Wave Phenomena — IB Physics HL Study Guide (HL Extension)

For: IB Physics HL candidates sitting IB Physics HL.

Covers: IB Topic 9 (HL only) — full SHM equations and energy, wave superposition and standing waves, single-slit diffraction and resolution (Rayleigh criterion), polarisation and Malus's law, Doppler effect.

You should already know: SL Topic 4 (Waves), Topic 9.1 builds on SL SHM.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the IB Physics HL style for educational use. They are not reproductions of past IBO papers.

1. Why Wave Phenomena Matters in HL

Topic 9 (HL only) deepens SL Topic 4 with mathematical rigour: full equations of SHM, single-slit diffraction patterns and resolution criteria, polarisation. About 7-10% of HL Paper 1+2 covers Topic 9, and Paper 3 (option) often draws on it.

2. Simple harmonic motion — HL form

For SHM with angular frequency $\omega$ and amplitude $A$: $$x(t)=A\cos (\omega t),v(t)=-A\omega \sin (\omega t),a(t)=-A{\omega }^2\cos (\omega t)=-{\omega }^{2}x$$

Velocity at any displacement: $$v=\pm \omega \sqrt{A^2-x^2}$$

So v_\max = A\omega at $x=0$ (equilibrium), $v=0$ at $x=\pm A$ (extrema).

Energy:

• Total: $E=\frac{1}{2}m{\omega }^2{A}^2=\frac{1}{2}k{A}^2$ (constant).
• KE at displacement $x$: $\frac{1}{2}m{\omega }^2(A^2-x^2)$.
• PE at displacement $x$: $\frac{1}{2}m{\omega }^2{x}^2$.

KE and PE oscillate at $2\omega$ (twice the position frequency); their sum stays constant.

3. Wave superposition

When two waves overlap, displacements add: $$y_{\text{total}}(x,t)=y_1(x,t)+y_2(x,t)$$

Constructive interference: peaks align with peaks, displacements add. Path difference = $n\lambda$ for integer $n$.

Destructive interference: peaks align with troughs, displacements cancel. Path difference = $(n+\frac{1}{2})\lambda$.

Standing waves form when two waves of equal frequency travel in opposite directions: $y(x,t)=2A\sin (kx)\cos (\omega t)$. Result: nodes (no motion) at $\sin (kx)=0$ and antinodes (max amplitude) at $\sin (kx)=\pm 1$.

4. Single-slit diffraction

Light passing a single slit of width $b$ creates a diffraction pattern with intensity minima at angles where $b\sin \theta =n\lambda$ ($n=1,2,...$).

The first minimum at $\sin {\theta }_1=\lambda /b$ marks the boundary of the central maximum (which contains ~84% of the energy).

Width of central max ≈ $2\lambda L/b$ at distance $L$ from the slit.

Narrower slit → more spread; wider slit → less spread. As $b$ approaches $\lambda$, diffraction dominates.

5. Resolution — Rayleigh criterion

Two point sources are just resolved when the central maximum of one coincides with the first minimum of the other: $${\theta }_R=1.22\frac{\lambda }{D}$$

where $D$ is the aperture diameter. Smaller $\lambda$ or larger $D$ → better resolution.

This explains why:

• Optical telescopes have huge mirrors (larger $D$).
• Electron microscopes use electrons (smaller $\lambda$ than light).
• Radio telescopes are huge (long radio $\lambda$ requires big $D$).

6. Polarisation

Light is a transverse EM wave with electric field oscillation in any direction perpendicular to motion. Polarised light has the E field oscillating in one plane.

Polariser: a filter that transmits only one polarisation direction. Unpolarised light through a polariser → polarised, with intensity halved.

Malus's law: when polarised light of intensity $I_0$ hits a second polariser at angle $\theta$ to the first axis: $$I=I_0{\cos }^2\theta$$

If $\theta =90°$ (crossed polarisers): $I=0$ (extinction).

7. Doppler effect

When source and observer move relative to each other, observed frequency differs from source frequency.

For sound (medium-based), with $v$ = speed of sound, $u_s$ = source speed, $u_o$ = observer speed: $$f_{\text{obs}}=f_{\text{source}}\cdot \frac{v+u_o}{v+u_s}$$

(Sign convention: positive $u_o$ is toward source; positive $u_s$ is away from observer. Adjust signs accordingly.)

For light at non-relativistic speeds: $\Delta f/f=-v/c$ (recession lowers frequency, redshift).

8. Worked Example

A car horn at 500 Hz approaches a stationary observer at 30 m/s. Speed of sound = 340 m/s. After passing, the car continues at 30 m/s away from observer.

(a) Find the observed frequency before and after passing. (b) An identical horn moves at 30 m/s toward another car going 20 m/s away. Find the observed frequency.

Solution.

(a) Approaching ($u_s$ negative — toward observer): $f_{\text{obs}}=500\cdot 340/(340-30)=500\cdot 340/310=548$ Hz.

After passing ($u_s$ positive — away): $f_{\text{obs}}=500\cdot 340/(340+30)=500\cdot 340/370=459$ Hz.

(b) Source moves toward observer (taking +$x$ as approach): $u_s=-30$. Observer moves away: $u_o=-20$. $f_{\text{obs}}=500\cdot (340-20)/(340-30)=500\cdot 320/310=516$ Hz.

9. Common Pitfalls

• Doppler signs: always set up with explicit motion direction. Approach reduces denominator OR adds to numerator (depending on convention).
• Diffraction vs interference: single-slit gives diffraction (single minima from $b\sin \theta =n\lambda$). Double-slit gives interference (maxima from $d\sin \theta =n\lambda$). Both have the diffraction envelope.
• Polariser intensity halving: the first polariser halves unpolarised light intensity (factor of 1/2). Subsequent polarisers use Malus.
• Standing wave nodes vs antinodes: nodes have no motion; antinodes have max motion. The reverse of pressure waves (nodes are pressure max, antinodes are pressure min).

10. Practice Questions

1. Light of $\lambda =600$ nm falls on a single slit of width 0.20 mm. Find the angular position of the first minimum and the width of the central maximum at 2.0 m distance.
2. Two stars at distance 10 ly are observed with a telescope of aperture 0.20 m. What is the minimum separation distance between the stars to be resolvable in visible light (550 nm)?
3. Polarised light of intensity 1000 W/m² passes through a second polariser at 30° to the first. Find the transmitted intensity.

11. Quick Reference Cheatsheet

• SHM: $x=A\cos (\omega t)$, $v=\pm \omega \sqrt{A^2-x^2}$, $a=-{\omega }^{2}x$.
• SHM energy: $E=\frac{1}{2}m{\omega }^2{A}^2$, PE = $\frac{1}{2}m{\omega }^2{x}^2$.
• Single slit minima: $b\sin \theta =n\lambda$.
• Rayleigh: ${\theta }_R=1.22\lambda /D$.
• Malus: $I=I_0{\cos }^2\theta$.
• Doppler (sound): $f_{\text{obs}}=f_s(v+u_o)/(v+u_s)$.

12. What's Next

Wave Phenomena leads into Topic 12 (Quantum & Nuclear) — wave-particle duality builds on diffraction. Use Ollie for any specific Doppler or diffraction problem: "How do I sign the Doppler equation for this situation?" or "Why does my single-slit pattern have a wider central max than the side maxima?"