Wave Motion

Spec mapping: OCR H556 Module 4.4 — Waves: progressive waves, transverse vs longitudinal, wavefronts and rays, energy transfer without net transfer of matter. (Refer to the official OCR H556 specification document for exact wording.)

Waves are one of the most pervasive phenomena in physics. From the ripples on a pond to the light reaching your eye from a distant galaxy, from the sound of a spoken word to the pulses travelling along a nerve fibre, waves carry energy and information across space without the bulk transport of matter. Module 4.4 of the OCR A-Level Physics A specification — Waves — asks you to understand wave motion at a level which will underpin everything you subsequently study in optics, quantum physics, communications and modern physics.

This lesson establishes the conceptual scaffolding: what a progressive wave actually is, the distinction between transverse and longitudinal waves, the representation of waves via wavefronts and rays, and the energy-without-matter principle that students at GCSE encounter as a slogan but at A-Level must be able to defend in detail.

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What Is a Wave?

Informally, a wave is a disturbance that travels through a medium (or through space) transferring energy from one place to another without transferring matter.

That definition contains three essential ideas, and each deserves scrutiny:

1. A disturbance — the wave is not the medium itself but a change in some property of the medium (or, in the case of electromagnetic waves, a change in field strengths in vacuum).
2. Travels — the disturbance propagates with a definite speed, which depends on properties of the medium.
3. Transfers energy without transferring matter — particles of the medium oscillate locally but do not travel with the wave.

This last point is the most common source of confusion for students beginning A-Level. When a water wave moves across a pond, a floating cork bobs up and down but does not travel with the wave. The wave carries energy horizontally; the cork's motion is (approximately) vertical. Similarly, when sound travels across a room, air molecules oscillate about their equilibrium positions by a tiny fraction of a millimetre — they do not drift from the speaker to your ear.

Exam Tip: If an OCR question asks you to "explain what is meant by a progressive wave", your answer must include both ideas: (1) energy is transferred through the medium, and (2) particles of the medium do not move with the wave — they oscillate about equilibrium.

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Progressive Waves

A progressive wave (sometimes called a travelling wave) is one in which the disturbance travels through space, carrying energy with it. This is contrasted with a stationary wave (Lesson 11), in which energy is stored but not transmitted.

For a progressive wave:

• Each particle of the medium oscillates about a fixed equilibrium position.
• Neighbouring particles oscillate slightly out of phase with one another, so the pattern of displacement appears to move.
• The speed of the wave is determined by the medium, not by the source.

Consider a long rope with one end shaken up and down. The shake creates a pulse which propagates along the rope. Every particle of the rope moves vertically — up and then down — and each particle starts its motion a fraction of a second after its neighbour. The result is a travelling hump that moves along the rope at a speed determined by the tension and mass per unit length of the rope.

flowchart LR
    S[Source oscillates] --> P1[Particle 1 pulled up]
    P1 --> P2[Particle 2 pulled up later]
    P2 --> P3[Particle 3 pulled up later still]
    P3 --> P4[Pattern appears to move]
    P4 --> E[Energy transferred along rope]
    P4 --> M[No net motion of matter]

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Transverse Waves

A transverse wave is one in which the oscillations (displacement of particles) are perpendicular to the direction of energy transfer.

Examples include:

• Waves on a rope or string — particles move up and down; the wave travels along the rope.
• Water waves — surface particles move approximately in vertical circles; the wave travels horizontally. (Strictly, surface water waves have both transverse and longitudinal components, but for A-Level they are treated as transverse.)
• All electromagnetic waves — the electric and magnetic field vectors oscillate perpendicular to the direction of propagation (see Lesson 3).
• S-waves (secondary seismic waves) — transverse body waves travelling through the Earth's interior.

A characteristic property of transverse waves is that they can be polarised — because the oscillations are perpendicular to propagation, they can be restricted to a single plane. This is the key experimental evidence that light is a transverse wave, as you will see in Lesson 4.

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Longitudinal Waves

A longitudinal wave is one in which the oscillations (displacement of particles) are parallel to the direction of energy transfer.

