Physics notes, chapter 5
Waves
Textbook Physics Form 4 KSSM (DLP), pages 170–229
What this chapter covers
- Explain how waves are produced and that waves transfer energy without transferring matter; classify waves (5.1)
- Compare transverse and longitudinal waves; define amplitude, period, frequency, wavelength and wave speed (5.1)
- Sketch and interpret displacement-time and displacement-distance graphs; use v = fλ (5.1)
- Explain damping and resonance with examples from daily life (5.2)
- Describe reflection of waves, draw wavefront diagrams and solve problems involving echoes (5.3)
- Describe refraction of waves, draw diagrams and use v₁/λ₁ = v₂/λ₂ (5.4)
- Describe diffraction and the effect of slit size and wavelength (5.5)
- Explain superposition and interference, and use λ = ax/D (5.6)
- State the characteristics, order and uses of electromagnetic waves (5.7)
5.1Fundamentals of waves
Wave — a disturbance made when a medium vibrates about a fixed position. Waves transfer energy from one place to another without transferring the matter of the medium.
An oscillating system (for example an iron bob on a spring) or a vibrating system (for example a tuning fork) produces waves. Oscillation and vibration are repeated motions about an equilibrium position in a closed path.
Tie a ribbon to a slinky spring and shake one end: the energy travels along the spring, but the ribbon only vibrates on the spot. The ribbon does not travel in the direction of the energy.
Ways to classify waves
| Classified by | Types | Key idea |
|---|---|---|
| Wave profile | Progressive wave | The wave profile moves with time in the direction of propagation (water ripples). |
| Stationary wave | The wave profile does not move with time (a plucked guitar string). Made when two identical progressive waves travel in opposite directions and meet. | |
| Need for a medium | Mechanical wave | Needs a medium: water waves, sound waves, seismic waves. |
| Electromagnetic wave | Needs no medium. Oscillating electric and magnetic fields at right angles: radio waves, light, gamma rays. | |
| Direction of vibration | Transverse / longitudinal | See below. |
Transverse and longitudinal waves
| Transverse wave | Longitudinal wave | |
|---|---|---|
| Vibration of particles | Perpendicular to the direction of propagation | Parallel to the direction of propagation |
| Made up of | Crests and troughs | Compressions and rarefactions |
| Examples | Water waves, light, radio waves | Sound waves |
Based on Physics Form 4 (DLP) textbook, Figures 5.10 and 5.11, p. 177.
Do not say a wave "carries the water along". Water particles in a ripple only move up and down; it is the energy that moves outwards.
Terms used to describe waves
| Term | Meaning | Unit |
|---|---|---|
| Amplitude, A | Maximum displacement of a particle from its equilibrium position | m or cm |
| Period, T | Time for one complete oscillation (or one complete wave cycle) | s |
| Frequency, f | Number of complete oscillations (or cycles) per second; f = 1/T | Hz |
| Wavelength, λ | Distance between two consecutive points that are in phase (crest to crest, compression to compression) | m |
| Wave speed, v | Distance travelled by the wave profile per second | m s⁻¹ |
v = fλ and f = 1/T
In one period the wave profile moves forward by one wavelength, so v = λ/T = fλ.
In one period the wave profile moves forward by one wavelength, so v = λ/T = fλ.
Example: a rope wave
A student shakes a rope so that 6 complete waves pass a point every 3 s, and the wavelength is 0.50 m.f = 6 ÷ 3 = 2 Hz. v = fλ = 2 × 0.50 = 1.0 m s⁻¹. T = 1 ÷ 2 = 0.5 s.
Wave graphs
Two graphs describe a wave. Read different things from each.
Displacement against timeGives amplitude A and period T (so f = 1/T).
Displacement against distanceGives amplitude A and wavelength λ.
To find wave speed you need both graphs: λ from the distance graph and T from the time graph, then v = λ/T.
Example: using both graphs
The time graph shows T = 0.4 s and the distance graph shows λ = 8 cm.v = λ/T = 8 ÷ 0.4 = 20 cm s⁻¹ (f = 1 ÷ 0.4 = 2.5 Hz).
Ripple tank
A ripple tank shows water waves. A plane dipper makes plane waves; a spherical dipper makes circular waves. A stroboscope flashes at the same frequency as the wave so the pattern looks frozen; measure the distance between bright fringes to get λ.
A wave has frequency 50 Hz and wavelength 0.20 m. Find its speed and period.
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v = fλ = 50 × 0.20 = 10 m s⁻¹. T = 1/f = 1/50 = 0.02 s.
5.2Damping and resonance
Damping — the decrease in amplitude of an oscillating system because it loses energy. The frequency stays the same; only the amplitude falls.
| Type | Energy is lost to |
|---|---|
| External damping | Overcoming friction or air resistance |
| Internal damping | Stretching and compression of the particles in the system itself |
The damping can be overcome by a periodic external force that supplies energy to replace the energy lost. The system is then in forced oscillation.
