The Wave Model is the foundation for the entire C-block of IB Physics HL. Once you can confidently use $v = f\lambda$, distinguish transverse from longitudinal waves, and read both displacement-time and displacement-position graphs, the rest of waves (Topic C.3 wave phenomena, C.4 standing waves, C.5 Doppler) becomes a matter of applying the same core ideas in new contexts.
This cheatsheet condenses the full Topic C.2 syllabus — wave quantities, the wave equation, mechanical vs electromagnetic waves, sound, the EM spectrum, and graph interpretation — into one page you can revise from. The classic IB traps (frequency at a boundary, particle vs wave speed, polarisation) are flagged in red. Scroll to the bottom for the printable PDF and the full Photon Academy notes library.
§1 — Wave Properties & Quantities C.2 SL + HL
Core wave equation
Wave equation:$$v = f\lambda = \dfrac{\lambda}{T}$$
Here $v$ is wave speed (m s$^{-1}$), $f$ is frequency (Hz), $\lambda$ is wavelength (m), and $T = 1/f$ is the period (s).
Key quantities
| Quantity | Definition |
|---|---|
| Amplitude $A$ | Maximum displacement from equilibrium. |
| Period $T$ | Time for one complete oscillation. |
| Frequency $f$ | Number of cycles per second; $f = 1/T$. |
| Wavelength $\lambda$ | Distance between two adjacent in-phase points. |
| Wave speed $v$ | Speed at which the wave pattern (energy) travels. |
TrickIn a given medium $v$ is fixed. If $f$ doubles, $\lambda$ halves. The wave speed does not change just because the source amplitude or frequency changes.
Trap$v$ is the speed of the wave pattern, NOT the speed of the particles. Particles oscillate about equilibrium; they do not travel with the wave.
NoteIntensity $I \propto A^2$. Doubling the amplitude multiplies the intensity by four. For a point source: $I \propto 1/r^2$.
§2 — Transverse vs Longitudinal Waves C.2 SL + HL
Transverse waves
- Particle motion is perpendicular to the direction of energy transfer.
- Can be polarised.
- Show crests and troughs.
- Examples: light, all EM waves, waves on a string, water surface waves.
Longitudinal waves
- Particle motion is parallel to the direction of energy transfer.
- Cannot be polarised.
- Show compressions and rarefactions.
- Examples: sound, seismic P-waves, slinky compressions.
Compressions & rarefactions
- Compression: particles closer than normal (high pressure).
- Rarefaction: particles further apart (low pressure).
- Distance between adjacent compressions $= \lambda$.
- On a displacement–position graph: C $=$ zero displacement with negative gradient; R $=$ zero displacement with positive gradient.
TrapOnly transverse waves can be polarised. If a wave is polarised, it must be transverse. Longitudinal waves oscillate parallel to propagation — there is no perpendicular plane to restrict.
§3 — Sound Waves C.2 SL + HL
Nature of sound
- Sound is mechanical and longitudinal.
- Requires a material medium — cannot travel in vacuum.
- Propagates as pressure variations.
- Speed in air at 20 °C $\approx 340\ \mathrm{m\,s^{-1}}$.
- Faster in liquids and solids than in gases.
- $f$ determines pitch; $A$ determines loudness.
- Audible range $\approx 20\ \mathrm{Hz} - 20\ \mathrm{kHz}$.
Approximate sound speeds
| Medium | Speed of sound |
|---|---|
| Air (20 °C) | $\approx 340\ \mathrm{m\,s^{-1}}$ |
| Water | $\approx 1500\ \mathrm{m\,s^{-1}}$ |
| Steel | $\approx 5000\ \mathrm{m\,s^{-1}}$ |
Sound is faster in solids and liquids than in gases because the stronger intermolecular forces transmit the pressure disturbance more quickly.
TrickBell-jar experiment: as air is removed, sound fades to zero, proving sound needs a medium. Light from the bulb persists, proving EM waves do not need a medium.
§4 — Electromagnetic Waves & the EM Spectrum C.2 SL + HL
Nature of EM waves
- Transverse: oscillating $\vec{E}$ and $\vec{B}$ fields, perpendicular to each other and to the direction of propagation.
- Do not need a medium — travel through vacuum.
- All EM waves travel at $c = 3.00\times 10^8\ \mathrm{m\,s^{-1}}$ in vacuum.
- Obey $c = f\lambda$ in vacuum.
- Slow down (and wavelength shortens) in a medium, but frequency is unchanged.
The EM spectrum (low $f$ → high $f$)
| Region | Typical wavelength |
|---|---|
| Radio | $\lambda \sim 1\ \mathrm{m} - 1\ \mathrm{km}$ |
| Microwave | $\lambda \sim 1\ \mathrm{cm}$ |
| Infrared | $\lambda \sim 10\ \mathrm{\mu m}$ |
| Visible | $\lambda \sim 400 - 700\ \mathrm{nm}$ |
| Ultraviolet | $\lambda \sim 100\ \mathrm{nm}$ |
| X-ray | $\lambda \sim 0.1\ \mathrm{nm}$ |
| Gamma | $\lambda < 0.01\ \mathrm{nm}$ |
TrapWhen an EM wave enters a medium: speed decreases, wavelength decreases, but frequency stays the same. Frequency is set by the source and never changes at a boundary.