The disturbance consists of alternating regions of compression (where particles are pushed together) and rarefaction (where particles are spread apart). The compressions and rarefactions travel through the medium at the wave speed, while each individual particle oscillates back and forth about its equilibrium position along the line of travel.

Examples include:

• Sound waves in air, water and solids — the most important example at A-Level.
• P-waves (primary seismic waves) — longitudinal body waves, the first to arrive from an earthquake because they travel faster than S-waves.
• Compression waves on a slinky spring — the classic laboratory demonstration.

Longitudinal waves cannot be polarised, because the oscillation is already along a single direction (the direction of travel); there is nothing further to restrict. This is a useful test: if you can polarise it, it is transverse; if you cannot, it is longitudinal.

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Comparing Transverse and Longitudinal Waves

Feature	Transverse	Longitudinal
Direction of oscillation	Perpendicular to propagation	Parallel to propagation
Can be polarised?	Yes	No
Examples	Light, EM waves, waves on a string, water waves, S-waves	Sound, P-waves, compression waves on a slinky
Can travel through vacuum?	Only EM waves	No
Visual appearance	Crests and troughs	Compressions and rarefactions
Displacement graphed against position	Sine curve	Sine curve (of displacement)

Note the last row: although longitudinal waves look very different physically, their displacement–position graph looks just the same as that of a transverse wave — a sinusoid. This is because we plot the signed displacement of each particle from equilibrium, whether that displacement is along the direction of travel or across it.

Exam Tip: The OCR specification explicitly requires you to "give examples of transverse and longitudinal waves" and to "describe the differences between transverse and longitudinal waves". Practise a one-sentence answer for each.

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Wavefronts and Rays

For two-dimensional and three-dimensional wave propagation, we need two further pieces of representation.

A wavefront is a surface (in 3-D) or a line (in 2-D) joining points of the wave at the same phase — for instance, all the points currently at the top of a crest. Wavefronts are usually drawn at intervals of one wavelength.

A ray is a line drawn perpendicular to the wavefronts, pointing in the direction of energy transfer. For light, the rays in geometric optics are the lines you draw with a ruler in ray-tracing diagrams.

flowchart TD
    A[Point source] --> B[Circular wavefronts]
    B --> C[Rays perpendicular to wavefronts]
    C --> D[Rays diverge radially]
    A --> E[Plane-wave source at infinity]
    E --> F[Straight, parallel wavefronts]
    F --> G[Parallel rays]

For a point source, wavefronts are concentric spheres (circles in 2-D), and rays are radial lines diverging outward — like the spokes of a wheel. For a plane-wave source (e.g. a distant star, treated as effectively at infinity), wavefronts are flat parallel planes (straight parallel lines in 2-D), and rays are parallel.

The wavefront-and-ray picture is the language of Huygens's construction, which we revisit in the refraction lesson: each point on a wavefront can be treated as a secondary source of spherical wavelets, and the new wavefront is the envelope of those wavelets a moment later. This is how the wave model produces straight-line propagation in homogeneous media (rays) and how it predicts bending at boundaries (refraction).

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How Waves Are Created

All waves originate in some oscillation at a source. A loudspeaker cone vibrating backwards and forwards compresses the air in front of it, creating a longitudinal sound wave. An electron oscillating in an aerial generates an electromagnetic wave. A stone dropped into a pond disturbs the water surface, creating a transverse ripple. The frequency of the source determines the frequency of the wave; the medium determines the speed; the wavelength is what you get from combining the two.

This is worth emphasising because students sometimes imagine that the wave speed is controlled by the source. It is not. A violin string and a cello string both produce sound waves in air at the same speed (about $340$340 m s$^{-1}$−1), even though the strings themselves vibrate at very different frequencies. The wavelengths of the resulting sound waves differ to match.

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Energy and Intensity

A wave transfers energy. For a wave of a given type and frequency, the energy transferred per unit time (the power $P$P) is proportional to the square of the amplitude $A$A:

$P \propto A^2$P∝A2

This is why a loud sound requires much more power than a quiet one — doubling the amplitude quadruples the power. It also explains why distant sources appear faint: as a wave spreads out from a point source, the energy is distributed over a larger surface area (a sphere of surface area $4\pi r^2$4πr2), and the intensity (power per unit area) falls as the inverse square of the distance:

$I = \frac{P}{4\pi r^2}$I=4πr2P​

You will use this inverse-square law again when you study electromagnetic radiation from stars (Module 5) and in the photoelectric-effect and quantum modules.

