Progressive and stationary waves (3.3.1)

The displacement of a point on a wave is its distance from the equilibrium (resting) position.

Displacement is a vector quantity, meaning it has both magnitude and direction. It can be positive or negative, depending on whether the point is above or below the equilibrium position.

Displacement is measured in metres ().

The amplitude is the maximum displacement of a point on a wave from its equilibrium position.

The amplitude indicates the energy of the wave; higher amplitudes correspond to waves carrying more energy.

Amplitude is measured in metres ().

The wavelength is the distance between two similar points on successive oscillations of a wave.

These points can be peak to peak, trough to trough, or any other point as long as the distance is measured from one point to the same part of the next wave. It is essentially the length of one full wave cycle.

Wavelength is measured in metres ().

The period of a wave is the time taken for one complete oscillation (or cycle). Time period is measured in seconds ().
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The frequency of a wave is the number of complete oscillations or wavelengths that pass a point per unit time. Frequency is measured in Hertz (), which are equivalent to ().

Frequency and time period are reciprocals of each other. They are related by:

The phase difference between particles oscillating in the same wave or between two particles in different waves can be described in one of three ways:

• Particles oscillating perfectly in sync (reaching maximum positive displacement simultaneously) are in phase, with a zero phase difference. Separated by one wavelength, the phase difference is radians); two wavelengths, radians), and so on.

• Particles oscillating oppositely (one at max positive, the other at max negative) are in antiphase, with a phase difference of ( radians).

• If the phase difference is any angle other than the waves are considered out of phase.

An oscilloscope is an instrument used to display and analyse the waveforms of electrical signals. It can be used as a DC or AC voltmeter.

An oscilloscope will typically show:

• Time on the X axis (called the time-base), usually in units of milliseconds per division (ms div⁻¹), where each line corresponds to a millisecond.
• Voltage on the Y axis, representing the amplitude of the wave.

Oscilloscope readings are often used to calculate the frequency of a wave. To do this:

1. Determine the period by reading the time for one complete wave cycle from the time-base setting.
2. Convert the period into seconds.
3. Use the relationship to calculate the frequency.

When dealing with wave graphs, be careful to read the axes labels, as the type of axis (time or distance) will determine what parameters can be interpreted from the graph.

A wave on a string might be represented by a distance–time graph, whereas oscilloscopes plot the voltage of an electrical signal against time.

For a constant wave speed, wavelength and frequency have an inversely proportional relationship:

Therefore (for a constant wave speed):

• Longer wavelengths: fewer wave cycles pass a point per second, resulting in a lower frequency.
• Shorter wavelengths: more wave cycles pass a point per second, resulting in a higher frequency.

There are two types of mechanical waves:

• transverse waves
• longitudinal waves.

The type of wave depends on the direction of the particle oscillations in relation to the direction of the wave propagation:

• Transverse: oscillations are perpendicular to the wave propagation.
• Longitudinal: oscillations are parallel to the wave propagation.

To represent transverse waves on a graph, we need two axes: one for the wave direction (X axis) and one for the displacement of the particles (Y axis). This is because transverse waves involve vibrations of particles that are perpendicular to the direction of energy transfer.

You should be able to label the significant features of the transverse wave:

• Peaks/crests: the points of maximum positive displacement
• Troughs: the points of maximum negative displacement
• Amplitude: the maximum displacement from the undisturbed state (zero on the Y axis).

Longitudinal waves have particle displacement in the same direction as wave travel.

A longitudinal wave can be plotted graphically. The Y axis can represent longitudinal particle displacement. The displacement of a particle at each position along the X axis from its equilibrium position.

This will produce a graph that looks like a transverse wave. However, as the axis explicitly states that it represents longitudinal displacement, the particles are still vibrating parallel to the direction of the wave.

The Y axis may also represent another parameter describing the wave, such as pressure for a sound wave.

You should be able to label the significant features of the longitudinal wave:

• Compressions: areas of high pressure where the particles are clustered closer together (think: the particles are compressed)
• Rarefactions: areas of low pressure where the particles are spread further apart (think: the particles are rarer in this area).

