1. Superposition

• The displacement of the oscillations that occur when two waves of the same type overlap and meet at the same spot is equal to the vector sum of the displacements of the individual waves. The superposition of waves is the name given to this occurrence.
• Superposition of waves only occurs between the same types of waves.

Figure 1 Two same waves when combined at the same spot then superposition occurrence

• Figure 1 shows the superposition of two water waves in phase and antiphase, each with the same amplitude, A.
• When the waves are in phase, the resulting amplitudes are 2A.
• When the waves are in antiphase, they are zero.
• Water can move more when waves overlap in phase, but it can also stay still when waves overlap in antiphase.
• Unwanted noise is reduced by applying the concept of wave superposition.

Figure 2  How noise cancellation work

• The idea is that a microphone picks up unwanted sound from a nearby source and feels it into noise-canceling equipment.
• The circuitry cancels out the original sound and the newly added waveform by inverting the noise, which is equivalent to shifting its phase by .
• Wearing headphones that reduce loud noises, which have the potential to harm ears, can shield a worker from the sound.

2. Interference

• The superposition of waves from two coherent sources of waves is referred to as interference.
• When waves are in phase, interference is constructive.
• When they are in antiphase, or out of phase, it is destructive.
• Only when two wave sources are coherent will there be a stable pattern or superposition (or interference)
• A coherent light source is a light source that emits light with a high degree of coherence, meaning that the light waves have a consistent phase relationship with each other. This results in a concentrated, directional beam of light with a high intensity.

⇒ Interference patterns from sound waves

• A single sustained note from a signal generator played through two loudspeakers creates interference patterns.
• The loudness of the sound heard by a person walking in front of the speakers varies from loud to quiet on a regular spacing due to the pattern of constructive and destructive interference.
• The path difference is the difference in distance travelled by the two waves produced by the loud speakers. Path difference is usually measured in multiples of wavelength.
• If the waves are in phase when they leave the speakers, they are in phase at any point where they have travelled the same distance or where their path difference is a whole number of wavelengths, nλ. These sound waves, initially in phase.
• If the waves are out of phase at any point where the path difference is a whole number of wavelengths plus a half wavelength, (n + ½)λ.

Figure 3 Path Difference

⇒ Constructive interference

• When two or more waves overlap and combine to generate a new wave with an enhanced amplitude, this occurrence is known as constructive.
• When waves are oriented such that their crests (high points) and troughs (low points) coincide, they have the same frequency, wavelength, and phase.
• The resultant wave in constructive interference has an amplitude larger than the sum of the amplitudes of the component waves.
• This is because the wave crests are accumulating to form a new wave bigger at the peak than previous waves.

⇒ Destructive Interference

• Destructive interference is a phenomenon in which two or more waves overlap and combine to form a new wave with a reduced amplitude.
• This occurs when the waves have the same frequency, wavelength, and phase, but are aligned in such a way that their crests (high points) and troughs (low points) cancel each other out.
• In destructive interference, the resulting wave has an amplitude that is smaller than the individual amplitudes of the original waves.
• This is because the crests of one wave align with the troughs of another wave, creating a new wave with a lower peak than the original waves.

⇒ Young’s double slit experiment

• Young’s double slit experiment demonstrates interference between coherent light sources, thus showing the wave nature of light.
• The experiment uses two coherent sources of light waves, produced from a single source of light, which then pass through two very narrow, parallel slits.
• The light diffracts (spreads) through the slits, producing an interference pattern of fringes on a screen.
• Interference occurs because the waves overlap and superpose in a stable pattern.

Figure 4 Interference in a ripple tank

• Light has such a short wavelength it is difficult to see its interference patterns.
• This is why Young’s double slit experiment works best using a blacked-out room and a very bright white light source, or a laser as a source of intense single-wavelength (monochromatic) light.
• The slits must be very narrow and less than 1mm apart. An interference pattern can be seen where patches of bright light alternate with regions of darkness.
• These correspond to areas of constructive and destructive interference.
• These patterns are called fringes.
• The interference fringes are visible on a screen placed at least 1mm away from the slits. If the screen is further away, the fringe separation increases but their appearance becomes fainter.

Figure 5 Young’s double slit experiment

⇒ The condition for constructive interference (bright fringes) is

s.sinθ = nλ    (1)

Where
S = spacing between the slits
θ = angle from the beam towards the screen
n = a whole number
λ = wavelength of light

• The extra distance travelled by the waves leaving S₂ is s.sinθ, and for constructive interference this distance (or path difference) must be a whole number of wavelengths, nλ.

⇒ We can also express the separation of the fringes in terms of the angle θ. Referring again to Figure 5, we can see that

• where X is the distance to the nth fringe from the midpoint
• D is the distance from the slits to the screen.

However, since D>>X, the angle θ is small.