A second consequence is that intensity scales as the square of amplitude:

$I \propto A^2$I∝A2

This relationship will reappear immediately in the Malus's-law lesson, where the cosine-squared dependence of transmitted intensity through a Polaroid filter is the direct consequence of amplitude being proportional to $\cos\theta$cosθ.

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Worked Example 1 — Classifying Wave Types

Q. Classify each of the following as transverse or longitudinal, and state whether it can be polarised: (a) ultrasound in water, (b) microwaves in air, (c) a ripple on a pond, (d) the vibration of a violin string, (e) a seismic P-wave.

A.

(a) Longitudinal, cannot be polarised — sound (including ultrasound) in fluids is always longitudinal. (b) Transverse, can be polarised — all electromagnetic waves, including microwaves, are transverse. (c) Transverse (to A-Level approximation), can be polarised — water waves are treated as transverse, with surface particles moving vertically. (d) Transverse, can be polarised — the string moves perpendicular to its length. (e) Longitudinal, cannot be polarised — P-waves are primary (pressure) waves, longitudinal in character. (S-waves, by contrast, are transverse shear waves.)

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Worked Example 2 — Why Sound Cannot Be Polarised

Q. A student claims that "if you stretch a horizontal slit narrow enough, sound passing through it will be polarised in the horizontal plane, just like light through a polaroid". Explain in two or three sentences why this claim is wrong.

A. Sound is a longitudinal wave: the particle oscillations are along the direction of propagation, not perpendicular to it. Polarisation is a restriction of perpendicular oscillations to a single plane; since longitudinal waves have no perpendicular oscillations to restrict, the concept does not apply. The narrow slit will diffract the sound (because the wavelength is comparable to the slit width) but it cannot polarise it.

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Worked Example 3 — Intensity from a Point Source

Q. A small loudspeaker emits sound with a power of $0.40$0.40 W uniformly in all directions. Estimate the sound intensity at a distance of $2.0$2.0 m from the loudspeaker, ignoring absorption by the air.

A. The loudspeaker is treated as a point source radiating into a sphere of radius $r = 2.0$r=2.0 m. The surface area of that sphere is $4\pi r^2 = 4\pi (2.0)^2 = 50.3$4πr2=4π(2.0)2=50.3 m$^2$2. Hence:

$I = \frac{P}{4\pi r^2} = \frac{0.40}{50.3} = 7.95 \times 10^{-3} \text{ W m}^{-2} \approx 8.0 \text{ mW m}^{-2}.$I=4πr2P​=50.30.40​=7.95×10−3 W m−2≈8.0 mW m−2.

At twice that distance ($4.0$4.0 m), the intensity would fall to one quarter, by the inverse-square law: about $2.0$2.0 mW m$^{-2}$−2.

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Synoptic Links

• Wave parameters and the wave equation (next lesson, Module 4.4) — the qualitative picture introduced here (oscillation, propagation, wavefront) is quantified in the next lesson by the wave equation $v = f\lambda$v=fλ and the notion of phase difference. The energy-amplitude relationship $I \propto A^2$I∝A2 established here is the foundation for Malus's law later in this module.
• Wave-particle duality (Module 4.5) — the electron diffraction and photoelectric effect of Module 4.5 only make sense once the wave picture of this lesson is secure. The de Broglie wavelength $\lambda = h/p$λ=h/p promotes the wave concept from a classical description of mechanical or electromagnetic disturbance to a quantum description of matter itself.
• Stationary waves (later in this module) — progressive waves on a string of finite length set up stationary waves by superposition with their own reflections. The transverse vs longitudinal distinction made here recurs there: stationary waves on strings are transverse, stationary waves in pipes are longitudinal sound waves.

