Topic 5: Waves and particle Nature of light

1. Particle movement in wave:

• particles to vibrate perpendicular to the direction energy is transferred
• The amplitude is the maximum displacement from the particle’s undisturbed position and the larger the amplitude is, the more energy is transferred.
• The distance between wave peaks is the wavelength (represented by λ and unit is meter)
• λ is the distance between two equivalent points in successive cycles.
• Figure 1 Wave terminology
• The frequency of a wave, is the number of cycles or vibrations per second.
  • Unit is hertz
  • Formula [math] f = \frac{1}{T} [/math]
  • Frequency Represented by f
  • T is represented by the period
• Period is the time for one complete cycle, measured in seconds.
• Since 1Hz is a very low frequency we also measure hertz in kHz (103Hz), MHz (106Hz) and GHz (109Hz).
• Figure 2 The period for one oscillation is 0.8 seconds

2. The wave equation:

• Different waves travel at very different speeds, but in each case the speed is calculated the same way using the equation
• [math] \text{Speed} = \frac{ \text{distance}}{\text{time}} \qquad (1) [/math]
• For a wave, the wavelength, λ, is the distance traveled in one cycle and the time to complete one cycle is the period, T. This means that:
• [math] \text{wave Speed} = \frac{\text{wavelength}} {\text{period}} \qquad (2) \\
  \text{speed of light(c)} = \frac{λ}{T} \qquad  (3) \\
  \text{speed of light(c)} = λ \frac{1}{T} \qquad   (4) [/math]
• The period is the time (in seconds) for one complete cycle
• The frequency,
• [math] f = \frac{1}{T} [/math]
• the equation4 becomes
• [math] \text{Speed of light (c)} = \text{Wavelegth} (λ) * \text{Frequency (f)} [/math]
• Now by units
• [math]  C( m.s^{-1}) = λ(m) . f(Hz) \qquad (5) [/math]

This is known as the wave equation

3. Longitudinal waves

• When the motion of the particles in a mechanical wave is back and forth along the direction of propagation, then create longitudinal waves.
• Longitudinal waves are characterized by compressions (high pressure) and rarefactions (low pressure) that propagate through the medium.
• The pressure variation is parallel to the direction of wave propagation.
• In a compression, the molecules are packed more tightly together, resulting in higher pressure.
• In a rarefaction, the molecules are spread out, resulting in lower pressure.
• Figure 3 Propagation of longitudinal wave
• A compression is a region where the particles are closest together.
• Displacement of Molecules:
  – Longitudinal waves cause the molecules to oscillate back and forth along the direction of wave propagation.
  – During a compression, the molecules are displaced towards the direction of wave propagation.
  – During a rarefaction, the molecules are displaced away from the direction of wave propagation.
  – The displacement of molecules is parallel to the direction of wave propagation.
• A rarefaction is a region where the particles are furthest apart.
• When sound travels through a solid, energy is transferred through intermolecular or inter-atomic bonds.
• Sound travels quickly through solids because the bonds are stiff and the atoms are close together.
• In gases the energy is transferred by molecules colliding.
• The speed of the sound depends on the speed of the molecules.
  – Longitudinal waves are also known as compressional waves or pressure waves.
  – The pressure variation and molecular displacement are in the same direction (parallel) in longitudinal waves.
  – Longitudinal waves can propagate through solids, liquids, and gases.
• Sound waves travel fastest in solids ([math]5100m^{-1} [/math] in aluminium)
• Less quickly in liquids (1500ms−1 in water).
• Even slower in gases (340ms−1 in air).

4. Transverse waves:

• Transverse waves are a type of wave where the displacement of the medium is perpendicular to the direction of wave propagation.
• Figure 3 Transverse waves
• Characteristics:
  – Displacement is perpendicular to wave propagation.
  – Wave propagation is perpendicular to the direction of displacement.
  – Can only propagate through solids (not liquids or gases).
  – Examples: light waves, water waves, seismic S-waves.
• Properties:
  – Wavelength (distance between successive peaks or troughs).
  – Frequency (number of oscillations per second).
  – Speed (dependent on medium properties).
  – Amplitude (maximum displacement from equilibrium)
• Types of Transverse Waves:
  – Electromagnetic waves (light, radio, etc.)
  – Mechanical waves (water, seismic, etc.)
  – Polarized waves (waves with restricted vibration directions)
• Differences from Longitudinal Waves:
  – Displacement direction (perpendicular vs parallel)
  – Medium requirements (solids only vs solids, liquids, and gases)

5. Draw and interpret graphs representing transverse and longitudinal waves:

⇒ Draw and interpret graph of transverse wave:

– Draw a sine or cosine curve to represent the wave.
– The horizontal axis represents the distance or position.
– The vertical axis represents the displacement or amplitude.
– The curve shows the displacement of the medium perpendicular to the direction of wave propagation.

