IB DP Physics: Travelling waves and the Doppler effect (DP Physics)

This guide covers sub-topics C.2 (Wave model) and C.5 (Doppler effect) from the IB Physics guide (first assessment 2025). C.2 is the same for SL and HL. In C.5 the nature of the effect, wavefront diagrams and the light-wave relationship are SL and HL; the formulas for sound and other mechanical waves are additional higher level (HL only). For quick recall use the revision notes, then test yourself with the practice questions. Find your weak spots first with the free 10-minute diagnostics.

Related pages: the course hub, the printable checklist and kinematics.

What this unit covers

Part	What you must be able to do	Tier
C.2	Distinguish transverse and longitudinal travelling waves	SL and HL
C.2	Use wavelength λ, frequency f, period T and speed v with v = fλ = λ/T	SL and HL
C.2	Describe the nature of sound and electromagnetic waves, and compare mechanical and electromagnetic waves	SL and HL
C.2	Describe how particles of the medium move as a wave passes, from graphs of displacement against position and time	SL and HL
C.2	Know the approximate orders of magnitude of wavelengths across the electromagnetic spectrum	SL and HL
C.5	Describe the Doppler effect for sound and light, using wavefront diagrams for a moving source or observer	SL and HL
C.5	Use Δf/f = Δλ/λ ≈ v/c for light when v is much smaller than c	SL and HL
C.5	Explain how shifts in spectral lines give the motion of stars and galaxies	SL and HL
C.5	Use f′ = f v/(v ± us) for a moving source and f′ = f(v ± uo)/v for a moving observer	HL only
C.5	Discuss medical and radar uses of the Doppler effect	SL and HL

C.2 Waves and what they carry

A travelling wave is a disturbance that moves through a medium, or through space, and carries energy from place to place. Each particle of the medium vibrates about a fixed rest position and is not carried along by the wave. The medium ends up with no net shift, yet energy has still been delivered elsewhere.

Transverse and longitudinal waves

In a transverse wave the particles oscillate at right angles to the direction the wave travels. Waves on a rope and all electromagnetic waves are transverse. In a longitudinal wave each particle vibrates along the same line as the direction of travel. Sound is the standard example. A longitudinal wave is a series of compressions, where particles are bunched together, and rarefactions, where they are spread apart.

The wave quantities

• Wavelength λ: the distance along the wave between two neighbouring points that are in step, for example crest to crest or compression to compression. Unit m.
• Frequency f: the number of complete oscillations passing a point each second. Unit Hz.
• Time period T: the time for one oscillation, T = 1/f. Unit s.
• Wave speed v: the speed at which the pattern moves.

In one period the pattern moves forward by one wavelength, so

v = λ/T = fλ

Worked example 1. A pulse generator shakes one end of a rope at 2.5 Hz and the wavelength on the rope is 0.80 m.

• v = fλ = 2.5 × 0.80 = 2.0 m s⁻¹.
• T = 1/f = 1/2.5 = 0.40 s.

The speed of a mechanical wave is fixed by the medium. The frequency is fixed by the source. So when a wave enters a new medium, f stays the same and λ changes in proportion to v.

Sound waves

Sound is a longitudinal mechanical wave. A vibrating loudspeaker cone pushes air forward to make a compression, then pulls back to leave a rarefaction. Each air particle oscillates to and fro along the line of travel. It needs a medium and cannot cross a vacuum. The audible range is roughly 20 Hz to 20 kHz.

Worked example 2. Use v = 340 m s⁻¹ for sound in air. Find the wavelength of a 512 Hz note, and the wavelengths at the two ends of the audible range.

• λ = v/f = 340/512 = 0.664 m.
• At 20 Hz: λ = 340/20 = 17 m.
• At 20 000 Hz: λ = 340/20 000 = 0.017 m = 17 mm.

Electromagnetic waves

Electromagnetic waves are transverse waves made of varying electric and magnetic fields, perpendicular to one another and to the way the wave moves. They do not need a medium and cross a vacuum. In a vacuum all of them travel at the same speed, c = 3.00 × 10⁸ m s⁻¹, so c = fλ for every EM wave.

Worked example 3. An FM station broadcasts at 94.5 MHz. Find the wavelength.

• λ = c/f = 3.00 × 10⁸ / 94.5 × 10⁶ = 3.17 m.

Mechanical against electromagnetic

Feature	Mechanical (for example sound, rope)	Electromagnetic
Medium needed	Yes	No
Type	Transverse or longitudinal	Transverse only
Speed	Set by the medium	c in a vacuum, the same for all
What oscillates	Particles of the medium	Electric and magnetic fields

Orders of magnitude of wavelength

The guide directs you to the Physics data booklet for these values. Treat them as rough, because the boundaries overlap.

Region	Approximate wavelength
Radio	10⁻¹ m and above
Microwave	10⁻³ to 10⁻¹ m
Infrared	10⁻⁶ to 10⁻³ m
Visible	about 4 × 10⁻⁷ to 7 × 10⁻⁷ m
Ultraviolet	10⁻⁸ to 10⁻⁷ m
X-ray	10⁻¹¹ to 10⁻⁸ m
Gamma ray	10⁻¹² m and below

Particle motion as a wave passes

A displacement-position graph is a snapshot of every particle at one instant; a displacement-time graph follows one particle. To find which way a particle moves, slide the snapshot a little way in the direction of travel and see what profile arrives at that particle.

Worked example 4 (transverse). A wave travels in the +x direction at 4.0 m s⁻¹ with λ = 2.0 m and amplitude 3.0 cm. At t = 0 the displacement is y = 3.0 sin(2πx/λ) cm.