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Specimen question modelled on the OCR H556 paper format

Question (9 marks): A long horizontal spring (slinky) is laid on a smooth table. A student holds one end and pushes it sharply forwards and backwards along its length, sending a wave along the spring.

(a) State what is meant by a progressive wave, making clear what is and is not transferred by the wave. [2]

(b) State whether the wave on the slinky in this experiment is transverse or longitudinal. Justify your answer in terms of the direction of particle oscillation. [2]

(c) A second student takes a piece of unsharpened pencil and rests it lightly on the slinky midway along its length, marking the spring with a small dab of paint. Describe and explain the motion of the painted region as the wave passes. Include a comment on its net displacement after the wave has fully passed. [3]

(d) Sound waves in air at room temperature travel at approximately $340$340 m s$^{-1}$−1 and a particular note has a frequency of $440$440 Hz. State the type of wave (transverse or longitudinal) and calculate its wavelength. Then explain in one sentence why this sound cannot be polarised. [2]

AO breakdown

Mark	AO	Awarded for
1	AO1	"Energy is transferred through the medium"
2	AO1	"No net transfer of matter" / particles oscillate about a mean
3	AO1	Identifies wave as longitudinal
4	AO2	Justifies via particle oscillation parallel to propagation
5	AO1	Describes oscillatory motion of painted region
6	AO2	Direction of oscillation matches longitudinal nature (parallel to spring)
7	AO2	Net displacement zero after wave passes
8	AO2	Sound identified as longitudinal; $\lambda = v/f = 340/440 = 0.77$λ=v/f=340/440=0.77 m
9	AO3	Polarisation requires perpendicular oscillation; longitudinal has none

AO split: AO1 = 4, AO2 = 4, AO3 = 1.

Mid-band response (4/9)

(a) A progressive wave is a wave that moves energy from one place to another.

(b) The wave is longitudinal because it goes along the spring.

(c) The painted bit moves back and forwards as the wave goes past.

(d) Sound is a longitudinal wave. Wavelength = $340/440 = 0.77$340/440=0.77 m. Sound cannot be polarised because it goes through air.

Examiner commentary: The next-band move is precision on the no net transfer of matter clause, and on the direction of particle oscillation as the discriminator between transverse and longitudinal. Mark 1 awarded; Mark 2 not awarded (no mention of matter or net displacement). Mark 3 awarded; Mark 4 not awarded — "goes along the spring" describes propagation, not oscillation, so the longitudinal justification is missing. In (c), Mark 5 awarded for back-and-forwards; Mark 6 not awarded explicitly because the candidate did not connect the direction to longitudinal nature; Mark 7 not awarded — no comment on net displacement. In (d), Mark 8 awarded for the correct wavelength; Mark 9 not awarded — "goes through air" is irrelevant; the candidate needs to invoke the perpendicular-oscillation requirement for polarisation.

Stronger response (7/9)

(a) A progressive wave is a wave in which energy is transferred through a medium without the bulk transport of matter. Particles of the medium oscillate about fixed equilibrium positions.

(b) The wave is longitudinal: the student pushes the slinky along its length, so the coils oscillate parallel to the direction in which the wave travels.

(c) The painted region of the spring oscillates back and forth along the length of the slinky as the wave passes through it. The amplitude of its motion is the wave amplitude; the period of its motion is the wave period. Once the wave has passed, the painted region returns to its original position — there is no net displacement.

(d) Sound is a longitudinal wave. $\lambda = v/f = 340/440 = 0.77$λ=v/f=340/440=0.77 m. Sound cannot be polarised because it is longitudinal.

Examiner commentary: To lift to top-band, the (d) justification needs to spell out why longitudinal precludes polarisation — namely that polarisation restricts perpendicular oscillations to a plane, and longitudinal waves have no perpendicular oscillation component. Marks 1-8 awarded; Mark 9 partial — the candidate has identified that longitudinality is the reason but has not explained the underlying mechanism. A confident, accurate answer with one synoptic gap.

Top-band response (9/9)

(a) A progressive wave is a disturbance that propagates through a medium (or, for electromagnetic waves, through vacuum), transferring energy without net transfer of matter. Each particle of the medium oscillates about its mean position; there is local motion of matter but zero net displacement averaged over a cycle.

