Wave motion

Learning goals

What students should be able to explain, calculate and investigate after completing Waves and Wave Motion.

• Distinguish transverse from longitudinal waves.
• Define and label the key characteristics of a wave.
• Use v = fλ to solve wave problems.
• Explain the principle of superposition and distinguish constructive from destructive interference.
• Explain resonance and the Doppler effect.

Key definitions

Precise terms, symbols and SI units used in Waves and Wave Motion.

Transverse and Longitudinal Waves

All waves transfer energy without transferring matter overall — individual particles (or field values) oscillate about a fixed rest position, but do not travel along with the wave.

In a transverse wave, oscillation is perpendicular to the direction of travel. A wave on a stretched string is the clearest mechanical example. In a longitudinal wave, particles move back and forth parallel to the direction of travel, producing compressions and rarefactions. Surface water waves are more complicated: particles have combined up-and-down and back-and-forth motion, so they should not be treated as purely transverse.

Wave Characteristics and Wave Speed

A wave is fully described by its amplitude, wavelength, period, and frequency.

A displacement–distance graph is a snapshot of the whole wave at one instant; horizontal spacing between matching points gives wavelength. A displacement–time graph follows one point as it oscillates; horizontal spacing between matching points gives period. The graphs can have the same shape, but their horizontal axes describe different things.

Superposition and Interference

When two waves overlap at the same point, their displacements add according to the principle of superposition.

When waves overlap, the resultant displacement is the algebraic sum of their individual displacements. Coherent sources maintain a constant phase difference. Constructive interference occurs where waves arrive in phase; destructive interference occurs where they arrive out of phase by half a cycle. After overlapping, the waves continue with their original properties.

Two narrow slits illuminated by the same source act as coherent sources. Their waves overlap on a screen, producing alternating bright and dark fringes: bright where the path difference is a whole number of wavelengths and dark where it is a half-odd number of wavelengths.

• Coherence means a constant phase difference and the same frequency.
• Increasing wavelength increases fringe spacing; increasing slit separation decreases fringe spacing; moving the screen farther away increases fringe spacing.
• The stable fringe pattern is evidence for the wave nature of light.

Resonance

Every physical system that can vibrate has one or more natural frequencies at which it vibrates most easily.

A vibrating system has a natural frequency. A periodic driving force transfers energy most efficiently when its driving frequency matches that natural frequency. The amplitude then becomes large: this is resonance. Damping removes energy and limits the maximum amplitude.

• Every oscillating system has one or more natural frequencies.
• Resonance occurs when the driving frequency matches a natural frequency.
• Energy transfer is then most effective and amplitude becomes large.

• Useful example: tuning a radio or producing a strong musical note.
• Potentially harmful example: excessive vibration of a bridge, machine or building.

The Doppler Effect

When a wave source and an observer are moving relative to each other, the observed frequency differs from the frequency actually emitted by the source.

If the source moves towards the observer (or the observer towards the source), successive wave crests are emitted closer together in the direction of approach, so the observed wavelength shortens and the observed frequency increases. If they are moving apart, the observed wavelength lengthens and the observed frequency decreases. This is commonly experienced with the changing pitch of a passing ambulance siren, and is also used in astronomy (redshift/blueshift of light from stars and galaxies) to measure relative motion. The formula f' = fc/(c ± u) given in the Formula Box applies specifically to a moving source and a stationary observer, where c is the wave's speed in the medium it is travelling through (not the speed of the source).

Diagrams and formulae

Key diagrams and equations for reviewing Waves and Wave Motion.

• Transverse wave (amplitude and wavelength labelled) vs longitudinal wave (compressions and rarefactions).
• Superposition: waves in phase add constructively (larger resultant); waves out of phase add destructively (smaller or cancelled resultant).
• Doppler effect: wavefronts bunch up ahead of a moving source (higher observed frequency) and spread out behind it (lower observed frequency).

Transverse wave (amplitude and wavelength labelled) vs longitudinal wave (compressions and rarefactions).

Superposition: waves in phase add constructively (larger resultant); waves out of phase add destructively (smaller or cancelled resultant).

Doppler effect: wavefronts bunch up ahead of a moving source (higher observed frequency) and spread out behind it (lower observed frequency).

Worked Example

Use v = fλ consistently: identify the frequency and wavelength, convert to SI units, substitute with units, and interpret the resulting wave speed.

Question: A wave has frequency 5.0 Hz and wavelength 0.40 m. Find its speed.

• Known: f = 5.0 Hz and λ = 0.40 m.
• Equation: v = fλ.
• Substitution: v = (5.0 s⁻¹)(0.40 m).
• Answer: v = 2.0 m s⁻¹.
• Interpretation: the disturbance travels 2.0 m along the medium each second.

