Waves
1. Types of Waves
By energy transfer:
- Progressive waves — transfer energy from one place to another.
- Stationary waves — do not transfer energy.
By direction of vibration:
- Transverse waves — vibration/motion of particles is perpendicular to the direction of travel of the wave / direction of propagation of energy. Can travel in a vacuum. e.g. EM spectrum, water waves.
- Longitudinal waves — vibration/motion of particles is parallel to the direction of travel of the wave / direction of propagation of energy. Require a medium to travel. e.g. sound waves.
2. Displacement–Distance Graphs
(snapshot of the whole wave at one instant in time)
Distance — distance moved along the wave.
Amplitude — maximum displacement of a particle from its mean position (scalar).
Wavelength (λ) — distance between 2 consecutive points in phase.
Phase difference
Tells how much of a wave is completed: 0° ⇒ 0% of a wave, 360° ⇒ 100% of a wave. 1 "transition" (¼ wave) ⇒ ¼×360° = 90°.
Using points on the wave above (λ = 4 m):
For points X (d = 1.65 m) & Y (d = 3.75 m) on a wave of λ = 4 m:
Vertical cross-section of a water wave moving left→right. At which labelled point is the water moving upwards with maximum speed?
Max speed happens where displacement = 0 (mean position) — not at the crest/trough (there v=0). Since the wave moves right, the point just behind the crest (on its rising side) is about to become the crest, so it is moving up through zero at that instant.
Answer: the zero-crossing point rising toward the next crest (labelled B in the original figure).
A wave has period T, wavelength λ and amplitude A, shown on a graph of displacement x against distance d. Which graph is correctly labelled?
Answer: A
A wave pulse moves right along a stretched rope. Which graph correctly shows the variation with time t of the displacement s of a fixed particle P on the rope, as the pulse passes it?
Trace what P does as the pulse (whatever shape it is) arrives, passes under it, then leaves — match the rope's shape at each instant to a value of s at the corresponding t.
Answer: D
X and Y are two points on the surface of water in a ripple tank. A source of constant frequency generates waves that travel past X then Y, causing them to oscillate.
Read off how many whole + partial wavelengths separate X and Y along the direction of travel, then convert using phase diff = (Δx/λ)×360°.
Answer: 270°
3. Displacement–Time Graphs
(one particle, tracked over time — not a snapshot of the whole wave)
Frequency (f) — number of complete waves formed in 1 second.
Practice: drawing a wave X° out of phase
Phase difference between two waves — general method
Works the same way whether comparing distance (x–d graph) or time (x–t graph) — just swap Δx/λ for Δt/T.
e.g. two waves of the same λ (same T) offset by 2 boxes out of 4 ⇒ (2/4)×360° = 180° either way.
The graph represents a sound wave: displacement/μm against time/ms, period 20 ms, 3 full cycles shown by 55 ms.
Answer: B
A whale produces sound of frequency 5 Hz, displayed on an oscilloscope.
Answer: D — 100 ms div⁻¹
Time-base = 5.0 ms/div, Y-gain = 5.0 mV/div. Four complete waves fill the screen.
Amplitude read as 2 divisions × 5.0 mV = 10 mV.
Answer: A — 10 mV, 50 Hz
4. Velocity of a Wave
Speed of deep-water waves: v² = gλ/2π. A student measures λ and f. Which graph gives a straight line through the origin?
Answer: D — f² against 1/λ
Displacement–time graph of a gas molecule as a continuous sound wave passes; period read as 500 μs. Speed of sound in the gas = 330 m/s.
A second molecule sits 0.165 m further from the source — exactly one whole λ away, so:
Answer: D (same displacement–time graph as the original, no shift)
A source has frequency f, produces sound of wavelength λ.
- Frequency of source = number of complete waves produced in 1 s by the source.
- Distance moved by a wavefront during n oscillations, in terms of λ = nλ.
Given time-base = 2.0 ms/cm and 3 divisions per cycle read off the CRO trace:
A second wave, same frequency & amplitude, phase difference 90° from the first ⇒ shifted by ¼ cycle (drawn as a red curve on the same axes, starting a quarter-cycle ahead of the grey original).
5. Transverse vs Longitudinal (exam wording)
Distinguish, by reference to vibrations of particles & direction of energy transfer:
Transverse: vibrations are perpendicular to the direction of energy transfer. Longitudinal: vibrations are parallel to the direction of energy transfer.
Polarised wave — vibration of particles is restricted to one plane only, that plane being perpendicular to the direction of propagation of the wave.
On a y/cm–x/m graph: amplitude read as 2.8 cm; phase difference between points A (x=0.1) and B (x=0.4) on a wave of λ = 0.8 m:
6. Finding Compressions & Rarefactions
Method: at any point on the displacement–distance graph, look at 3 particles — one at that point, one slightly to its left, one slightly to its right — and mark their displacement (velocity) vectors according to your sign convention.
→ · ← (converging) = compression (pressure > atmospheric)
← · → (diverging) = rarefaction (pressure < atmospheric)
7. Intensity of a Wave
Definition: intensity is the power incident per unit area. Scalar. Units W m⁻².