(b) The wave is longitudinal. The student's "sharply forwards and backwards along its length" motion causes the coils of the slinky to oscillate parallel to the direction of propagation, producing alternating compressions and rarefactions along the spring.

(c) The painted region oscillates back and forth along the length of the slinky — parallel to the direction in which the wave travels — confirming the longitudinal nature observed in (b). Its amplitude equals the wave amplitude; its angular frequency $\omega = 2\pi f$ω=2πf equals that of the source. Crucially, after the wave has fully passed, the painted region returns to its original equilibrium position with zero net displacement: the wave has transferred energy along the spring but the painted matter itself has not been displaced as a whole.

(d) Sound in air is a longitudinal wave. Using $v = f\lambda$v=fλ:

$\lambda = \frac{v}{f} = \frac{340}{440} = 0.77 \text{ m}.$λ=fv​=440340​=0.77 m.

Sound cannot be polarised because polarisation is, by definition, the restriction of perpendicular oscillation to a single plane; a longitudinal wave has no perpendicular oscillation component to restrict. The concept of polarisation only has content for transverse waves.

Examiner commentary: Full marks. The discriminators that lift this to top-band: (i) the explicit no net transfer of matter clause in (a), with the "averaged over a cycle" caveat; (ii) the precise causal chain in (b) — particle oscillation direction → longitudinal classification — rather than the loose "it goes along the spring"; (iii) the zero-net-displacement statement in (c), explicit rather than implied; (iv) in (d), a definitional justification for why longitudinal waves cannot be polarised (no perpendicular component to restrict), not just a restatement of the fact.

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Common errors and mark-loss patterns

Pedagogical observations from teaching A-Level wave motion, with no fabricated examiner-report percentages.

• Claiming that "particles travel with the wave" — particles oscillate about fixed equilibrium positions; only the pattern of disturbance travels. OCR mark schemes consistently penalise this in part (a)-style "describe a progressive wave" questions.
• Treating sound as transverse — sound in fluids (air, water) is always longitudinal. The misconception often arises because the displacement-vs-position graph of a longitudinal wave is sinusoidal, leading students to infer that the particles must be moving "up and down".
• Stating that "all waves can be polarised" — only transverse waves can be polarised.
• Mixing up P-waves and S-waves — P (primary, pressure) waves are longitudinal and faster; S (secondary, shear) waves are transverse and slower. The mnemonic is in the names.
• Assuming wave speed is set by the source — wave speed depends on the medium (tension, density, bulk modulus, permittivity etc.); the source controls only the frequency. The wavelength is the dependent variable, set by $\lambda = v/f$λ=v/f.
• Describing water waves as "purely transverse" — they are approximately so at A-Level but technically have both components. Do not claim mathematical perfection unless asked.
• Confusing wavefront with ray — wavefronts join points of equal phase; rays are perpendicular to wavefronts and indicate the direction of energy transfer.
• Forgetting the inverse-square law for intensity from a point source — many candidates remember $P \propto A^2$P∝A2 but not $I = P/(4\pi r^2)$I=P/(4πr2), costing marks in any "compare loudness at two distances" question.

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Going further

Christiaan Huygens's 1678 Traité de la Lumière (published 1690) proposed the wave model of light at a time when Isaac Newton's competing corpuscular model dominated. Huygens's construction — each point on a wavefront acts as a secondary source of spherical wavelets — predicts straight-line propagation in homogeneous media (the laws of geometric optics) but also correctly handles refraction at boundaries, an early piece of evidence in favour of the wave picture. The decisive case for waves was made by Thomas Young's double-slit experiment in 1801 (which you will study later in this module) and then sealed by James Clerk Maxwell's equations in the 1860s, which derived electromagnetic waves from first principles and predicted their speed in vacuum to be the measured speed of light.

