Physics - waves

Path Difference for Constructive Interference

Constructive interference occurs when the path difference between two waves is an integer multiple of the wavelength (nλ)

Path Difference for Destructive Interference

Destructive interference occurs when the path difference between two waves is an odd multiple of half wavelengths ((n + ½)λ)

Condition for Coherence

Two sources are coherent if they emit waves with the same frequency and a constant phase difference

Importance of Coherence

Coherence is required to produce a stable and observable interference pattern

Progressive Wave

A wave in which energy is transferred from one place to another through oscillations of particles

Stationary Wave

A wave formed by the superposition of two waves of the same frequency travelling in opposite directions, with no net energy transfer

Amplitude in Progressive Waves

The amplitude is the same for all points in an ideal wave (ignoring energy losses)

Amplitude in Stationary Waves

The amplitude varies with position, being zero at nodes and maximum at antinodes

Frequency in Progressive Waves

All points oscillate at the same frequency, equal to the source frequency

Frequency in Stationary Waves

All points between nodes oscillate at the same frequency

Phase in Progressive Waves

Phase varies continuously with position along the wave

Phase in Stationary Waves

All points between adjacent nodes oscillate in phase, but points in neighbouring sections are in antiphase

Phase Difference in Progressive Waves

Depends on position along the wave and can take any value

Phase Difference in Stationary Waves

Zero between points in the same segment, π between adjacent segments

Energy Transfer in Progressive Waves

Energy is transferred through the medium

Energy Transfer in Stationary Waves

There is no net energy transfer along the wave

Polarisation

The restriction of oscillations of a transverse wave to one plane perpendicular to the direction of travel

Condition for Polarisation

Only transverse waves can be polarised because their oscillations are perpendicular to the direction of propagation

Effect of Polarisation

Reduces intensity of a wave depending on the alignment of the transmission axis

Unpolarised Wave

A wave with oscillations in multiple planes perpendicular to the direction of travel

Plane Polarised Wave

A wave with oscillations in only one plane

Fundamental Mode (First Harmonic)

The lowest frequency at which a stationary wave forms, with one antinode between two nodes (𝑓1​=v/2L​ OR f1 = [1/2L]x[root(t/u)])

First Harmonic Condition

The length of the system is equal to half a wavelength

General Harmonic Formula

The frequency of the nth harmonic is n times the fundamental frequency (λn​=2L​/n)

Node

A point where displacement is always zero due to destructive interference

Antinode

A point where displacement is maximum due to constructive interference

Number of Antinodes (nth Harmonic)

Equal to n

Number of Nodes (nth Harmonic)

Equal to n + 1

Node Spacing

The distance between adjacent nodes is half a wavelength

Antinode Spacing

The distance between adjacent antinodes is half a wavelength

General Pattern of Stationary Waves

Increasing harmonic number increases the number of nodes and antinodes and decreases wavelength

Energy Distribution in Stationary Waves

Energy is stored within segments between nodes and does not propagate along the medium

Formation Condition for Stationary Waves

Requires two waves of equal frequency and similar amplitude travelling in opposite directions

Phase Behaviour at Nodes

Particles on either side of a node oscillate in antiphase

Zero Displacement at Nodes

Caused by continuous destructive interference

Maximum Displacement at Antinodes

Caused by continuous constructive interference

Phase Difference and Path Difference Relationship

Phase difference is proportional to path difference, with a full wavelength corresponding to a phase difference of 2π radians

Phase Condition for Interference

Constructive interference occurs at phase difference of 2πn and destructive interference at (2n + 1)π radians

Path Difference

The difference in distance travelled by two waves from their sources to a point

Coherence

The property of waves where they maintain a fixed phase difference and have the same frequency, and wavelength

Define diffraction

The spreading out of waves as they pass through a gap or around an obstacle.

Under what condition is diffraction most significant?

When the wavelength of the wave is similar in size to the width of the gap or obstacle.

Effect on waves when the gap is much wider than the wavelength

Little to no noticeable diffraction; waves pass through largely unaffected.

Effect on waves when the gap is smaller than the wavelength

Most of the wave energy is reflected back toward the source.

Appearance of a monochromatic single slit diffraction pattern

A bright central maximum that is double the width of other fringes, flanked by alternating dark and bright fringes of decreasing intensity.

Cause of bright fringes in a single slit pattern

Constructive interference where waves arrive in phase.

Cause of dark fringes in a single slit pattern

Destructive interference where waves arrive completely out of phase.

The central maximum in a white light single slit diffraction pattern

A broad, bright white fringe where all wavelengths interfere constructively.

Appearance of non-central maxima in white light diffraction

They appear as spectra with violet closest to the central maximum and red furthest away.

Why red light appears further from the center than violet in white light diffraction

Red has a longer wavelength than violet, so it diffracts at a larger angle.

Effect of increasing slit width on the central maximum

The central maximum becomes narrower and its intensity increases.

Effect of increasing wavelength on the central maximum

The central maximum becomes wider and its intensity decreases.

Comparison of fringe spacing: Red vs. Blue laser

Red light produces wider fringe spacing because it has a longer wavelength and diffracts more.

Difference in fringe intensity: Double-slit vs. Single-slit

Double-slit fringes have roughly equal intensity, while single-slit fringes decrease rapidly in intensity moving away from the center.

The "Single-slit envelope" in double-slit interference

The single-slit diffraction pattern determines the overall intensity limits (the envelope) of the smaller double-slit interference fringes.