Resonance — happens when a periodic force is applied at the natural frequency of the system. The system oscillates with its natural frequency and with maximum amplitude.
- Tuning forks: when fork P is struck, fork Q of the same natural frequency vibrates and gives out sound without being touched.
- Barton's pendulum: the pendulum with the same length (same natural frequency) as the driver pendulum swings with the largest amplitude.
| Effect of resonance | Example |
|---|---|
| Harmful | The Tacoma Narrows bridge (1940) collapsed when wind made it oscillate with large amplitude; the Millennium Footbridge in London swayed when about 2 000 people walked on it. |
| Useful | Tuning of musical instruments. |
Resonance needs the same frequency, not the same amplitude. A large driver force at the wrong frequency gives only a small response.
Why does a swing that is left alone eventually stop?
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Damping: it loses energy to air resistance and friction at the pivot, so its amplitude decreases until it stops.
5.3Reflection of waves
Wavefront — a line joining points that are in phase (same displacement, same distance from the source). The direction of propagation is perpendicular to the wavefront, and the distance between two consecutive wavefronts is one wavelength.
| Characteristic | After reflection |
|---|---|
| Angle | Angle of incidence = angle of reflection |
| Wavelength | No change |
| Frequency | No change |
| Speed | No change |
| Direction of propagation | Changes |
Drawing a reflection
- Draw the incident ray perpendicular to the incident wavefronts.
- Draw the normal at the point where the ray meets the reflector.
- Draw the reflected ray so that r = i.
- Draw reflected wavefronts perpendicular to the reflected ray, with the same spacing (wavelength) as the incident ones.
The angle of incidence is measured between the ray and the normal, not between the ray and the reflector. It also equals the angle between the wavefront and the reflector.
Uses
- Ultrasound scanning of a foetus or internal organs.
- Parabolic dish of a satellite reflects radio waves onto the antenna.
- SONAR: finds fish and the depth of the sea from echoes. Seismic reflection finds oil and gas under the seabed.
Example: echo sounding (textbook p. 193)
An ultrasonic pulse takes 120 ms to go from the ship to the seabed and back. Speed in water = 1 500 m s⁻¹; frequency 25 kHz.The distance travelled is 2d: 2d = vt = 1 500 × 0.120 = 180 m, so depth d = 90 m.
λ = v/f = 1 500 ÷ 25 000 = 0.06 m.
A ship sends a sonar pulse and receives the echo 0.08 s later. Speed in water is 1 500 m s⁻¹. What is the depth?
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2d = 1 500 × 0.08 = 120 m, so d = 60 m. (Remember the pulse travels down and up.)
5.4Refraction of waves
Refraction — the change in direction of waves when their speed changes as they move from one medium (or depth) into another.
Water wave speed depends on depth: faster in deep water, slower in shallow water. Sound speed depends on air density (temperature); light speed depends on optical density.
| Characteristic | Deep to shallow | Shallow to deep |
|---|---|---|
| Speed | Decreases | Increases |
| Wavelength | Decreases | Increases |
| Frequency | No change | No change |
| Direction | Towards the normal (i > r) | Away from the normal (i < r) |
Frequency is fixed by the source, so it never changes at a boundary. Since v = fλ with f constant, v₁/λ₁ = v₂/λ₂.
Example: refraction calculation
A wave of wavelength 2 cm and speed 8 cm s⁻¹ moves from a shallow region into a deep region where its speed is 12 cm s⁻¹.8/2 = 12/λ₂, so λ₂ = 12 × 2 ÷ 8 = 3 cm.
Refraction in daily life
- Day: air near the ground is hotter, sound travels faster there and bends away from the ground, so distant sound is weak.
- Night: air near the ground is colder, sound bends towards the ground and is heard more clearly.
- Sea waves: shallow water at a cape slows the waves and bends the wavefronts to follow the shore; energy converges at the cape, so amplitude is larger there than in the bay.
Water waves of frequency 10 Hz have wavelength 3.0 cm in deep water and 1.8 cm in shallow water. Find the speed in each.
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Deep: v = 10 × 3.0 = 30 cm s⁻¹. Shallow: v = 10 × 1.8 = 18 cm s⁻¹. Frequency is 10 Hz in both.
5.5Diffraction of waves
Diffraction — the spreading of waves when they pass through a gap or round the edge of a barrier.
| Characteristic | After diffraction | Reason |
|---|---|---|
| Wavelength | No change | Speed and frequency are the same |
| Frequency | No change | Same source |
| Speed | No change | Same medium |
| Amplitude | Decreases | Energy is spread over a wider region |
| Direction | One direction to many directions | Wavefronts spread out |
Factors: diffraction is more obvious when the gap is narrower (close to the wavelength) and when the wavelength is longer.
Examples
- Water: a gap in an embankment makes a calm region, good for docking ships and recreation.
- Light: holograms on bank cards use diffraction of light.
- Sound: you hear someone round a corner, even though you cannot see them; elephants use long-wavelength infrasound to communicate over long distances.
