Wave properties (Topic 5A)

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:

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).

A standing wave is a wave pattern that results when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other.

In a standing wave, points of zero displacement are called nodes and points of maximum displacement are called antinodes, and these are created at fixed intervals.

Unlike progressive waves, standing waves do not transfer energy along the medium; instead, they appear to stand still in place, oscillating in time at specific points. There is no wave propagation.

Standing waves commonly form in confined spaces, such as in a string, air column, or resonance tube, where waves reflect and interact.

The principle of superposition states that when two or more waves meet at a point, the resulting displacement is the sum of the individual displacements from each wave at that point.

This can lead to constructive interference (where waves add up to create a larger amplitude) or destructive interference (where waves cancel each other out).

In resonance tubes and musical instruments like organ pipes, superposition creates standing or stationary waves.

When sound waves travel back and forth in a confined space (like a tube), they reflect at the ends and meet other waves travelling in the opposite direction.

This superposition causes the wave pattern to ‘stand still’ with specific points, called nodes (where there’s no movement or oscillations) and antinodes (where movement or oscillation is greatest).

In an organ pipe, blowing into it creates a sound wave that reflects off the closed end, producing a stationary wave pattern within the pipe. Each pipe length produces a unique pitch based on its stationary wave pattern.

In experiments, two speakers connected in stereo to the same frequency generated by a signal generator create sound waves that superpose in the air. By walking directly in front of the speakers, one can experience interference firsthand.

• Constructive Interference: When waves meet in phase, their amplitudes add up, resulting in louder sounds.
• Destructive Interference: When waves meet out of phase, their amplitudes cancel, resulting in softer sounds or silence.

This alternating pattern of loud and quiet regions is a clear indication of superposition, as sound intensity varies depending on your position relative to the speakers. This effect is used in acoustics and sound engineering to manage sound placement in concert halls and theatres.

The principle of superposition can also be demonstrated with diffraction gratings:

1. They have many slits close together, which enhances the interference pattern by producing bright and sharp lines.
2. Light diffracts through each slit, causing multiple waves to overlap and superpose.
3. The distance between bright fringes in the pattern depends on the wavelength of the light.

The principle of superposition can be demonstrated with Young’s double-slit experiment:

1. Light from a single source passes through two narrow slits.
2. The two waves emerging from these slits then overlap, leading to interference on a screen.
3. The overlapping light waves create bright regions (constructive interference) and dark regions (destructive interference), forming a pattern of fringes.
4. Bright fringes occur where the waves reinforce each other, while dark fringes occur where they cancel out.

The principle of superposition applies to all waves, not just sound and visible light. including the full electromagnetic spectrum.

An example of this is in a microwave oven, where reflected microwaves superpose to form standing waves with fixed high-energy (antinodes) and low-energy (nodes) points. Antinodes heat food most effectively. Rotation is essential for even cooking, as stationary food would have cold spots (nodes).

To create superposition in controlled experiments, two microwave transmitters are used, each emitting a wave that meets in the same region. A microwave detector can then identify areas of constructive interference (high signal strength) and destructive interference (low signal strength).

A stationary wave can also be created using a single transmitter and a reflector. The microwave reflects back towards the transmitter, superposing with incoming waves and forming standing waves with nodes and antinodes.

When two or more waves occupy the same space at the same time, they combine to form a single resultant wave.

The displacement of the resultant wave at any point is the sum of the displacements of the individual waves at that point. This is known as the principle of superposition.

Graphically, superposition can be represented by plotting the individual waves and then drawing a new line for their resultant. For each point on the wave, add the displacements of the individual waves to create the black line, which shows the resulting wave. This technique visually demonstrates the principle of superposition.

• Destructive interference: occurs when the peaks of one wave align with the troughs of another wave, resulting in cancellation or reduction in amplitude. Complete destructive interference happens when two equal but opposite waves cancel each other out.

• Constructive interference: occurs when the peaks (crests) of one wave align with the peaks of another wave, and their amplitudes add up to create a larger amplitude.

Interference occurs when two or more waves of the same type meet in space, and their resultant displacement is the sum of the displacements of each wave. This follows the principle of superposition, where the combined effect depends on the relative phase and amplitude of the waves involved.

The resultant wave may have a larger or smaller amplitude than the individual waves.

Types of interference:

• Constructive interference: occurs when two waves with the same frequency and amplitude are in phase (their peaks and troughs line up). The amplitude of the resultant wave is doubled.
• Destructive interference: occurs when two waves are in anti-phase (peaks of one wave align with troughs of the other). This causes the waves to cancel each other out, resulting in a wave with zero amplitude.

Coherent waves have the same frequency and a constant phase difference. This means that the peaks and troughs of the waves consistently match up at regular intervals. When waves are coherent, they produce stable, observable interference patterns because their phase relationship remains constant over time.

Non-coherent waves do not maintain a constant phase relationship. If two waves are not coherent, their phase relationship fluctuates. This means they do not consistently reinforce or cancel each other, which prevents consistent interference patterns from forming.