Then, the small angle approximation gives:

tanθ = sinθ        (3)

By equation 1

Compare with equation 3

Then compare equation 2 and 5

We can use this formula to predict the fringe spacing, w, between two adjacent bright fringes. When n = 1 the distance X becomes the spacing, w, between adjacent bright fringes. w=X, So equation 6 become

3. Stationary waves

• A stationary (or standing) wave is a wave formed by the superposition of two progressive waves of the same frequency and amplitude travelling in opposite directions.
• Oscillates in a fixed position, with nodes (points of zero amplitude) and antinodes (points of maximum amplitude) that do not move.

Figure 6 Stationary waves

• When waves reflect off a rigid boundary, the reflected waves are out of phase with the original wave by 180° and travelling in the opposite direction, but have the same amplitude and frequency.
• Stationary waves only form on a guitar string at specific frequencies.
• Where the waves are in phase, the displacements add to form a peak or a trough of double the original amplitude.
• Where the waves are in anti-phase, their displacements cancel out.
• For stationary waves, the positions of maximum and minimum amplitude remain in the same places, with particles, at antinodes, vibrating rapidly between their positions of maximum displacement. This is why the waves don’t appear to progress along the string.

⇒ Nodes

• Zero amplitude: Nodes are points where the wave amplitude is zero.
• No displacement: The wave does not displace the medium at nodes.
• Fixed points: Nodes are fixed in position and do not move.
• Half-wavelength apart: Nodes are typically half a wavelength apart.

⇒ Antinodes

• Maximum amplitude: Antinodes are points where the wave amplitude is maximum.
• Maximum displacement: The wave displaces the medium the most at antinodes.
• Fixed points: Antinodes are fixed in position and do not move.
• Half-wavelength apart: Antinodes are typically half a wavelength apart.

• Nodes and antinodes are always alternating in a stationary wave.
• The distance between two consecutive nodes or antinodes is half the wavelength.
• The phase of the wave changes by 180° between nodes and antinodes.

Visualizing nodes and antinodes can help you understand various phenomena, such as:

• Vibrating strings (musical instruments)
• Sound waves in pipes (acoustics)
• Electromagnetic waves (cavities, waveguides)
• Quantum mechanics (particle in a box)

⇒ How stationary waves form

• Figures 7 (a) to (d) show how stationary waves form on a guitar string.
• Wave 1 and wave 2 are progressive waves travelling in opposite directions along the string with the same frequency and amplitude.
• The amplitude of the wave 3 is the amplitude of the two waves superimposed.
• The diagrams show a sequence of snapshots of the wave at different times during one complete cycle.

Figure 7 Stationary waves form

⇒ Harmonics

• A harmonic is a mode of vibration that is a multiple of the first harmonic.
• The frequency of the vibrational in found using
• where f = frequency of the harmonic in Hz (Hertz)
• 
• v = speed of wave in ms^-1 (meter per second)
• λ = wavelength of harmonic in m (meter)
• The first harmonic on a string includes one antinode and two nodes.
• Stringed instruments have nodes at each end of the string as these points are fixed.
• For a guitar string of length l, the wavelength of the lowest harmonic is 2l.
• This is because there is one loop only of the stationary wave, which is a half wavelength. Therefore, the frequency is,

Figure 8 First stationary harmonic wave

• The second harmonic has three nodes and two antinodes; the wavelength is l and frequency is

Figure 9 Second stationary harmonic wave

• The third harmonic has four nodes and three antinodes; the wavelength is and frequency is:

Figure 10 Third stationary harmonic wave

⇒ The first harmonic on a string

• For waves travelling along a string in tension, the speed of a wave is given by

• where:
• T = tension in the string in N (newton)
• μ = mass per unit length of the string in kgm^-1 (kilogram per meter)
• The first harmonic for a stationary wave on a string is half a wavelength. The wavelength of the first harmonic is 2for a string of length .
• By equation 8

Now comparing the equation 11 and 12

So

This equation gives the frequency of the first harmonic

where: f = frequency in Hz

T = tension in the string in N

μ = mass per unit length of the string in kgm^-1

l= the length of the string length in m

⇒ Stationary waves: Sound waves and musical instruments

• Stationary wave diagrams for sound waves show the amplitude for particles vibrating longitudinally in the air column.
• The amplitude is greatest at the open end of pipes, where there is an antinode.
• The particles cannot vibrate at a closed end and so there is a node.
• If the pipe has two open ends, the stationary wave has at least two antinodes, at either end of the pipe.

Figure 11 Figure 12 Sound waves in open-ended tubes with stationary waves. An antinode is located at the open end and a node is located at the closed end when there is only one open end.