Sound is heard round a corner because sound has a much longer wavelength than light, not because sound is "stronger". Light has such short wavelengths that it hardly diffracts at an ordinary doorway.
A gap is made narrower while the waves are unchanged. What happens to the diffraction and to the wavelength?
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Diffraction (spreading) becomes more obvious. The wavelength does not change.
5.6Interference of waves
Principle of superposition — when two waves overlap, the resultant displacement is the sum of the displacements of the two waves.
Coherent sources — two sources with the same frequency and a constant phase difference. Interference is the superposition of waves from coherent sources.
| Superposition of | Result | Type |
|---|---|---|
| Crest + crest (A + A) | Displacement 2A | Constructive interference |
| Trough + trough (−A − A) | Displacement −2A | Constructive interference |
| Crest + trough (A − A) | Displacement 0 | Destructive interference |
| Water | Light | Sound | |
|---|---|---|---|
| Constructive | Large amplitude (bright area on screen) | Bright fringe | Loud sound |
| Destructive | Zero amplitude (dark / grey area) | Dark fringe | Soft sound |
The formula λ = ax / D
- a = distance between the two coherent sources (or double slits)
- x = distance between two adjacent antinodal (or nodal) lines, or adjacent bright fringes
- D = perpendicular distance from the sources to where x is measured
λ = ax / D or x = λD / a
- x is larger when λ is larger or D is larger.
- x is smaller when a is larger.
Example: Young's double-slit experiment (textbook p. 217)
a = 0.4 mm = 0.4 × 10⁻³ m, x = 4.5 mm = 4.5 × 10⁻³ m, D = 3.0 m.λ = ax/D = (0.4 × 10⁻³)(4.5 × 10⁻³) ÷ 3.0 = 6.0 × 10⁻⁷ m.
Convert mm to m before substituting. All of a, x and D must be in metres.
Uses of destructive interference
- The bulbous bow of a ship makes waves that cancel the hull waves, so there is less drag.
- Anti-reflection coatings on spectacle lenses cancel reflected light.
- Noise-cancelling headphones make sound that cancels the surrounding noise.
In a double-slit experiment the screen is moved further from the slits. What happens to the fringe separation x?
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x = λD/a, so x increases when D increases (the fringes are further apart).
5.7Electromagnetic waves
Young showed in 1801 that light is a wave; Maxwell proposed in 1862 that light is an electromagnetic wave; Hertz produced radio waves in 1887.
Electromagnetic waves are transverse waves made of an electric field and a magnetic field that oscillate at right angles to each other and to the direction of propagation.
- Need no medium; can travel through a vacuum.
- Speed in vacuum c = 3.00 × 10⁸ m s⁻¹; slower in any other medium.
- Can be reflected, refracted, diffracted and show interference.
- Use c = fλ (in vacuum or air).
From radio waves to gamma rays: wavelength decreases while frequency and energy increase. Gamma rays and X-rays carry large amounts of energy and must be handled carefully.
| Wave | Applications |
|---|---|
| Radio wave | Radio and TV broadcasting, long-distance communication, Wi-Fi and Bluetooth, airport body scanning |
| Microwave | Satellite communication, mobile phones, radar and speed traps, microwave ovens |
| Infrared | Grills and toasters, night vision, TV remote controls, drying paint, treating muscle pain |
| Visible light | Seeing, photography, photosynthesis, lasers and optical fibres |
| Ultraviolet | Hardening tooth fillings, checking banknotes, treating jaundice, sterilising, purifying water |
| X-ray | Finding broken bones, checking welds, airport baggage scanning |
| Gamma ray | Killing cancer cells, sterilising medical equipment and food |
Example: wavelength in a medium
Red light of wavelength 700 nm in air (c = 3.00 × 10⁸ m s⁻¹) enters glass where its speed is 2.00 × 10⁸ m s⁻¹.The frequency is unchanged: f = 3.00 × 10⁸ ÷ 700 × 10⁻⁹ = 4.29 × 10¹⁴ Hz.
λ in glass = v/f = 2.00 × 10⁸ ÷ 4.29 × 10¹⁴ = 4.67 × 10⁻⁷ m (467 nm). Equivalent: λ₂ = 700 × (2.00 ÷ 3.00) = 467 nm.
Put these in order of increasing frequency: ultraviolet, radio wave, infrared, gamma ray.
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Radio wave, infrared, ultraviolet, gamma ray.
Summary
| Remember | Key fact |
|---|---|
| Wave speed | v = fλ; f = 1/T; c = fλ for electromagnetic waves |
| Wave types | Vibration perpendicular / parallel to wave direction |
| Damping | Amplitude falls, frequency constant |
| Resonance | Driving frequency = natural frequency |
| Reflection | i = r; λ, f and v unchanged |
| Refraction | f unchanged; v and λ change; v₁/λ₁ = v₂/λ₂ |
| Diffraction | λ, f, v unchanged; amplitude falls |
| Interference | Constructive: crest + crest; destructive: crest + trough |