Superposition (4.4.3)

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.

In a two-slit interference experiment, the waves from each slit travel slightly different distances to reach a point on the screen.

The wave from slit S2 has to travel further than the wave from slit S1 to reach the same point; the difference in these distances is the path difference.

This path difference determines whether the point on the screen will appear as a bright or dark fringe.

• If it is a bright fringe, this means constructive interference is occurring and the path difference is an even number of half wavelengths

• If it is a dark fringe, this means destructive interference is occurring and the path difference is an odd number of half wavelengths

Maxima and minima in an interference pattern:

• Maxima (bright fringes):

At positions where the path difference is an integer multiple of the wavelength, constructive interference occurs, creating a bright fringe.

These bright fringes are called maxima, and they are numbered by an order number , where is the central maximum, is the first order maximum on either side, and so on.

• Minima (dark fringes):

At positions where the path difference is a half-integer multiple of the wavelength, destructive interference occurs, creating a dark fringe.

These dark fringes, known as minima, are situated between the maxima on the interference pattern.

Young’s double-slit experiment is a classic demonstration of two-source interference using light waves. It shows how light, behaving as a wave, can produce an interference pattern.

1. A monochromatic light source shines through a single slit first, causing the light to diffract and spread out.
2. This light then passes through two narrow, closely spaced slits (A and B), which act as two coherent sources of light waves.
3. The light waves from each slit overlap and interfere as they travel to a screen, creating an interference pattern.

The interference of light waves from the two slits creates a pattern of bright and dark fringes on the screen.

• Bright fringes occur when the waves arrive in phase.
• Dark fringes occur where the waves arrive out of phase.

Because both slits are illuminated by the same initial light source, the light waves from slits A and B are coherent, having the same frequency and a constant phase difference. This coherence is essential for a stable, visible interference pattern.

In response to the limitations of Newton’s corpuscular theory of light, Christiaan Huygens developed the wave theory of light in the late 1600s.

According to Huygens, light behaves as a wave rather than as a stream of particles. So he introduced a concept now known as Huygens’ principle:

• Every point on a wavefront acts as a source of secondary wavelets that spread out in all directions at the same speed as the original wave.

• The new wavefront at any later time is found by drawing a surface tangential to these wavelets.

This principle enables the prediction of how waves propagate, including diffraction and refraction.

Advantages of the wave theory:

• Huygens’ wave model explained diffraction and interference, phenomena that Newton’s corpuscular theory could not account for.
• For example, Young’s double-slit experiment (1801) provided experimental confirmation of Huygens’ ideas, showing that light produces an interference pattern when passing through two slits; behaviour that is characteristic of waves.

The double-slit interference equation relates the distance between adjacent fringes (known as fringe width), the wavelength of the light, the distance between the slits, and the distance from the slits to the screen:

where:

• is the fringe width (the distance between two adjacent bright or dark fringes) in metres,
• is the wavelength of the light in metres,
• is the distance from the slits to the screen in metres, and
• is the distance between the two slits in metres.

A diagram is typically used to show the slit separation, the screen distance and the resulting interference fringes on the screen. This visual helps clarify the relationship between these variables in the experiment.

This equation is valid under the condition that slit separation, is much smaller than the screen distance, This condition ensures that the light waves from each slit reach the screen with a minimal change in angle.