Learn these 3 relationships:
I = P/4πr² for a source of constant power P. How does amplitude a of the vibrating air molecules vary with r?
Answer: A — a ∝ 1/r
Which statement is correct?
Answer: A
A sound wave's displacement–time graph is given; intensity is then halved. Which graph correctly shows the new wave?
Answer: D — same frequency, reduced amplitude (A, B, C all wrongly change the frequency or scale the amplitude incorrectly)
Fig 5.1: x/mm–t graph, T = 0.60 s. Fig 5.2: x/mm–d graph at one instant, λ = 4.0 cm, amplitude decreasing with distance (2 mm at source → 1.1 mm at 6.0 cm).
Losing power? Yes — the wave is losing power as it moves away from the source, because the amplitude decreases with distance.
Fig 4.1: x/mm–t graphs of two waves P (amplitude 3.0 mm) and Q (amplitude 2.0 mm) at the same point, T = 0.60 s. Speed of both waves = 20 cm/s.
At t = 0.45 s the two waves superpose (add algebraically); reading each wave's displacement off Fig 4.1 at that instant and summing gives the resultant displacement.
8. The Doppler Effect
Definition: the change in observed frequency due to relative motion between an observer and a source.
Redshift (worked example)
Redshift: wavelength of light stretched to a longer wavelength due to the separation velocity between galaxies.
Galaxy A emits blue light λ_s = 450 nm; stationary Galaxy B detects it at λ₀ = 780 nm (red). c = 3×10⁸ m/s.
Vehicle emits const-frequency microwaves, moves toward a stationary observer at const speed (f₀>F_s), then accelerates and holds a higher steady speed (f₀ even higher), then decelerates to a stop (f₀→F_s).
Answer: C — observed frequency rises, remains steady, then falls back to F_s (as v_s→0, f₀→f_s)
A source of constant f_s moves at constant speed directly toward, then passes, then moves directly away from an observer.
Answer: B — a step up to a steady high f₀ while approaching, then a step down to a steady low f₀ (below f_s) once receding
Jet travels at 0.8v (v = speed of sound) toward a stationary observer; emitted frequency 100 Hz.
Answer: D — 500 Hz
Siren f_s = 1000 Hz moves at 20 m/s toward a stationary observer; speed of sound = 330 m/s.
Answer: B — (1000×330)/(330−20)
Car moves at constant speed toward a stationary observer; horn f_s = 510 Hz, observer hears f₀ = 550 Hz; speed of sound = 340 m/s.
Child moves in a circle at 7.5 m/s, blowing a whistle (f_s = 950 Hz) starting at point P (moving toward a distant observer) and stopping at point Q (moving away). Speed of sound = 330 m/s.
Variation heard: frequency decreases from 970 Hz (at P) through 950 Hz (at the point level with the observer, X) down to 930 Hz (at Q) — i.e. it falls smoothly from the max value, through f_s itself, to the min value f₀(min) = 330/(330+7.5)×950 = 930 Hz.
I is intensity (power per unit area), K a unitless constant, v = speed of sound, ρ = density of air, f = frequency, A = amplitude. Show both sides have the same SI base units.
Describe the Doppler effect: the change in observed frequency due to relative motion between source and observer.
A distant star moves away from a stationary observer — effect on the light observed? Light will be redshifted — the wavelength stretches to a longer value / the observed frequency is less than the source frequency.
9. The Electromagnetic Spectrum
- Group of transverse waves.
- All travel at 3×10⁸ m/s in a vacuum.
| Region | Wavelength range /m | Typical use |
|---|---|---|
| Radio waves | 10³ < λ ≤ 10⁰ | FM radio / RC cars |
| Microwaves | 10⁰ < λ ≤ 10⁻³ | Microwave ovens / network signals |
| Infrared | 10⁻³ < λ ≤ 10⁻⁶ | Infrared cameras / TV remotes |
| Visible light | ≈ 10⁻⁶ | We use this to see! |
| Ultraviolet | 10⁻⁶ < λ ≤ 10⁻⁸ | Disinfection |
| X-rays | 10⁻⁸ < λ ≤ 10⁻¹⁰ | X-ray scans |
| Gamma rays | 10⁻¹⁰ < λ ≤ 10⁻¹² | Cancer treatment / detecting leaks / PET scans |
1 wavelength spans 1.0 cm on screen; time-base = 0.20 μs/cm.
Answer: A — 2.5 MHz, radio wave region
Answer: C — X-rays, infrared
Which row shows the correct order-of-magnitude wavelengths for microwaves, infrared, ultraviolet, X-rays?
Answer: C — 10⁻², 10⁻⁶, 10⁻⁸, 10⁻¹⁰ m respectively
Property of EM waves not shared with other transverse waves: travel in a vacuum at 3×10⁸ m/s.
Naming regions B, E, F (with typical wavelength):
Frequency for a wavelength of 500 nm:
10. General Wave Properties
Wavefront: an imaginary line joining all points in phase on different waves.
