Lecture 10: Waves

Waves are defined as repeated periodic disturbances in a medium, which can propagate energy and momentum from one location to another without the transfer of matter.
This definition applies to various types of waves, including:
- Disturbances in solids: These are represented by vibrations in a medium, like the vibrations of a guitar string when played, which create sound.
- Changes in pressure: For example, sound waves are a result of variations in air pressure, which travel through air or other gases.
- Motion of electric fields: Electromagnetic waves, including visible light, radio waves, and X-rays, consist of oscillating electric and magnetic fields, which propagate through space.

Traveling Waves: These waves propagate through space or a medium, transferring energy without the movement of matter along with it. An example is the ripples created on the surface of a pond when a stone is thrown in.
Mechanical Waves: These waves require a medium (solid, liquid, or gas) to propagate. Sound waves in the air are a primary example; they cannot travel in a vacuum as there are no molecules to transmit the sound energy.

Longitudinal Waves:
- In longitudinal waves, the disturbances occur parallel to the direction of wave travel.
- Example: Sound waves or P-waves during earthquakes, where compressions and rarefactions move along the same direction as the wave.
Transverse Waves:
- In transverse waves, disturbances occur perpendicular to the direction of wave travel.
- Example: Electromagnetic waves, such as light waves, or S-waves during earthquakes.
- Key characteristics include:
  - Crests: The highest points of the wave where energy is concentrated.
  - Troughs: The lowest points of the wave, representing the points of minimum energy.
  - Amplitude: The maximum height of the wave from rest position. A greater amplitude signifies greater energy transfer.

Wavelength ( $λ$ ): The spatial period of the wave, measured as the distance between two similar points on the wave (e.g., crest to crest or trough to trough).
Period: This is the time taken for one complete cycle of the wave, often measured in seconds. It is the reciprocal of frequency.
Phase: This defines the position within the wave cycle, expressed in degrees (e.g., 0° for start, 360° for complete cycle) or radians (where $2π$ radians corresponds to one full cycle).

Wave Equation: This fundamental equation links wave speed ( $v$ ), frequency ( $f$ ), and wavelength ( $λ$ ):
- $v = f imes λ$
- For instance, to find the wavelength of a radio wave at 96 MHz:
  1. Convert frequency: $96 ext{ MHz} = 96 imes 10^6 ext{ Hz}$
  2. Given speed of light in air: $3 imes 10^8 ext{ m/s}$
  3. Calculate wavelength:
    $λ = rac{v}{f} = rac{3 imes 10^8 ext{ m/s}}{96 imes 10^6 ext{ Hz}} = 3.125 ext{ m}$

When multiple waves pass through a point, the resulting disturbance is the sum of individual wave disturbances.
- Constructive Interference: This occurs when waves are in phase (peaks align with peaks), resulting in increased amplitude and greater energy at that point. On a diagram, this is shown as waves arriving at the same point simultaneously.
- Destructive Interference: This occurs when waves are out of phase (peak aligns with trough), leading to cancellation and zero disturbance at the point of overlap.

Every point on a wavefront can be considered a source of secondary waves, which spread out and superpose to form a new wavefront. This principle helps explain phenomena such as reflection, diffraction, and refraction of waves.

This classic experiment demonstrates wave interference and is conducted using coherent light sources to create overlapping light waves.
It produces alternating bright and dark fringes as a result of constructive and destructive interference between the waves.
The use of lasers is preferred for clarity and precision in the observed results.
Interference Pattern: The pattern of light and dark bands depends on wave coherence, frequency, and the path differences traveled by the light.
Fringe Spacing Equation: $d imes ext{sin}(θ) = nλ$ , where:
- $d$ : the distance between the slits
- $θ$ : the angle at which a particular fringe is observed
- $n$ : the order of the fringe
- $λ$ : the wavelength of the light used.

A diffraction grating is a device containing numerous closely spaced slits, used to produce sharp and bright interference fringes via constructive interference of light waves.
Due to the specific geometry, constructive interference occurs at distinct angles, leading to bright fringes that are observable in the pattern produced.

Example questions that can help demonstrate the application of wave theory in practical scenarios include:
1. Calculate Wave Speed: Given the wavelength and frequency of a wave, derive the speed.
2. Wave Behavior Analysis: Identify incorrect statements regarding various properties and behaviors of waves.
3. Interference Patterns Exploration: Analyze how changes in wavelength or distance between slits affect the spacing of maxima in interference patterns.