Figure 4 Distance-displacement graph for transverse wave

⇒ Draw and interpret graph of longitudinal wave:

– Draw a series of compressions and rarefactions to represent the wave.
– The horizontal axis represents the distance or position.
– The vertical axis represents the pressure or density.
– The compressions are represented by the high-pressure regions, and the rarefactions are represented by the low-pressure regions.

Figure 5 Series of compressions and rarefaction with sine wave

⇒ Standing/Stationary waves:

– Draw a sine or cosine curve with nodes (points of zero displacement) and antinodes (points of maximum displacement).
– The nodes are represented by the points where the curve crosses the horizontal axis.
– The antinodes are represented by the peaks and troughs of the curve.
– The distance between two consecutive nodes or antinodes is half the wavelength.

Figure 6 Standing/stationary waves

6. CORE PRACTICAL 6: Determine the speed of sound in air using a 2-beam oscilloscope, signal generator, speaker and microphone.

• Equipment:
  – 2-beam oscilloscope
  – Signal generator
  – Speaker
  – Microphone
  – Ruler or measuring tape
  – Calculator
• Figure 7 experiment to measure the speed of sound
• Procedure:
  1. Set up the equipment:
     – Connect the signal generator to the speaker.
     – Connect the microphone to the oscilloscope (Channel 1).
     – Connect the signal generator to the oscilloscope (Channel 2).
  2. Generate a signal:
     – Set the signal generator to produce a sine wave (e.g., 1 kHz).
     – Adjust the amplitude to a suitable level.
  3. Measure the distance:
     – Place the speaker and microphone apart at a fixed distance (e.g., 50 cm).
     – Measure the distance accurately using a ruler or measuring tape.
  4. Display the waves:
     – Display the microphone signal (Channel 1) on the oscilloscope.
     – Display the signal generator signal (Channel 2) on the oscilloscope.
  5. Measure the time delay:
     – Measure the time delay between the two signals (Δt) using the oscilloscope.
  6. Calculate the speed of sound:
     – Use the formula:

     [math] \text{Speed of sound } (v) = \frac{\text{distance } (d)}{\text{time delay } (\Delta t)} [/math]

     – Plug in the values and calculate the speed of sound.

• Tips and Variations:
  – Use a longer distance for more accurate results.
  – Repeat the experiment with different frequencies.
  – Use a different type of wave (e.g., square wave).
  – Investigate the effect of temperature or humidity on the speed of sound.
• Remember to:
  – Take careful measurements and note the uncertainty.
  – Analyze the results and discuss any limitations or sources of error.
  – Compare your result with the accepted value of the speed of sound in air (approximately 343 m/s at room temperature and pressure).

7. Understanding of wavefront, coherence, path difference, superposition, interference and phase:

⇒Wavefront:

• A wavefront is the surface of a wave where the phase is constant. It’s the locus of points where the wave has the same phase.
• Think of it like a “wave layer” where all points have the same oscillation or vibration.
• Figure 8 Lens and wavefront rotation
• – Constant phase: All points on a wavefront have the same phase angle.
  – Wave propagation: Wavefronts move through space as the wave propagates.
  – Orthogonality: Wavefronts are perpendicular to the direction of wave propagation.
  – Shape: Wavefronts can be planar (flat), spherical, or more complex shapes.

⇒Coherence:

• Coherence refers to the ability of waves to maintain a fixed phase relationship over time and space. In other words, coherent waves have a constant phase difference between them.
• There are two types of coherence:
  1. Temporal coherence: The ability of a wave to maintain its phase over time.
     

