavelength is which corresponds to two antinodes and one node in between the ends.

The wavelength of a stationary wave on a stretched spring is where is the number of antinodes. Therefore, the next longest wavelengths after are:

It is important to note that the wavelength of a stationary wave refers to the wavelength of the two progressive waves that form it.

Stationary waves generated by sound in a closed tube take the same shape as those on a string fixed at both ends.

However, stationary waves produced by sound at the open end of a tube have an antinode at the open end and a node at the closed end.

• The longest wavelength stationary wave that can be formed in an open tube is where is the length of the tube, and has no nodes or antinodes in between the ends.
• The second longest wavelength is which corresponds to one antinode and one node in between the ends.

The wavelength of a stationary wave in an open tube is where is the number of antinodes and must always be odd. The next longest wavelengths are:

The wavelength of a stationary wave refers to the wavelength of the two progressive waves that form it. It is twice the distance between two nodes or between two antinodes.

For a wave on a stretched string or a closed tube with only one antinode, the distance between two nodes is the distance between the endpoints, which is the length of the string. The wavelength is therefore twice the length of the string:

For a stationary wave on a stretched spring or a closed tube with two antinodes along its length, the distance between two nodes (or antinodes) is half the length of the string, so the wavelength is equal to the length of the string:

For a stationary wave formed in a tube with one open end with no nodes or antinodes along its length, the distance between the node at the closed end and the antinode at the open end is equal to the length of the tube, so the wavelength is four times the length of the tube:

Harmonics are the different stationary-wave patterns that a system can support, each with a specific wavelength and frequency determined by the boundary conditions.

The first harmonic is the fundamental mode of vibration, and the higher harmonics are multiples of this fundamental frequency, corresponding to patterns with more nodes and antinodes along the system:

• The lowest-frequency stationary wave is also the one with the longest wavelength.
• The fundamental mode of vibration is referred to as the first harmonic.
• The first harmonic of a stretched string of length has one antinode and a wavelength of The fundamental frequency is denoted

The second harmonic of a stretched string is a stationary wave with two antinodes.

The second harmonic has half the wavelength of the first harmonic, so it has double the fundamental frequency,

The third harmonic of a stretched string is a stationary wave with three antinodes.

The third harmonic has a third of the wavelength of the first harmonic, so it has triple the fundamental frequency,

The harmonic has a frequency of

When the string on an instrument is plucked, both the first harmonic and the higher harmonics interfere to produce the characteristic sound of the instrument.