Waves: Progressive Waves, the Doppler Effect and Polarisation

This guide covers Topic 7, Waves, in full — subtopics 7.1 Progressive waves, 7.2 Transverse and longitudinal waves, 7.3 Doppler effect for sound waves, 7.4 Electromagnetic spectrum and 7.5 Polarisation — from Cambridge International AS & A Level Physics 9702, 2025–2027 series. This is AS Level content, and an understanding of colour from Cambridge IGCSE/O Level Physics or equivalent is assumed.

Before studying this

This topic starts a new strand of the AS course (waves), independent of the mechanics topics 1–6. No specific prior 9702 topic is assumed beyond a basic IGCSE/O Level familiarity with colour and light.

Syllabus coverage

CAMBRIDGE INTERNATIONAL AS & A LEVEL PHYSICS 9702 — AS Level, Topic 7

7.1 Progressive waves — describing wave motion illustrated by ropes, springs and ripple tanks; the terms displacement, amplitude, phase difference, period, frequency, wavelength and speed; using a cathode-ray oscilloscope (CRO) time-base and y-gain to determine frequency and amplitude; deriving and using v = fλ; understanding that energy is transferred by a progressive wave; recalling and using intensity = power/area and intensity ∝ amplitude².

7.2 Transverse and longitudinal waves — comparing transverse and longitudinal waves; analysing and interpreting graphical representations of both.

7.3 Doppler effect for sound waves — understanding that a moving source relative to a stationary observer changes the observed frequency; using f₀ = fₛv/(v ± vₛ) for the observed frequency.

7.4 Electromagnetic spectrum — all electromagnetic waves are transverse and travel at speed c in free space; the approximate wavelength ranges of the principal regions from radio waves to γ-rays; wavelengths 400–700 nm are visible.

7.5 Polarisation — polarisation as a phenomenon of transverse waves; recalling and using Malus’s law (I = I₀cos²θ) for the intensity of plane-polarised light after a polarising filter.

Describing a progressive wave

A progressive wave transfers energy from one place to another without transferring matter. Key terms:

• Displacement — how far a point on the wave is from its undisturbed position
• Amplitude — the maximum displacement
• Period T — time for one complete oscillation
• Frequency f — oscillations per second, f = 1/T
• Wavelength λ — the distance between corresponding points on successive oscillations
• Phase difference — how far out of step two points or waves are, usually expressed as an angle or fraction of a cycle

The wave equation relates speed, frequency and wavelength:

v = fλ

Intensity is power transferred per unit area, and for a progressive wave is proportional to the square of the amplitude:

intensity = power/area,   intensity ∝ amplitude²

Using a CRO to find frequency and amplitude. A cathode-ray oscilloscope (CRO) displays a signal’s voltage against time. Two settings let you read off the wave’s frequency and amplitude directly from the trace:

• The y-gain (in volts per division, V/div) converts the trace’s vertical height into a voltage: amplitude = (number of divisions from the centre line to the peak) × (y-gain).
• The time-base (in seconds — or milliseconds/microseconds — per division, s/div) converts the trace’s horizontal spacing into a time: measure the number of divisions for one complete cycle, multiply by the time-base to get the period T, then use f = 1/T.

Worked example. A trace shows a peak 2.5 divisions above the centre line, with the y-gain set to 2.0 V/div, and one complete cycle spanning 4.0 divisions with the time-base set to 5.0 ms/div. Amplitude = 2.5 × 2.0 = 5.0 V. Period T = 4.0 × 5.0 ms = 20 ms = 0.020 s, so frequency f = 1/T = 1/0.020 = 50 Hz.

Seeing wave motion: ropes, springs and ripple tanks

Subtopic 7.1 starts with describing wave motion as shown by vibrations in ropes, springs and ripple tanks. In each one, the thing to notice is the same: each part of the medium oscillates about a fixed position, while the wave (and the energy it carries) moves along.

A rope. Lay a long rope (or a length of rubber tubing) along the floor, fix or hold the far end, and move your end up and down once: a single pulse travels along the rope. Keep moving your end up and down steadily and a continuous progressive wave travels along it. A ribbon tied to the rope moves up and down as the wave passes but does not travel along the rope. Each part of the rope moves perpendicular to the direction the wave travels, so this is a transverse wave. The frequency of the wave is the frequency at which you move your hand.

A spring (a “slinky”). Stretch a long, loose spring along a bench and move one end steadily backwards and forwards along the spring’s length. Regions where the coils are closer together (compressions) and further apart (rarefactions) travel along the spring. A coil marked with tape oscillates backwards and forwards about its rest position but does not move along with the wave. The coils move parallel to the direction the wave travels, so this is a longitudinal wave — the same pattern of compressions and rarefactions as a sound wave. Moving the same end from side to side instead gives a transverse wave on the same spring, which shows that “transverse” and “longitudinal” describe the direction of oscillation relative to the direction of travel, not the medium.