      

     Figure 9 Temporal coherence of waves

  2. Spatial coherence: The ability of a wave to maintain its phase across different points in space.
• Coherence is crucial in various wave-related phenomena, such as:
  – Interference: Coherent waves can interfere constructively or destructively.
  – Diffraction: Coherent waves can produce diffraction patterns.
  – Optics: Coherence is essential for optical phenomena like laser operation and holography.
  – Acoustics: Coherence is important for sound wave propagation and acoustic imaging.

⇒Path difference:

• Path difference refers to the difference in distance traveled by two waves from their source to a point of observation.
• This difference can lead to phase differences and interference.
• Figure 10 Path difference between two waves
• – Definition: Path difference is the difference in distance traveled by two waves.
  – Symbol: [math] Δx (or Δl) [/math]
  – Units: Meters (or any length unit)
  – Calculation: [math] Δx = x_1 – x_2 [/math] (where [math] x_1 \,  \text{and} x_2 [/math]  are the distances traveled by the two waves)

⇒Superposition:

• Superposition is the combination of two or more waves overlapping in space and time. This leads to interference and can result in constructive or destructive interference.

  
  Figure 11 super position of waves

• Here are some key aspects of superposition:
  – Definition: Superposition is the addition of two or more waves.
  – Principle: The superposition principle states that the resulting wave is the sum of the individual waves.

⇒ Interference:

• Interference is the resulting wave pattern formed by the superposition of two or more waves. It can be constructive (reinforcing) or destructive (cancelling).
  – Definition: Interference is the resulting wave pattern due to superposition.
• Figure 12 Waves interference
• – Constructive interference: Waves add up to form a stronger wave.
  – Destructive interference: Waves cancel each other out.
  – Interference pattern: The resulting wave pattern due to interference.
  – Interference fringes: The alternating bright and dark bands in an interference pattern.

⇒ Phase:

• Phase refers to the angular relationship between two waves or a wave’s current state and a reference state (usually the wave’s initial state). It’s a fundamental concept in wave physics.
• Here are some key aspects of phase:
  – Definition: Phase is the angular difference between two waves or a wave’s current state and a reference state.
  – Units: Radians (or degrees)
  – Symbol: φ (phi)
  – Range: 0 to 2π (or 0 to 360 degrees)

8. Phase difference and path difference

• The relationship between phase difference and path difference is a fundamental concept in wave physics.
• Phase difference (Δφ) and path difference (Δx) are related by:
• [math] Δφ = (2π/λ) × Δx [/math]
• where λ is the wavelength of the wave.
• This equation shows that:
  – A path difference of one wavelength ( Δx = λ) corresponds to a phase difference of 2π radians (or 360 degrees).
  – A path difference of half a wavelength (Δx = λ/2) corresponds to a phase difference of π radians (or 180 degrees).
• Figure 14 Path difference
• Using this relationship, you can:
  – Calculate the phase difference from a given path difference.
  – Determine the path difference from a given phase difference.
  – Analyze wave interference and diffraction patterns.
• Some key applications include:
  – Interference experiments (e.g., double-slit experiment)
  – Diffraction gratings.
  – Optical interferometry.
  – Wave propagation in various media.

9. Standing wave

• Standing waves, sometimes called stationary waves, are created by the superposition of two progressive waves of equal frequency and amplitude moving in opposite directions.
• If two speakers connected to the same signal generator face each other, a standing wave will exist between them (Figure 15).
• Figure 15 Standing waves
• At P, the midpoint between the speakers, the waves, having travelled the same distance at the same speed, will always be in phase and interfere constructively.
• At A, [math] \frac{\lambda}{4} [/math] from P, the distance from speaker [math] S_1[/math] has increased by a quarter of a wavelength, while that from speaker [math] S_2[/math] has decreased by the same distance.
• The path difference [math] (S_1 A − S_2 A) [/math]is therefore half a wavelength, the waves are in antiphase and destructive interference takes place.
• Similarly at B, half a wavelength from P, the waves will be back in phase and produce a sound of maximum intensity.
• A person walking along the line between the speakers will detect a series of equally spaced maxima and minima along the standing wave.
• The points of zero amplitude within a standing wave are called nodes and the maxima are called antinodes.
• Figure 15 shows that the separation of adjacent nodes or antinodes is always half a wavelength.
• Standing waves differ from travelling waves in the following ways:
  • Standing waves store energy, whereas travelling waves transfer energy from one point to another.
  • The amplitude of standing waves varies from zero at the nodes to a maximum at the antinodes, but the amplitude of all the oscillations along a progressive wave is constant.
  • The oscillations are all in phase between nodes, but the phase varies continuously along a travelling wave.
• In general, for stringed instruments, the frequency is greater for:
  • shorter strings
  • strings with greater tension
  • strings that have a lower mass per unit length – that is, thinner strings of the same material or strings made from a lower density material.

