This note covers all the wave‑related requirements of Section 7 of the Cambridge syllabus (9702). It links the seven fundamental wave quantities to energy transport, wave speed in different media, and the experimental techniques you will encounter in the exams.
| Term | Symbol | Definition / Typical Formula | SI Unit |
|---|---|---|---|
| Displacement | y (or s) | Instantaneous distance of a particle from its equilibrium position | metre (m) |
| Amplitude | A | Maximum magnitude of the displacement (|y| ≤ A) | metre (m) |
| Phase | ϕ | Argument of the sinusoidal function; ϕ = kx − ωt + ϕ₀ | radian (rad) or degree (°) |
| Phase Difference | Δϕ | Angular separation between two points: Δϕ = kΔx − ωΔt | rad (or °) |
| Period | T | Time for one complete oscillation at a fixed point | second (s) |
| Frequency | f | Number of cycles per second; f = 1/T | hertz (Hz = s⁻¹) |
| Wavelength | λ | Distance between successive points that are in phase (e.g. crest‑to‑crest) | metre (m) |
| Wave‑number | k | Spatial angular frequency; k = 2π/λ | rad m⁻¹ |
| Angular Frequency | ω | Temporal angular frequency; ω = 2πf = 2π/T | rad s⁻¹ |
| Wave Speed | v | Rate at which a given phase propagates; v = fλ = ω/k = λ/T | metre per second (m s⁻¹) |
| Intensity | I | Power transmitted per unit area; I = ½ ρ v ω² A² ∝ A² | watt per square metre (W m⁻²) |
| Wave Type | Particle Motion | Common Example | Typical Diagram |
|---|---|---|---|
| Transverse | Particle motion ⟂ to direction of propagation | String wave, water‑surface ripple | Figure 1 – a string with arrows up/down while the wave travels right |
| Longitudinal | Particle motion ‖ to direction of propagation | Sound in air, compression wave in a spring | Figure 2 – alternating compressions and rarefactions along the axis |
Both wave types can be described by the same sinusoidal function; only the physical meaning of the displacement variable changes (vertical displacement for transverse, longitudinal displacement for longitudinal).
General relation:
\$\Delta\phi = k\,\Delta x - \omega\,\Delta t\$
Example: Points separated by one wavelength (Δx = λ) at the same instant (Δt = 0) have Δϕ = kλ = 2π → they are in phase.
From the definition of wavelength (a 2π phase change):
\$k = \frac{2\pi}{\lambda}\qquad\text{[rad m⁻¹]}\$
From the definition of period (a 2π phase change in time):
\$\omega = \frac{2\pi}{T}=2\pi f\qquad\text{[rad s⁻¹]}\$
General relationship linking spatial and temporal characteristics:
\$v = f\lambda = \frac{\lambda}{T} = \frac{\omega}{k}\qquad\text{[m s⁻¹]}\$
Media‑specific formulas (required for AO2):
When source and/or observer move relative to the medium, the observed frequency \(f'\) changes:
\$f' = f\;\frac{v \pm v{o}}{v \mp v{s}}\$
where
Worked example (ambulance): A siren of \(f = 700\) Hz is heard at \(f' = 770\) Hz by a stationary observer as the ambulance approaches at 30 m s⁻¹. Using \(v = 340\) m s⁻¹, confirm the speed.
| Region | Typical Wavelength (λ) | Typical Frequency (f) | Common Uses |
|---|---|---|---|
| Radio | 10⁻¹ – 10³ m | 10⁶ – 10⁹ Hz | Broadcast, radar, communication |
| Microwave | 10⁻³ – 10⁻¹ m | 10⁹ – 10¹¹ Hz | Cooking, satellite links |
| Infra‑red | 7 × 10⁻⁷ – 10⁻³ m | 10¹¹ – 10¹⁴ Hz | Thermal imaging, remote sensing |
| Visible | 4 × 10⁻⁷ – 7 × 10⁻⁷ m | 4.3 × 10¹⁴ – 7.5 × 10¹⁴ Hz | Human eye, colour photography |
| Ultraviolet | 10⁻⁸ – 4 × 10⁻⁷ m | 10¹⁵ – 10¹⁶ Hz | Sterilisation, fluorescence |
| X‑ray | 10⁻¹¹ – 10⁻⁸ m | 10¹⁶ – 10¹⁹ Hz | Medical imaging, crystallography |
| Gamma‑ray | <10⁻¹¹ m | >10¹⁹ Hz | Radioactive decay, cancer treatment |
All electromagnetic waves travel at the same speed \(c\) in vacuum; the differences lie in wavelength and frequency, which determine their interaction with matter.
\$I = I_{0}\cos^{2}\theta\$
Practical example: placing two Polaroid sheets in the path of a laser beam. Rotating the second sheet changes the transmitted intensity according to the angle θ between their transmission axes.
\$I = \frac{1}{2}\,\rho\,v\,\omega^{2}A^{2}\qquad\text{[W m⁻²]}\$
Consequences:
| Apparatus | Wave Type | Quantities Measured | Typical Method |
|---|---|---|---|
| Ripple tank | Transverse water surface | λ (crest spacing), v (by timing a marker), A (probe displacement) | Strobe light or video analysis; measure distance travelled in a known time. |
| String‑pulse set‑up | Transverse on a stretched string | v (distance/time), T (period of a standing‑wave pattern) | Photogate or high‑speed camera; count pulses. |
| Closed air column (organ pipe) | Longitudinal sound | f (tuning fork), λ (resonant length), v (via v = fλ) | Microphone + CRO; adjust column length until resonance. |
| Cathode‑ray oscilloscope (CRO) | Electrical analogue of any periodic wave | f, T, A (voltage amplitude) | Read time/div and volts/div settings; use cursors for accuracy. |
| Polaroid set‑up | Polarised light (EM wave) | I as a function of angle θ | Rotate second Polaroid, record intensity with a light‑meter. |
| Doppler‑shift apparatus | Sound | Observed frequency f′ for known source speed | Speaker on a cart, microphone + CRO; compare f′ with f. |
Design skill (AO2): You should be able to sketch a simple set‑up, explain which two of \(f, \lambda, v\) you will measure directly, and show how to calculate the third using the appropriate formula.
\$v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{120}{2.5\times10^{-3}}}\approx 2.19\times10^{2}\ \text{m s}^{-1}\$
\$\lambda = \frac{v}{f} = \frac{2.19\times10^{2}}{80}\approx 2.74\ \text{m}\$
\(\omega = 2\pi f = 2\pi(80)=5.03\times10^{2}\ \text{rad s}^{-1}\)
Convert amplitude: \(A = 1.2\ \text{mm}=1.2\times10^{-3}\ \text{m}\).
\$\$I = \frac12\,\rho\,v\,\omega^{2}A^{2}
=\frac12(1.2)(2.19\times10^{2})(5.03\times10^{2})^{2}(1.2\times10^{-3})^{2}
\approx 1.1\ \text{W m}^{-2}\$\$
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