Reflection laws:
- Incident wave, reflected wave and the normal all lie in the same plane.
- ∠i = ∠r
11. Refraction
The bending of a wave as it passes from a rarer to a denser medium or vice versa — caused by a velocity change.
Deep → shallow:
- frequency stays the same
- λ decreases
- v↓ = fλ↓ ⇒ wave bends
Shallow → deep:
- frequency stays the same
- λ increases
- v↑ = fλ↑ ⇒ wave bends the other way
12. Diffraction
Spreading of a wave into a region it would not reach if it travelled in straight lines — i.e. spreading into its geometric shadow — as it passes a narrow slit or edge.
Plane wavefronts pass through a gap (P before, Q after).
Answer: C — amplitude changes (velocity, frequency and wavelength are all unchanged by diffraction)
Lattice spacing ≈ 10⁻¹⁰ m. Diffraction is most significant when slit size ≈ wavelength.
Answer: D — X-rays (λ range 10⁻⁸–10⁻¹⁰ m, matching the lattice spacing)
Answer: A — diagrams X and Y both illustrate diffraction (spreading round an edge / through a gap)
13. Polarisation
The process of confining the oscillations of particles in one direction only (and always perpendicular to the direction of travel).
How to polarise light
Use a polaroid sheet/filter, which acts as a polariser.
- Oscillations parallel to the axis of the slit → pass through completely.
- Oscillations perpendicular to the axis of the slit → completely blocked.
A polaroid filter is made of long chains of organic molecules; the axis of the slit is perpendicular to the chains.
Two types:
- Partial polarisation — vibrations not fully restricted to one direction (e.g. reflection off water).
- Full / complete plane polarisation — vibrations fully restricted to one direction only.
Malus' Law
θ = angle between the electric field vector and the axis of the (second) slit.
- I ∝ a² (recall)
- amplitude ∝ electric field strength ⇒ a ∝ E, a² ∝ E²
- I ∝ a² and a² ∝ E² ⇒ I ∝ E²
Light reflected from a water surface is partially plane polarised. Describe briefly how you would demonstrate this.
- Make the reflected light incident on a polaroid filter.
- Rotate the filter about its axis.
- You will see maximums and minimums of intensity (never falling fully to zero — proving it's only partially polarised).
Vertically polarised light (I₀) hits filter 1 (axis vertical), then filter 2 (axis 45° to vertical), then filter 3 (axis horizontal). Show the final intensity is not zero.
Filter's transmission axis is initially vertical (transmitted intensity = incident intensity).
(a) Angle to the vertical that gives no transmitted beam: 90° or 270°.
(b) Ratio I/I₀ = 0.75 when the filter is at angle θ:
Ratio (amplitude transmitted)/(amplitude incident):
Rotating so θ (to the vertical) increases from 20°, the transmitted intensity first falls (max at 0°, min at 90°) then rises again — it returns to its starting value (cos² symmetric about 90°) at θ = 160° (since 90°+(90°−20°) = 160°).
Meaning of polarisation: the process of confining oscillations of the electric field vector to one direction only, while still perpendicular to the direction of propagation.
Why light can be plane polarised but sound cannot: light waves are transverse, sound waves are longitudinal.
Filter A (vertical axis) polarises unpolarised light to intensity I₀; filter B (rotatable) starts horizontal. After rotation, intensity emerging from B is ¼I₀ — find the rotation angle from B's starting position.
- Only transverse waves polarise; the E-field is what's confined to one plane.
- I = I₀cos²θ (Malus' Law) — only for already-polarised light.
- Unpolarised light through the first filter always loses half its intensity, regardless of angle.
- I vs θ graph is symmetric about 90° and 270° — two angles either side of a symmetry line always give the same intensity.
14. The Cathode-Ray Oscilloscope (C.R.O)
Time-base setting — basically the setting for the time axis (s/div).
Time-base = 1 ms/cm; Y-plate sensitivity = 0.2 mV/cm. Trace spans 4 cm per full wave, peak height 2 cm.
What each screen shows
- Transverse: vibration ⊥ travel, travels in vacuum. Longitudinal: vibration ∥ travel, needs a medium.
- Phase diff (distance) = (Δx/λ)×360° | Phase diff (time) = (Δt/T)×360°
- v = fλ
- I = P/A | I ∝ a² | I ∝ 1/r² | I₁/a₁² = I₂/a₂² | I₁r₁² = I₂r₂²
- Doppler: f₀ = v/(v∓v_s) × f_s (− for approach, + for separation)
- EM spectrum (long→short λ): Radio, Microwave, Infrared, Visible, Ultraviolet, X-ray, Gamma — all at 3×10⁸ m/s in vacuum
- Refraction: bending due to velocity change; f constant, λ & v change together
- Diffraction: most significant when slit size ≈ λ; λ, f, v all unchanged
- Malus' Law: I = I₀cos²θ (already-polarised light only); unpolarised light halves through the first filter
- CRO: Y-sensitivity sets the displacement axis, time-base sets the time axis