• Musical instruments include stringed instruments, where stationary waves form on the string when it is plucked or bowed.
• Stationary waves also form when the air column in organ pipes or wind instruments is forced to vibrate.
• The harmonic frequencies available to players will vary with different instruments, even if the air column is the same length, depending on whether the instrument has one or both ends open.
• Instrument design: Musical instrument design exploits stationary wave properties to optimize sound production. For example:
  • Guitar bodies are shaped to resonate with specific frequencies.
  • Violin soundposts and bridges transfer vibrations to the body.
  • Drumheads are tuned to specific frequencies.
  • Examples of musical instruments utilizing stationary sound waves.
  • String instruments: Guitars, violins, cellos, and more.
  • Wind instruments: Flutes, clarinets, and brass instruments use air columns to produce stationary waves.
  • Percussion instruments: Drums and marimbas rely on stationary waves in the drumhead or resonating chamber.

⇒ Stationary waves: microwaves

• Interactions between microwaves and other electromagnetic waves can result in stationary waves.
• A magnetron is the device used in microwave ovens to create microwaves.
• Microwaves are directed from the magnetron and reflect off metallic interior surfaces because they reflect off metals, which helps to guarantee that they distribute uniformly throughout the oven.
• Still, food still tends to form stagnant waves, which causes meals to be overcooked at the antinodes and undercooked at the nodes. That’s the reason the turntable on most microwave ovens rotates.
• Grated cheese or chocolate spread 6 cm apart on a platter and baked in a microwave oven without the turntable on.
• These locations match the first harmonic’s antinodes.
• We may infer from this data that the microwaves have a wavelength of around 12 cm.

4.Diffraction

Diffraction is a fundamental concept in physics that describes the bending of waves around obstacles or the spreading of waves through openings.
• It’s a crucial aspect of wave behavior, and it has many practical applications in various fields.

Figure 13 Diffraction of waves

ripple tank may be used to demonstrate diffraction, which is most noticeable when the wavelength is about the same as that of the obstruction or gap.
• Diffracted waves propagate into shadow areas surrounding the obstruction or gap, but their wavelength remains unchanged.

⇒ Diffraction of light

• Visible light has a relatively small wavelength (between 400 and 700*10^-9 m), diffraction only matters at very tiny slits.
• A fringe pattern appears as a bright centre fringe encircled by smaller, less brilliant fringes on either side of the narrow slit where light is diffracting. These fringes, also known as maxima, are the result of light’s constructive interference.
• The light’s destructive interference is the cause of the dark areas between (minima). The reason for the interference pattern is because light diffracting from one area of the slit overlaps and interferes with light diffracted from other areas of the slit.
• Destructive interference arises when there is a half-wavelength path difference between the top and bottom halves of the slit and light intensity is zero.
  • The width of the central diffraction maximum depends on the wavelength of light and the slit width.
  • At the edges of the central maxima, light leaving the top half of the slit interferes destructively with light leaving the bottom half, So

• Or the first minimum occurs at an angle given by
• 

light increases, the central maximum becomes wider because, as λ increases, λ/a = sinθ increases.
• As the slit width increases, the central maximum becomes narrower because, as a increases,  λ/a = sinθ decreases.

Figure 14 Diffraction fringes from a single slit

• A diffraction grating is a device that splits light into its constituent colors, creating a spectrum.
• It consists of a series of parallel slits or grooves on a surface, which cause diffraction and interference of light waves.
• The slits are very narrow so that the light diffracts through a wide angle.
• The pattern is a result of light overlapping (or superposing) and interfering from a great number of slits.

Figure 15 The path difference for a diffraction grating

• For the first order maxima:
• Interference is constructive
• The path difference of light from adjacent slits is λ
• Light travels at angle to the direction of incident light.
• Applying this information to the triangle abc,

• Where λ is the wavelength of light in m
• is the angle between the 0th and 1st order maxima
• d is the spacing between slits in m.
• For second order fringes, the path difference is 2λ and light travels at angle θ2 to the direction of incident light. The equation becomes:
• 
• In general, the condition for maxima to occur is given by
• 
• Where n is a whole number, and the order of the maximum.

⇒ Applications of diffraction gratings

• Spectroscopy: Analyzing the spectrum of light to determine the composition of a substance.
• Astronomy: Studying the spectrum of light from stars and other celestial objects.
• Lasers: Using diffraction gratings to select specific wavelengths for laser applications.
• Optical instruments: Using diffraction gratings in telescopes, microscopes, and other optical instruments.
• Color sorting: Sorting materials based on color using diffraction gratings.
• Holography: Creating holograms using diffraction gratings.
• Optical communication: Using diffraction gratings in fiber optic communication systems.
• Biomedical applications: Using diffraction gratings in biomedical research and medical devices.
• Material analysis: Analyzing the properties of materials using diffraction gratings.
• Quantum mechanics: Studying quantum phenomena using diffraction gratings.

  Some specific examples include:

  • CD and DVD players: Using diffraction gratings to read data from discs.
  • Spectrometers: Analyzing the spectrum of light to identify substances.
  • Telescopes: Using diffraction gratings to study the spectrum of light from distant objects.
  • Microscopes: Using diffraction gratings to study the properties of materials.
  • Laser pointers: Using diffraction gratings to select specific wavelengths for laser applications.