Practical examples:

• Coherent sources: a laser is an example of a coherent light source, where light waves have the same frequency and fixed phase difference, producing clear interference patterns.
• Incoherent sources: light from filament lamps is incoherent and produces waves with random phase relationships, which disrupt stable interference.

Path difference is the difference in the distances travelled by two waves from their sources to a particular point where they meet.

Path difference is crucial in determining whether the waves will interfere constructively or destructively when they meet. It is often expressed in terms of wavelength.

When two waves from different sources travel and meet at a point, the difference in the distances they travel (path difference) dictates the phase relationship between them.

• Constructive Interference occurs if the path difference is an integer multiple of the wavelength This is because each full wavelength difference corresponds to being “back in phase.”

• Destructive Interference happens when the path difference is a half-integer multiple of the wavelength: This leads to the peaks of one wave aligning with the troughs of the other, causing cancellation.

Phase difference is the difference in phase angle (measured in degrees or radians) between two waves that meet at the same point.

A phase difference reflects how ‘in step’ or ‘out of step’ two waves are. If two waves have no phase difference (0° or 0 radians), they are perfectly in phase, meaning their peaks and troughs align.

Path difference translates directly to phase difference because each full wavelength of path difference corresponds to a 360° (or radians) phase difference.

Examples

• A path difference of corresponds to a phase difference of leading to constructive interference.

• A path difference of corresponds to a phase difference of leading to destructive interference.

In this way, the phase difference in radians between two waves due to a path difference in metres is given by:

where is the wavelength in metres.

Path difference is the difference in the distance travelled by two waves from their respective sources to a point of overlap. It is generally measured in metres or multiples of the wavelength of the waves.

Constructive interference occurs when the path difference between two waves is an integer multiple of the wavelength.

Mathematically, this is represented as:

where is the path difference in metres, is the wavelength in metres, and is an integer, e.g. 0, 1, 2, 3…

Destructive interference occurs when the path difference between two waves is an odd multiple of half the wavelength.

This is represented as:

where is an integer.

Phase difference refers to the angular difference between two waves as they reach a particular point. It is usually measured in radians or degrees.

When two points or waves have a phase difference of 360° (or radians), they are in phase. This means the crests and troughs are aligned so they happen simultaneously:

• When the phase difference is 180° (or radians), the waves are in anti-phase. This means the crest of one wave aligns with the trough of another.
• If the phase difference is any angle other than 0°, the waves are considered out of phase.

Constructive interference occurs when the phase difference between two waves is an even multiple of radians (180 degrees), meaning the waves are in phase:

Destructive interference occurs when the phase difference is an odd multiple of radians (180 degrees), meaning the waves are in anti-phase:

Where is the phase difference in radians and is an integer, e.g. 0, 1, 2…

Sound waves are longitudinal waves, meaning they consist of alternating compressions (regions of high pressure) and rarefactions (regions of low pressure).

When two sound sources emit the same frequency and are in phase, these compressions and rarefactions can align or misalign, leading to interference patterns that we perceive as changes in volume (louder or quieter sounds).

• Constructive interference occurs when two compressions or two rarefactions from the sound waves overlap. This overlapping increases the amplitude, which we perceive as a louder sound at points where compressions or rarefactions align.

• Destructive interference happens when a compression from one wave aligns with a rarefaction from the other wave, effectively cancelling out the pressure difference.
  This reduces the amplitude, resulting in a quieter sound or even silence at points where compressions and rarefactions cancel each other out.

An example of this is noise-cancelling headphones, which reduce unwanted sounds by emitting an anti-phase wave that cancels out the compressions and rarefactions of external sound waves.

Microwaves are a type of electromagnetic wave, which means they are transverse waves and consist of oscillating electric and magnetic fields.

In a two-source interference setup for microwaves, the wavefronts from two sources overlap, producing interference patterns that can be detected with specialised equipment.

A movable microwave detector measures interference patterns, typically from two sources or a single source passed through double slits. Moving the detector registers changes the signal amplitude, revealing regions of constructive and destructive interference.

• Constructive interference occurs where the waves from the two sources meet in phase, resulting in maximum amplitude detected by the receiver. This corresponds to points where the path difference is an integer multiple of wavelengths (e.g., ) and the detector registers a strong signal.

• Destructive interference occurs when the waves meet out of phase (with a path difference of half a wavelength, such as ) causing the waves to cancel each other. At these points, the detector picks up little or no signal, indicating a minimum in the interference pattern.

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.

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:

Spherical waves are waves originating from a point source that spread equally in all directions, forming spherical wavefronts.

As a spherical wave moves further from the source, it covers a larger area. This area expands according to the surface area of a sphere:

where:

• is the distance from the source in metres.

By substituting this formula for area into the equation for intensity, we can see that intensity is inversely proportional to the distance from the spherical wave source:

When the distance from a point source doubles, the intensity reduces to a quarter of its original value.