⇒ Overtones and harmonics:

• the simplest standing wave has a single antinode at the midpoint.

  
  Figure 16 standing wave with node and anti-node

• The frequency of the note emitted from such a wave is called the fundamental frequency of the string.
• By plucking the string off center it is possible to create several standing waves on the same string.
• Figure 16 shows the fundamental mode and two other possible waves.
• The fundamental vibration has the longest wavelength (λ = 2l) and the others reduce in sequence.
• The notes emitted by vibrations other than the fundamental are called overtones. Overtones that have whole number multiples of the fundamental frequency are harmonics.
• The sounds we hear from a guitar, for example, are a complex mixture of harmonics and are noticeably different from the same tune played on a violin.
• The property that enables us to distinguish different musical instruments is the quality, or timbre, of the note.

10. the equation for the speed of a transverse wave

⇒Stringed instruments:

• Stringed instruments such as guitars, violins and pianos all produce standing waves on strings stretched between two points.
• When plucked, bowed or struck, the energy in the standing wave is transferred to the air around it and generates a sound.
• Because the string interacts with only a small region of air, the sound needs to be amplified – either by a resonating sound box or electronically.
• The principle of stringed instruments can be demonstrated using a sonometer (Figure 16.14).
• Figure 17 Sonometer
• When the string is plucked at its midpoint, the waves reflected from each end will interfere to set up a standing wave in the string.
• As both ends are fixed, they must be nodes, so the simplest standing wave will have one antinode between two nodes – that is, the length of the string will be half a wavelength.
• Using the expression [math] v = \sqrt{\frac{T}{\mu}} [/math] and the wave equation,[math] v = f \lambda [/math] , the frequency of the note emitted by the wire in this mode will be [math] f = v \lambda = \frac{1}{2l} \sqrt{\frac{T}{\mu}} [/math]

11. CORE PRACTICAL 7: Investigate the effects of length, tension and mass per unit length on the frequency of a vibrating string or wire.

• Aim:
  “Investigate how the length, tension, and mass per unit length of a vibrating string or wire affect its frequency”.
• variables:
  – Independent variables:

  – Length (L)
  – Tension (T)
  – Mass per unit length (μ)

• – Dependent variable:

  – Frequency (f)

• – Controlled variables:

  – String material
  – Diameter
  – Temperature

• Apparatus:
  – String or wire (e.g., guitar string)
  – Vibrating apparatus (e.g., guitar, violin, or a simple vibrating setup)
  – Ruler or meter stick
  – Tensioning device (e.g., tuning peg, screw)
  – Masses (e.g., weights)
  – Frequency measurement tool (e.g., oscilloscope, frequency counter)

  
  Figure 18 Investigation of frequency of vibrating string

• Procedure:
  1. Measure and record the initial length, tension, and mass per unit length of the string.
  2. Vibrate the string and measure its frequency using the frequency measurement tool.
  3. Change one variable (e.g., length, tension, or mass per unit length) and measure the new frequency.
  4. Repeat step 3 for different values of the variable, keeping the other variables constant.
  5. Plot graphs to show the relationship between each variable and frequency.
• Expected results:
  – Frequency increases with:

  – Increasing tension
  – Decreasing length
  – Decreasing mass per unit length

  – Frequency decreases with:

  -Decreasing tension
  – Increasing length
  – Increasing mass per unit length

• Theory:
• The frequency of a vibrating string or wire is given by:
• [math]f = \frac{1}{2L} \times \sqrt{\frac{T}{\mu}}
  [/math]
• where f is the frequency, L is the length, T is the tension, and μ is the mass per unit length.
• By analyzing this equation, we can see that:
  – Increasing tension increases frequency
  – Increasing length decreases frequency
  – Increasing mass per unit length decreases frequency
• This practical investigation helps you understand the relationships between these variables and how they affect the frequency of a vibrating string or wire.