At undergraduate level the qualitative wave picture is replaced by the wave equation, a partial differential equation:

$\frac{\partial^2 \psi}{\partial t^2} = v^2 \frac{\partial^2 \psi}{\partial x^2}$∂t2∂2ψ​=v2∂x2∂2ψ​

for a wave $\psi(x,t)$ψ(x,t) travelling at speed $v$v. The transverse-vs-longitudinal distinction becomes a property of the vector field $\vec{\psi}$ψ​ — whether it points perpendicular or parallel to the direction of propagation. For electromagnetic waves, Maxwell's equations force the field vectors to be transverse to the direction of energy transfer (Poynting's vector $\vec{S} = \vec{E} \times \vec{B}/\mu_0$S=E×B/μ0​). This is not a contingent fact about electromagnetism; it is a consequence of the divergence equations $\nabla \cdot \vec{E} = 0$∇⋅E=0 and $\nabla \cdot \vec{B} = 0$∇⋅B=0 in vacuum.

Oxbridge interview prompts that probe wave motion conceptually:

1. "You stand at the edge of a lake and drop a stone into still water. A few seconds later a small twig 5 m out begins to bob up and down. Has any water actually travelled from where you dropped the stone to the twig? Justify your answer in two sentences." Strictly no — the water particles oscillate about their mean positions; the wave pattern travels outward, but the water itself does not undergo net displacement. The twig bobs because the local water under it is oscillating vertically.
2. "If sound is longitudinal, why does a sine-wave graph appear on the oscilloscope when you record it with a microphone?" The oscilloscope is plotting the signed displacement (or pressure variation) of the air at the microphone diaphragm against time. Whether the underlying oscillation is parallel or perpendicular to the propagation, the trace of that signed quantity vs time is still a sinusoid.
3. "Why do S-waves not propagate through the Earth's outer core, and what does that tell us about the outer core?" S-waves are transverse — they rely on shear restoring forces. Fluids cannot sustain static shear, so transverse mechanical waves do not propagate through them. The "shadow zone" for S-waves observed on the opposite side of the Earth from a deep earthquake is direct evidence that the outer core is liquid.

Recommended reading: Vibrations and Waves by A. P. French (MIT introductory series) is a classic at undergraduate level. For a historical view, the early chapters of QED (Feynman) trace the wave-vs-particle debate from Huygens to Schrödinger.

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A-Level misconceptions

• "Waves transfer matter" — they do not. Each particle oscillates about a mean position, and the net displacement of any particle over a complete cycle is zero. Only energy (and information / momentum) is transferred along the direction of propagation.
• "Wave speed depends on the source" — it does not, for mechanical waves. Speed is a property of the medium (tension, density, modulus, permittivity). The source controls only frequency; wavelength is then forced by $v = f\lambda$v=fλ.
• "Longitudinal waves do not have a sinusoidal graph" — they do. The displacement-vs-position graph of a longitudinal wave is sinusoidal in signed parallel-to-propagation displacement.
• "Sound can be polarised by a very narrow slit" — it cannot. A slit diffracts sound but cannot polarise it; longitudinal waves have no perpendicular oscillation component to restrict.
• "All waves can travel through vacuum" — only electromagnetic waves can. Mechanical waves (sound, water waves, seismic waves) require a medium.
• "Wavefronts and rays are the same thing" — they are perpendicular to each other. Wavefronts join points of equal phase; rays point in the direction of energy transfer.
• "Intensity falls linearly with distance from a point source" — it falls as $1/r^2$1/r2 because the energy is spread over a sphere whose surface area grows as $4\pi r^2$4πr2. Amplitude correspondingly falls as $1/r$1/r.

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Summary

• A progressive wave transfers energy through a medium (or vacuum) without net transfer of matter; particles oscillate about fixed mean positions.
• Transverse waves: oscillation perpendicular to propagation. Examples: EM waves, waves on a string, S-waves. Can be polarised.
• Longitudinal waves: oscillation parallel to propagation. Examples: sound, P-waves, slinky compression waves. Cannot be polarised.
• Wavefronts join points of equal phase; rays are drawn perpendicular to wavefronts and indicate the direction of energy transfer.
• Speed depends on the medium; frequency depends on the source; wavelength is forced by $v = f\lambda$v=fλ.
• Intensity from a point source: $I = P/(4\pi r^2)$I=P/(4πr2), and $I \propto A^2$I∝A2.