All EM waves share similar properties:

• Transverse waves: The vibrations of the wave (the electric and magnetic fields) occur perpendicular to the wave’s direction of travel.
• Travel in vacuum: Unlike sound waves (which require a medium such as air or water), EM waves can travel through the emptiness of space.

• Speed in vacuum: All EM waves travel at the speed of light in a vacuum. It is the fastest possible speed.
• Speed in air: The speed of EM waves is slightly slower in air than in a vacuum, but for most purposes it is still considered to be
• In denser materials (e.g. water, glass), the speed of EM waves decreases due to interaction with the material’s particles.

Polarisation occurs only in transverse waves, with vibrations perpendicular to the direction of propagation.

When a wave is polarised, its vibrations are restricted to one plane perpendicular to its direction of travel.

Longitudinal waves cannot be polarised because their vibrations occur parallel to the direction of propagation – they are already oscillating in one plane.

In electromagnetic waves, by convention, polarisation refers to the direction of the electric field oscillation rather than the magnetic field.

For example, if an electromagnetic wave is vertically polarised, its electric field oscillates in the vertical plane, even if the magnetic field oscillates horizontally.

Unpolarised light, like that from the sun or a lightbulb, has vibrations that occur in all directions perpendicular to the direction of wave travel.

To convert unpolarised light into polarised light, it must pass through a polarising filter, which restricts oscillations to a single plane, creating plane-polarised waves.

• If unpolarised light passes through a filter, only the component aligned with the filter’s transmission axis will pass through, blocking all other directions.

• If polarised light encounters a filter with a perpendicular transmission axis, no light passes through because the filter blocks the aligned oscillations.

Polarisation can also occur through reflection, refraction, and scattering.

Microwaves are polarised using a metal grille instead of a filter. A metal grille polariser works differently from a polarising filter:

• The free electrons in the metal bars of the grille align and move to block any electric field component parallel to the grille bars. This only allows the perpendicular electric field component to pass through.
• This is opposite to the way a polarising filter works: A polarising filter restricts waves from oscillating in the direction aligned with its transmission axis.

This setup is particularly useful in physics labs to study polarisation principles since microwaves have longer wavelengths compared to visible light , so high precision equipment is not required – the metal grilles have large gaps and are easily constructed.

The effect of polarisation on light intensity can be investigated by placing two polarising filters, A and B, one after the other:

Filter A polarises the initially unpolarised light in a specific direction, allowing only the light oscillations along its transmission axis to pass:

• If Filter B has its transmission axis aligned parallel to Filter A, it will allow all the polarised light from A to pass through.
• In this parallel arrangement, the transmitted polarised light is as at its maximum intensity.

Rotating Filter B gradually reduces the amount of light that can pass through, lowering the intensity:

• When the transmission axes are perpendicular, Filter B blocks all the light polarised by Filter A, resulting in zero transmitted intensity.
• This changing alignment means the intensity varies periodically based on the rotation angle of B.

The intensity of light shining through two polarisers shows a sinusoidal pattern. The graph below shows the transmitted light intensity as a function of the rotation angle of the second filter, assuming the first filter is held at a fixed angle.

Intensity is maximum when the filters are aligned or and minimum when they are perpendicular or

Radio and television services are transmitted using either horizontally polarised or vertically polarised signals.

The orientation of the reception aerial is crucial for optimal signal reception:

• Flat (horizontal) mounting is used for receiving horizontally-polarised signals.
• Vertical mounting is required for vertically-polarised signals.

The correct orientation of the aerial ensures that it is aligned with the plane of polarisation of the incoming waves, allowing for maximum signal strength.

Polarisation is a process by which the vibrations of a transverse wave are restricted to a single direction.

The polarisation direction is always perpendicular to the direction of wave propagation.

Only transverse waves can be polarised because their vibrations naturally occur in directions perpendicular to their travel.

Longitudinal waves cannot be polarised as their vibrations occur in the same direction as the wave travels, and so the vibrations are already restricted to one direction.

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.