Practical / Investigation

Experimental planning, measurement and evaluation for Waves and Wave Motion.

Investigation: Investigating resonance using secondary sources / simple demonstration

• Simple demonstration: hang several pendulums of different lengths from a common horizontal string; set one swinging and observe that only the pendulum with a matching natural frequency (same length) picks up significant motion from the small driving vibrations transmitted along the string.
• Secondary-source task: research a real-life example of resonance (e.g. the Tacoma Narrows Bridge collapse, or resonance in musical instruments) and explain the underlying physics in terms of driving frequency matching natural frequency.
• Nature of Science link: this illustrates how a physical principle (resonance) applies across very different contexts — from a simple demonstration in the classroom to major engineering case studies.

• Use pendulums with clearly different lengths plus one matching pair. Start one driver pendulum and compare the response amplitudes.
• Keep bob masses, suspension conditions and initial driver amplitude reasonably consistent; identify the matching natural frequency as the cause of energy transfer.
• For a quantitative extension, drive one oscillator through measured frequencies and plot steady-state amplitude against driving frequency.

• Direct a monochromatic light source through a double slit onto a screen and observe the central and neighbouring bright fringes.
• Measure across several fringe spacings and divide, rather than measuring one very small spacing.
• Keep the slit plate parallel to the screen and follow laser safety: never view the direct or reflected beam.

Quick-Check Questions

Short retrieval questions covering the essential ideas in Waves and Wave Motion.

1. Give one example each of a transverse wave and a longitudinal wave.

2. Define wavelength and state its unit.

3. State the principle of superposition.

4. What condition causes resonance?

5. What happens to the observed frequency of a wave source moving towards an observer?

Common mistakes and exam tips

Common misconceptions and exam-technique checks for Waves and Wave Motion.

• Avoid: Thinking particles travel with a progressive wave. Correct idea: Particles oscillate about equilibrium while energy and the wave pattern travel through the medium.
• Avoid: Saying wave energy is proportional to amplitude. Correct idea: For otherwise comparable waves, transferred energy is proportional to amplitude squared.
• Avoid: Restricting resonance to sound. Correct idea: Resonance can occur in any oscillating system when a driving frequency matches a natural frequency.

• Distinguish transverse and longitudinal motion using the direction of oscillation relative to wave travel.
• For Doppler calculations, identify which of source and observer is moving before selecting signs.
• In a resonance explanation, name both the driving frequency and the natural frequency.

Exam-style questions

Five exam-style questions with worked answers on Waves and Wave Motion.

Higher Level — Required Content

Required Higher Level content for this topic.

The Doppler effect is the apparent change in frequency caused by relative motion between a source and an observer.

• Use the minus sign when the source approaches and the plus sign when it recedes.
• Define every symbol and choose the sign from the physical situation before substituting.

End-of-unit revision

Formulae, definitions, experiments and mixed revision questions for Waves and Wave Motion.

• v = fλ
• f = 1/T
• f' = fc/(c ± u) (Doppler effect, moving source, stationary observer)

• Transverse vs longitudinal wave
• Amplitude, wavelength, period, frequency
• Superposition
• Resonance
• Doppler effect

• Transverse and longitudinal wave shapes, correctly labelled
• Constructive and destructive interference
• Doppler effect wavefront pattern around a moving source

• Demonstrating resonance (e.g. pendulum/string demonstration) and researching a real-life resonance example

• Confusing transverse and longitudinal waves
• Thinking particles travel with the wave
• Forgetting energy depends on amplitude squared
• Mixing up +/− in the Doppler formula

1. A sound wave has a frequency of 440 Hz and travels at 340 m s⁻¹. Calculate its wavelength.

2. Explain, in terms of particle motion, why sound cannot travel through a vacuum but light can.

3. A source emits sound of frequency 500 Hz and moves towards a stationary observer at 30 m s⁻¹. Using c = 340 m s⁻¹, calculate the observed frequency.

4. Explain why soldiers are traditionally told to break step when crossing a bridge.

1. λ = v/f = 340 ÷ 440 ≈ 0.77 m.

2. Sound is a mechanical (longitudinal) wave and needs particles of a medium to pass the vibration along; light is an electromagnetic (transverse) wave and does not need a medium, so it can travel through the vacuum of space.

3. f' = fc/(c−u) = 500×340/(340−30) = 170000/310 ≈ 548.4 Hz.

4. Marching in step creates a strong periodic driving force at a fixed frequency; if this matches the bridge's natural frequency, resonance could build up dangerously large oscillations, so breaking step avoids a sustained matching driving frequency.

The table below shows which equations from this unit appear in the official State Examinations Commission Formulae and Tables book, and where.

Wave characteristics, superposition and resonance are for all students; the mathematical Doppler model is identified as Higher Level content.