Properties of Waves and Light

Water waves can transmit a large amount of energy. Examples include:
- Water waves observed at water parks, lakes, and oceans.
- Tsunami news reports.
Types of Waves:
- Mechanical Waves: Examples include vibrations on a string or sound from a ringing bell.
- Seismic Waves: Produced by earthquakes.
Definition of a Wave:
- A wave is a moving disturbance that transports energy from one place to another without necessarily transporting matter.
Periodic Waves:
- A simple wave, or periodic wave, repeats at regular intervals.
- Example: A mechanical wave can be generated by shaking a string (disturbance travels along the string).

Wave Front: The front edge of a wave.
Crest: The upper half of the wave.
Trough: The lower half of the wave.
Cycle: One complete crest and one complete trough.
Amplitude ( $A$ ): The extreme value (maximum or minimum) of the wave.
Wavelength ( $\lambda$ ):
- The distance between one positive amplitude and the next positive amplitude (or one negative to the next negative amplitude).
Phase:
- Two points on the wave that have the same cycle position (e.g., two successive crests); determined by the wave's offset from a reference point.
Period ( $T$ ): The time it takes for one wave cycle to pass a specific point.
Frequency ( $f$ ): The number of cycles that pass a specific point per unit time, measured in Hertz ( $Hz$ ).
Unit for Frequency: $1 \text{ Hz} = 1 \text{ cycle/second}$ .

Speed of a wave can be defined using the distance traveled during one cycle:
- $v = \frac{\Delta x}{\Delta t}$ where $\Delta x = \lambda$ and $\Delta t = T$ .
Speed Equation:
- $v = \frac{\lambda}{T}$
Relationships:
- $f = \frac{1}{T}$
- Also, $T = \frac{1}{f}$
- Substituting into the wave speed equation gives:
- $v = f\lambda$
Universal Wave Equation:
- $v = f\lambda$
- Indicates speed is directly proportional to both frequency and wavelength.

Light typically travels in straight lines, treated under the ray approximation, which models light propagation as straight rays.
Reflection: The change in direction upon encountering an obstacle.
Angle of Incidence ( $\theta_i$ ): Angle between incoming ray and the normal.
Angle of Reflection ( $\theta_r$ ): Angle between reflected ray and the normal.
Law of Reflection: States that the angle of incidence equals the angle of reflection: $\theta<em>i = \theta</em>r$ .
Specular Reflection: Reflection from a flat surface where all rays reflect in the same direction.
Diffuse Reflection: Reflection from a rough surface where rays reflect in multiple directions.

A wave transports energy, not matter.
The speed of a wave is determined by its wavelength and frequency via $v = f\lambda$ .
Reflection measured through angles of incidence and reflection against the normal.

Refraction: When light enters a transparent material (e.g., glass or water), some is reflected and some transmitted.
Refracted Ray: The part of light that passes through the medium.
Reflected Ray: The part of the light that bounces back from the surface.
Optical Density: Measures a medium’s ability to absorb electromagnetic wave energy; denser mediums slow wave speed.
Index of Refraction ( $n$ ): Ratio of light’s speed in vacuum ( $c$ ) to its speed in another medium ( $v$ ): $n = \frac{c}{v}$ .
If a ray passes from a medium with lower $n$ to a higher $n$ , it bends toward the normal. Conversely, it bends away when going from a higher $n$ to a lower $n$ .

n1 sin(θ1) = n2 sin(θ2)
θ₁ is the angle of incidence, and θ₂ is the angle of refraction.
Total Internal Reflection: Occurs when light moves from a denser to a less dense medium at angles greater than the critical angle.
This phenomenon occurs when light tries to pass from a more optically dense medium to a less dense one.
Critical Angle ( $\mu c$ ): The threshold angle for which total internal reflection occurs.
$\sin\left(\theta c\right)=\frac{n1}{n2}$
θ₍c₎ is the critical angle, where n_1 is the refractive index of the denser medium and n_2 is the refractive index of the less dense medium.

Fibre Optics: Utilizes total internal reflection to transport light signals efficiently across distances without significant loss.
Dispersion: The separation of light into its constituent wavelengths when passing through a medium, relevant to discussions on light diffraction and interference.

Diffraction: Bending of waves around obstacles or through openings.
Greater diffraction occurs when wavelengths are comparable to the size of the opening.
Interference: When waves overlap and combine, leading to regions of higher and lower intensity (constructive and destructive interference).

Constructive Interference: Occurs when waves are in phase, creating a larger amplitude.
Destructive Interference: Occurs when waves are out of phase, reducing or canceling each other's amplitude.

Produced in scenarios like water waves, sound waves, and light waves— visible patterns that can establish properties such as wavelength and frequency.
The presence of nodal lines, where destructive interference occurs, indicates regions of zero amplitude.
By analyzing the path length differences, you can utilize two-point-source interference patterns to derive formulas for calculating wavelengths:
- $\Delta s = d \sin(\theta)$ (where $\Delta s$ is path length difference, $d$ is slit separation, and $\theta$ is angle).
- E.g., for constructive interference, $d \sin(\theta) = m\lambda$ .
- E.g., for destructive interference, $d \sin(\theta) = (m + \frac{1}{2})\lambda$ .

Throughout history, light has been conceptualized both as a wave (Huygens) and a particle (Newton).
Huygens’ Principle: Every wave front is made up of secondary wavelets that propagate at the speed of the wave.
Newton's Particle Theory: Posited light travels as particles (corpuscles) without a required medium for propagation and could explain rectilinear propagation, but struggled with phenomena such as diffraction.

Experimental evidence showing light behaving in cases as a wave (in interference) or particle (photoelectric effect) led to the concept of wave-particle duality.

Young’s Double-Slit Experiment demonstrated that coherent light sources lead to observable patterns of alternating bright and dark fringes, confirming light’s wave properties.
Conditions for interference include coherent sources and proper slit spacing.
Separation between bright fringes can be calculated using $D = \frac{L\lambda}{d}$ where $D$ is fringe distance, $L$ is distance to screen, $\lambda$ is wavelength, and $d$ is slit separation.

Calculate Wavelength and Slit Separation: Use given data from the experimental setup to determine the properties of light through interference patterns.

Light exhibits both wave and particle characteristics, evident from experimental findings across multiple contexts, providing a robust foundation for the