Waves & Superposition

1. Progressive Waves

A progressive (or travelling) wave is a disturbance that transfers energy from one point to another without the transfer of matter. The particles of the medium oscillate about their fixed equilibrium positions.

There are two main types:

• Transverse Waves: The oscillations of the medium's particles are **perpendicular** to the direction of energy transfer.
• *Examples:* All electromagnetic waves (light, radio waves), ripples on water, waves on a string.
• A key property is polarization, where the oscillations are restricted to a single plane. Unpolarised light has oscillations in all planes perpendicular to the direction of propagation. Passing it through a polarising filter (polariser) aligns these oscillations into one plane. This is crucial evidence that light is a transverse wave.
• Longitudinal Waves: The oscillations of the medium's particles are **parallel** to the direction of energy transfer. They consist of a series of **compressions** (regions of high pressure) and **rarefactions** (regions of low pressure).
• *Examples:* Sound waves, ultrasound, p-waves in earthquakes.

#### Key Wave Parameters:

• Displacement (x): The distance of a point on the wave from its equilibrium position. SI unit: metre (m).
• Amplitude (A): The maximum displacement from the equilibrium position. SI unit: metre (m).
• Wavelength (λ): The shortest distance between two points in phase (e.g., from crest to crest). SI unit: metre (m).
• Period (T): The time taken for one complete oscillation of a point on the wave. SI unit: second (s).
• Frequency (f): The number of complete oscillations per unit time. **f = 1/T**. SI unit: hertz (Hz).
• Wave Speed (v): The speed at which energy is transferred. Governed by the **wave equation**: **v = fλ**.
• Intensity (I): The power transmitted per unit area perpendicular to the direction of energy transfer. For any wave, **Intensity ∝ Amplitude²**.

2. The Electromagnetic (EM) Spectrum

EM waves are transverse waves that do not require a medium to travel. They all travel at the speed of light in a vacuum, c ≈ 3.00 × 10⁸ m/s. The spectrum is arranged by increasing frequency (and decreasing wavelength):

Radio → Microwaves → Infrared → Visible Light → Ultraviolet → X-rays → Gamma rays

3. Superposition and Interference

The Principle of Superposition states that when two or more waves of the same type overlap at a point, the resultant displacement at that point is the vector sum of the individual displacements of the waves.

Interference is the effect produced by the superposition of waves. For interference to be observable, the sources must be coherent, meaning they emit waves with a constant phase difference and have the same frequency.

• Constructive Interference: Occurs when waves meet **in phase** (e.g., crest meets crest). The path difference between the waves is an integer multiple of the wavelength (nλ, where n = 0, 1, 2...). The resultant amplitude is the sum of individual amplitudes, leading to maximum intensity.
• Destructive Interference: Occurs when waves meet in **antiphase** (180° or π radians out of phase, e.g., crest meets trough). The path difference is an odd integer multiple of half a wavelength ((n+½)λ, where n = 0, 1, 2...). The resultant amplitude is the difference between individual amplitudes, leading to minimum (often zero) intensity.

*Practical Application:* Noise-cancelling headphones generate a sound wave in antiphase to ambient noise, causing destructive interference and cancelling it out.

4. Diffraction

Diffraction is the spreading of waves as they pass through a gap or around an obstacle. The effect is most significant when the size of the gap or obstacle is comparable to the wavelength of the wave (λ ≈ gap size).

5. Two-Source Interference: Young's Double-Slit Experiment

This experiment provides definitive evidence for the wave nature of light. A coherent light source (like a laser) illuminates two very narrow, closely spaced slits. The light diffracts at each slit, and the two sets of diffracted waves interfere. An interference pattern of alternating bright and dark fringes is observed on a screen.

• Bright fringes (maxima): Formed by constructive interference.
• Dark fringes (minima): Formed by destructive interference.

The fringe spacing (x) is given by the formula:

λ = ax/D

• λ = wavelength of the light (m)
• a = distance between the centres of the slits (m)
• x = distance between the centres of adjacent bright (or dark) fringes (m)
• D = distance from the slits to the screen (m)

*Exam Trap:* Ensure you use consistent units (usually metres). 'x' is the fringe *separation*, not the width of a single fringe.

6. The Diffraction Grating

A diffraction grating consists of a large number of equally spaced parallel slits. It produces much sharper and more widely spaced interference maxima than a double slit, making it ideal for measuring wavelengths accurately.

The condition for maximum intensity is given by:

d sinθ = nλ

• d = the grating spacing (distance between adjacent slits, m). If the grating has N lines per metre, then d = 1/N.
• θ = the angle of diffraction for a particular maximum.
• n = the order of the maximum (n = 0 is the central maximum, n = 1 is the first order, etc.).
• λ = wavelength of the light (m).

7. Stationary (Standing) Waves

A stationary wave is formed by the superposition of two progressive waves of the same frequency, amplitude, and speed, travelling in opposite directions.

Unlike progressive waves, stationary waves do not transfer energy; they store energy in oscillating segments.

• Nodes: Points of zero amplitude where there is always destructive interference.
• Antinodes: Points of maximum amplitude where there is constructive interference.

The distance between two adjacent nodes (or two adjacent antinodes) is λ/2. The distance between a node and an adjacent antinode is λ/4.

*Examples:*

• Stretched String: A guitar string fixed at both ends forms a stationary wave. The fixed ends must be nodes.
• Microwaves: Microwaves reflecting inside an oven form a stationary wave pattern. This is why turntables are used in older models to cook food evenly, moving it through the nodes and antinodes.
• Sound in Pipes: In wind instruments, like the flutes sold in the markets of Lahore, stationary sound waves are formed within the air column.