Comprehensive theory, key formulas, diagrams, and memory aids for Waves.
A wave is a disturbance that transfers energy from one point to another without the permanent transfer of matter. The particles of the medium oscillate about their equilibrium positions, passing energy along.
Waves are classified by their direction of oscillation relative to the direction of energy travel:
Waves are further classified as: - Mechanical waves: Require a medium to travel through (sound, water waves, seismic waves). - Electromagnetic waves: Can travel through a vacuum (light, radio, X-rays, gamma rays). - Progressive (travelling) waves: Transfer energy through space. - Stationary (standing) waves: Store energy; formed by superposition of two waves of the same frequency travelling in opposite directions.
| Quantity | Symbol | Definition | Unit |
|---|---|---|---|
| Amplitude | $A$ | Maximum displacement from equilibrium | m |
| Wavelength | $\lambda$ | Distance between successive identical points (crests, troughs) | m |
| Period | $T$ | Time for one complete oscillation | s |
| Frequency | $f$ | Number of complete oscillations per second | Hz |
| Wave speed | $v$ | Speed of energy transfer through the medium | m sโปยน |
The fundamental wave equation: $$v = f\lambda$$
This applies to all types of waves. For electromagnetic waves in a vacuum: $v = c = 3.00 \times 10^8$ m sโปยน.
Phase: Two points are in phase if they have the same displacement and velocity at all times (separated by a whole number of wavelengths). They are antiphase (180ยฐ or $\pi$ rad out of phase) if displaced equally but in opposite directions.
When two or more waves overlap, the total displacement at any point is the algebraic sum of the individual displacements. This is the Principle of Superposition.
When two waves superpose: - Constructive interference: Waves in phase (or path difference = $n\lambda$) โ amplitude increases โ maximum intensity. - Destructive interference: Waves exactly out of phase (or path difference = $(n+\frac{1}{2})\lambda$) โ amplitude decreases โ minimum intensity (zero if equal amplitudes).
For interference to produce a stable pattern, the two sources must be coherent โ they must emit waves of the same frequency and maintain a constant phase relationship. Lasers produce coherent light; ordinary bulbs do not.
graph TD
A[Two Coherent Wave Sources] --> B{Phase Relationship at Point}
B -->|In phase: path diff = nฮป| C[Constructive Interference: Bright/Loud]
B -->|Antiphase: path diff = n+ยฝ ฮป| D[Destructive Interference: Dark/Quiet]
Diffraction is the spreading of waves as they pass through a gap or around an obstacle. It is most pronounced when the gap width is comparable to the wavelength.
Single-slit diffraction of light produces a characteristic pattern: a bright central maximum that is twice as wide as the side maxima, with alternating dark minima on each side. The minima occur at angles where $d\sin\theta = n\lambda$ ($n = 1, 2, 3...$), where $d$ is slit width.
Thomas Young's experiment (1801) provided the first convincing evidence that light is a wave. Two coherent slits separated by distance $d$ produce an interference pattern on a screen at distance $D$:
$$\lambda = \frac{ay}{D}$$
Where: - $a$ = slit separation (m) - $y$ = fringe separation (distance between adjacent bright fringes, m) - $D$ = distance from slits to screen (m)
Bright fringes are equally spaced. The pattern shows that light undergoes constructive and destructive interference โ behaviour exclusive to waves.
A stationary wave forms when two waves of equal amplitude and frequency travel in opposite directions and superpose. Unlike progressive waves, stationary waves do not transfer energy.
Key features: - Nodes: Points of zero displacement (permanent destructive interference). Adjacent nodes are $\lambda/2$ apart. - Antinodes: Points of maximum displacement (permanent constructive interference). They occur midway between nodes. - All points between two adjacent nodes oscillate in phase with each other, but with different amplitudes. - Points in adjacent segments oscillate in antiphase.
The string's length $L$ must accommodate whole numbers of half-wavelengths: $L = \frac{n\lambda}{2}$
| Harmonic | Relationship | Frequency |
|---|---|---|
| Fundamental (1st) | $L = \lambda/2$ | $f_1 = v/2L$ |
| 2nd Harmonic | $L = \lambda$ | $f_2 = v/L = 2f_1$ |
| 3rd Harmonic | $L = 3\lambda/2$ | $f_3 = 3v/2L = 3f_1$ |
Reflection: Angle of incidence = angle of reflection (both measured from the normal).
Refraction: When a wave crosses a boundary between media, it changes speed, which changes its direction (unless it hits normal to the boundary). Snell's Law: $n_1\sin\theta_1 = n_2\sin\theta_2$. The refractive index $n = c/v$, where $v$ is the wave speed in the medium.
Total internal reflection occurs when light travels from a denser medium to a less dense medium at an angle exceeding the critical angle $\theta_c$: $$\sin\theta_c = \frac{n_2}{n_1} = \frac{1}{n}$$ (for $n_2 = 1$, air)
This is used in optical fibres and binoculars.
Polarisation is the restriction of transverse wave oscillations to a single plane. Only transverse waves can be polarised. Two polaroids at 90ยฐ to each other transmit zero intensity (complete extinction). Polarisation confirms the transverse nature of light.