A ripple tank. A ripple tank is a shallow, transparent-bottomed tray of water. A bar that just touches the surface, vibrated by a small motor, produces straight (plane) wavefronts; a small ball-shaped dipper produces circular wavefronts. A lamp above the tank shines through the water onto a screen or sheet of paper below. The crests of the ripples focus the light, so they show up as bright lines and the troughs between them as darker lines. The distance between neighbouring bright lines corresponds to one wavelength.

• A stroboscope flashing at the same frequency as the vibrating bar makes the pattern appear still, because each crest moves exactly one wavelength between flashes. This lets you measure the wavelength, then find the speed from v = fλ.
• Measure across several wavelengths and divide, to reduce the uncertainty. The pattern on the screen is larger than the real waves, so either measure against a ruler placed in the tank or scale the reading by the image’s magnification.
• A small floating object bobs up and down as the waves pass but is not carried along with them: the wave transfers energy, not water. At this level water surface waves are treated as transverse.

The ripple tank is also the standard way to show reflection, refraction and diffraction of waves (see Superposition for diffraction).

Transverse vs. longitudinal waves

In a transverse wave, particle displacement is perpendicular to the direction of energy transfer (e.g. light, water surface waves). In a longitudinal wave, particle displacement is parallel to the direction of energy transfer (e.g. sound). Graphically, transverse waves show displacement against distance as a familiar sine-wave shape; longitudinal waves are often shown as regions of compression and rarefaction along the direction of travel.

The Doppler effect

When a source of sound moves relative to a stationary observer, the observed frequency differs from the source frequency — higher if the source approaches, lower if it recedes:

f₀ = fₛv / (v ± vₛ)

where v is the speed of sound, vₛ is the source speed, and the sign in the denominator is chosen according to whether the source approaches (subtract) or recedes (add). Only the case of a stationary observer and moving source is required at this level.

The electromagnetic spectrum

All electromagnetic waves are transverse and travel at the same speed c in free space. In order of increasing frequency (decreasing wavelength), with their approximate wavelength ranges:

Region	Approximate wavelength range
Radio waves	> 0.1 m
Microwaves	1 mm – 0.1 m
Infrared	700 nm – 1 mm
Visible light	400–700 nm
Ultraviolet	10 nm – 400 nm
X-rays	10 pm – 10 nm
γ-rays	< 10 pm

These ranges overlap at their boundaries and vary slightly between sources — learn the order and rough scale (each region is roughly 10–1000 times the wavelength of its neighbour) rather than exact cutoff values, which the syllabus does not fix precisely.

Polarisation

Polarisation — restricting the oscillation of a transverse wave to a single plane — is a phenomenon exclusive to transverse waves; longitudinal waves cannot be polarised, which is itself evidence for light’s transverse nature. Malus’s law gives the intensity remaining after plane-polarised light passes through a polarising filter at angle θ to the plane of polarisation:

I = I₀ cos²θ

Worked example. Plane-polarised light of intensity 8.0 W m⁻² passes through a polarising filter oriented at 30° to the plane of polarisation.

I = I₀cos²θ = 8.0 × cos²(30°) = 8.0 × 0.75 = 6.0 W m⁻²

Common mistakes

• Confusing phase difference with path difference. Phase difference is an angle or fraction of a cycle; path difference is a distance — they are related (via wavelength) but not the same quantity.
• Using intensity ∝ amplitude without squaring it. Intensity is proportional to amplitude squared.
• Applying the Doppler formula to a moving observer and stationary source — this syllabus only requires the stationary-observer, moving-source case.
• Forgetting that Malus’s law here only calculates the effect of a filter on already plane-polarised light, not the effect of the first filter on unpolarised light (which is not required at this level).

Quick revision checklist

• Wave motion shown by a rope (transverse), a spring (longitudinal or transverse) and a ripple tank
• Wave terms: displacement, amplitude, period, frequency, wavelength, phase difference
• v = fλ, and intensity ∝ amplitude²
• Transverse vs. longitudinal waves, with graphical representation
• The Doppler effect formula for a moving source and stationary observer
• The electromagnetic spectrum, in order, with the visible range
• Malus’s law, I = I₀cos²θ

Related resources

• Superposition: Stationary Waves, Diffraction and Interference — the next AS topic, building directly on wave terms introduced here
• Deformation of Solids: Stress, Strain and the Young Modulus — the previous AS topic
• Cambridge AS & A Level Physics hub

Written against Cambridge International AS & A Level Physics 9702, 2025–2027 series. Always check the current syllabus for your examination year.