§5 — Mechanical vs Electromagnetic Waves C.2 SL + HL
| Property | Mechanical | Electromagnetic |
|---|---|---|
| Medium needed? | Yes | No |
| Wave type | Transverse OR longitudinal | Transverse only |
| Speed in vacuum | Does not propagate | $c = 3.00 \times 10^8\ \mathrm{m\,s^{-1}}$ |
| Speed depends on | Medium (density, elasticity) | Medium ($v < c$ in matter) |
| Polarisable? | Only if transverse | Yes (all EM waves) |
| Examples | Sound (L), strings (T), water (T) | Light, radio, X-rays, gamma |
NoteAll travelling waves — mechanical or EM — transfer energy without any net transfer of matter. Intensity $I = P/A$; for an isotropic point source $I \propto 1/r^2$.
§6 — Graph Interpretation C.2 SL + HL
Displacement–position graph ($y$ vs $x$)
- Snapshot of the whole wave at one instant $t$.
- Horizontal axis in metres $\Rightarrow$ read off wavelength $\lambda$.
- Vertical axis $\Rightarrow$ read off amplitude $A$.
- If the wave moves to the right, each point will next move in the direction of the point behind it.
Displacement–time graph ($y$ vs $t$)
- Motion of one particle over time.
- Horizontal axis in seconds $\Rightarrow$ read off period $T$.
- Vertical axis $\Rightarrow$ read off amplitude $A$.
- $f = 1/T$, then use $v = f\lambda$ if a wavelength is given.
TrickDistinguish the two graphs by the units on the horizontal axis. Metres → $\lambda$; seconds → $T$. A typical Paper 1 question gives one type and asks you to read values from the other — never mix them up.
§7 — Exam Attack Plan All sections
When you see this in the question — reach for that:
| Question trigger | Reach for |
|---|---|
| "Find $\lambda$, $f$ or $v$" | $v = f\lambda$ in the relevant medium. |
| "Wave enters a denser medium" | $v$ ↓, $\lambda$ ↓, $f$ unchanged. |
| "Can the wave be polarised?" | Yes only if transverse. |
| "Sound in a vacuum?" | No — sound is mechanical and needs a medium. |
| "Doubling the amplitude — what happens to intensity?" | $I \propto A^2 \Rightarrow$ × 4. |
| "Speed of light in a vacuum" | $c = 3.00 \times 10^8\ \mathrm{m\,s^{-1}}$. |
| $y$–$x$ graph reading | Read $\lambda$ from horizontal axis. |
| $y$–$t$ graph reading | Read $T$ from horizontal axis, then $f = 1/T$. |
| "Order of EM spectrum" | Radio < microwave < infrared < visible < UV < X-ray < gamma. |
| "Particle speed vs wave speed" | Particles oscillate, do not travel with the wave. |
Worked Example — IB-Style Wave Model Problem
Question (HL Paper 2 style — 6 marks)
A loudspeaker emits a sound of frequency 850 Hz into air at 20 °C, where the speed of sound is 340 m s$^{-1}$. The sound wave then enters water, where the speed of sound is 1500 m s$^{-1}$. (a) Calculate the wavelength of the sound in air. (b) State and explain what happens to the frequency as the wave enters the water. (c) Calculate the wavelength of the sound in water.
Solution
- Wavelength in air: rearrange $v = f\lambda$ to give $\lambda = v/f$. (M1)
- $\lambda_\text{air} = 340 / 850 = 0.400\ \mathrm{m}$. (A1)
- Frequency is set by the source (the loudspeaker) and is unchanged at the boundary. Only $v$ and $\lambda$ change. (R1)
- Wavelength in water with the same frequency: $\lambda_\text{water} = v_\text{water}/f$. (M1)
- $\lambda_\text{water} = 1500 / 850 = 1.76\ \mathrm{m}$ (3 s.f.). (A1)
- State: the wavelength increases because sound is faster in water and the frequency is fixed. (A1)
Examiner's note: The most common mistake is writing "$f$ also increases because the wave is faster". Frequency is set by the source — it never changes at a boundary in IB Physics. State this explicitly to score the R1 mark; otherwise the (b) part is lost.
Common Student Questions
What changes when a wave enters a denser medium?
The wave speed and the wavelength change. The frequency does NOT change — frequency is set by the source. For sound entering a denser medium, both speed and wavelength typically increase (sound is faster in solids and liquids than in gases). For light entering a denser medium (e.g. air to glass), speed and wavelength both decrease while frequency stays the same.
Why can only transverse waves be polarised?
Polarisation restricts oscillation to a single plane perpendicular to the direction of propagation. Transverse waves oscillate perpendicular to the direction of energy transfer, so there is a plane to restrict. Longitudinal waves oscillate parallel to the direction of propagation, so there is no perpendicular plane to limit — they cannot be polarised. If a wave can be polarised, it must be transverse.
What is the difference between wave speed and particle speed?
Wave speed $v = f\lambda$ is the speed at which the wave pattern (and the energy) travels through the medium. Particle speed is how fast individual particles oscillate about their equilibrium positions. Particles do not travel with the wave — they oscillate in place. Wave speed is constant in a given medium; particle speed varies sinusoidally and is maximum when the particle passes through equilibrium.
How do I tell a displacement-position graph from a displacement-time graph?
Look at the horizontal axis units. Metres → it's a displacement–position graph (a snapshot of the whole wave at one instant). Seconds → it's a displacement–time graph (motion of one particle over time). From the $y$ vs $x$ graph you read off the wavelength $\lambda$; from the $y$ vs $t$ graph you read off the period $T$. Mixing these up is the most common error in IB Physics Paper 1 wave questions.
Does intensity depend on amplitude or frequency?
Intensity is proportional to amplitude squared: $I \propto A^2$. Doubling the amplitude quadruples the intensity. For a point source, intensity also follows an inverse-square law with distance: $I \propto 1/r^2$. Frequency does not directly appear in the intensity formula at IB Physics level — but it sets the energy per photon for EM waves, which is a separate Topic E concept.
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