• T = λ/v = 2.0/4.0 = 0.50 s.
• At t = 0 the particles at x = 0, 0.5, 1.0, 1.5 and 2.0 m have y = 0, +3.0, 0, −3.0, 0 cm.
• After a quarter period (0.125 s) the pattern has moved 0.50 m to the right. The value that was at x = 0 is now at x = 0.5 m, and so on. So y at x = 0, 0.5, 1.0, 1.5, 2.0 m is −3.0, 0, +3.0, 0, −3.0 cm.
• The particle at x = 0 has gone from 0 to −3.0 cm, so it is moving down. The particle at x = 0.5 m has gone from +3.0 to 0, so it is also moving down.

Worked example 5 (longitudinal). Take the same wave as longitudinal, with displacement measured positive in the direction of travel. At t = 0, y = +3.0 mm at x = λ/4 and −3.0 mm at x = 3λ/4.

• At x = λ/2 the displacement is zero. Particles just to the left are displaced to the right (positive), and particles just to the right are displaced to the left (negative). They are pushed towards x = λ/2, so it is the centre of a compression.
• At x = 0 and x = λ the neighbours are displaced away from the point, so these are centres of rarefactions.

C.5 The Doppler effect

The Doppler effect is the change in the observed frequency, and so wavelength, of a wave when the source and the observer are moving relative to each other. You hear it as a siren passes: higher pitch on approach, lower as it recedes.

Wavefront diagrams

A wavefront joins points in the same phase, such as successive compressions. For a stationary source they are concentric circles.

Moving source. Each wavefront is emitted from a point further along the path. The circles are no longer centred on the same point. Ahead of the source they are closer together, so the wavelength is shorter and the observed frequency higher. Behind it they are further apart, giving longer wavelength and lower frequency. The wave speed in the medium is unchanged.

Moving observer. The source is stationary, so the wavefronts stay evenly spaced and λ is unchanged. An observer moving towards the source meets the wavefronts more often, so the observed frequency is higher. Moving away, the fronts arrive less often and the frequency is lower.

Light: Δf/f = Δλ/λ ≈ v/c

For light the speed is the same for every observer, so the sound-wave approach is replaced by a relationship valid when the relative speed v is much smaller than c:

Δf/f = Δλ/λ ≈ v/c

Here Δf and Δλ are the changes in the observed frequency and wavelength, and f and λ are the values for a stationary source. A source moving away gives a longer wavelength (redshift). A source moving towards you gives a shorter wavelength (blueshift).

Worked example 6. The hydrogen line of rest wavelength 486.1 nm is seen at 489.6 nm in light from a distant galaxy. Find the speed along the line of sight.

• Δλ = 489.6 − 486.1 = 3.5 nm.
• v = c Δλ/λ = 3.00 × 10⁸ × 3.5/486.1 = 2.16 × 10⁶ m s⁻¹.
• The wavelength has increased, so the galaxy is moving away.
• Check: v/c = 0.0072, much less than 1, so the approximation holds.

Spectral lines in astronomy

Each element emits and absorbs light at fixed wavelengths. Compare a star’s lines with the laboratory pattern. If every line has moved to longer wavelength by the same fraction, the body is receding. If they have moved to shorter wavelength, it is approaching. The fractional shift gives the line-of-sight speed. A line that shifts back and forth periodically shows the star orbiting another body.

HL only: sound and mechanical waves

For waves in a medium the speed v is that of the wave relative to the medium. The guide uses us for the speed of the source and uo for the speed of the observer. Problems will not involve both moving.

Moving source: f′ = f v/(v ± us). Use v − us when the source moves towards the observer, and v + us when it moves away.

Moving observer: f′ = f(v ± uo)/v. Use v + uo when the observer moves towards the source, and v − uo when moving away.

Worked example 7 (HL). A siren of frequency 800 Hz moves at 25 m s⁻¹. Use v = 340 m s⁻¹.

• Towards you: f′ = 800 × 340/(340 − 25) = 800 × 340/315 = 863 Hz.
• Away from you: f′ = 800 × 340/(340 + 25) = 745 Hz.
• Wavelength ahead: λ′ = (v − us)/f = 315/800 = 0.394 m. Behind: 365/800 = 0.456 m. At rest: 340/800 = 0.425 m.

Worked example 8 (HL). A 1200 Hz source is fixed. An observer moves towards it at 15 m s⁻¹.

• f′ = 1200 × (340 + 15)/340 = 1200 × 355/340 = 1250 Hz.

Worked example 9 (HL, finding the speed). A source of 440 Hz is heard at 470 Hz as it approaches.

• 470 = 440 × 340/(340 − us), so 340 − us = 440 × 340/470 = 318.3.
• us = 340 − 318.3 = 21.7 m s⁻¹.

Medical and radar uses

A radar speed gun sends out microwaves, and the reflection from a moving vehicle is shifted by an amount that increases with its speed. In medicine, Doppler ultrasound reflects high-frequency sound from moving blood. The frequency shift of the echo shows how fast blood is flowing and in which direction, which can reveal narrowed vessels. In each case the reflecting object first acts as a moving observer and then as a moving source. You need the idea, not a calculation.

Common errors

• Claiming the particles are carried forward by the wave, when each only vibrates about its rest position.
• Placing a compression at maximum displacement. It sits where displacement is zero.
• Writing that the Doppler effect changes the speed of sound. A moving source changes the wavelength, not the wave speed.
• Mixing up signs. Towards means higher frequency, so for a source the denominator gets smaller.

Official syllabus

Source: Physics guide, first assessment 2025, International Baccalaureate Organization (Diploma Programme), sub-topics C.2 Wave model and C.5 Doppler effect.