Describing progressive waves
Section: Waves | Syllabus: Cambridge AS Level Physics (9702)
What is a Progressive Wave?
A progressive wave is a periodic disturbance that transfers energy without transferring matter. This periodic disturbance is usually the regular vibration, or oscillation, of particles of the medium (material) through which the wave travels.
Key ConceptYou can make a progressive wave using a rope. If you move one end rapidly from side to side, the wave moves from your hand towards the other end of the rope, but the rope itself does not move this way. You are using the wave to transfer energy, but not matter, from your hand to the other end.
Transverse Waves
In a transverse wave, the oscillations are perpendicular to the direction of energy transfer.
When a wave is produced in a rope by moving the hand from side to side, the particles oscillate from side to side, passing through the equilibrium position. The particles do not move along the rope, only from side to side. The distance of a particle from the equilibrium position at any point in time is called its displacement.
FIG 7.1: Transverse Wave PropertiesShow a sinusoidal transverse wave with clearly labelled features: wavelength (λ) measured between two adjacent crests, amplitude (A) measured from equilibrium to crest, equilibrium position (horizontal line), direction of oscillation (vertical arrows), and direction of energy transfer (horizontal arrow). Mark crests (peaks) and troughs.
Examples of Transverse Waves- Waves in vibrating strings of musical instruments
- Surface water waves
- Electromagnetic waves (light, radio waves)
- Waves on a rope
Wavelength and Amplitude
We can describe the form of all waves using the terms wavelength and amplitude.
Wavelength (λ) The distance between any two adjacent equivalent points on the wave. Usually measured from one wave peak to the next, but it could be measured from any point on the wave to the next equivalent point. SI unit: metre (m) Amplitude (A) The maximum displacement of the wave from the equilibrium position. The equilibrium position is the location of the particles when there is no disturbance due to the wave. SI unit: metre (m) ImportantOn a diagram, the amplitude of a wave is measured from the mid-point line of the wave to the top of a peak or the bottom of a trough. It is not the peak-to-trough distance.
Speed, Frequency and Period
Wave Speed (v) v = (distance covered/time) SI unit: m s⁻¹ Frequency (f) The number of complete oscillations per unit time. SI unit: s⁻¹ or hertz (Hz) Period (T) The time taken for one complete oscillation to pass a fixed point. SI unit: second (s)Speeds of Different Waves
| Wave and Medium | Speed / m s⁻¹ |
|---|---|
| Water waves caused by wind | 2.5 – 15 |
| Sound waves in air at sea level at 20°C | 343 |
| Sound waves in sea water at 4 m depth at 20°C | 1500 |
| EM waves in optical (fibre optic) cable | 2.00 × 10⁸ |
| EM waves in air or vacuum | 3.00 × 10⁸ |
The speed of electromagnetic waves in a vacuum is one of the fundamental constants in the Universe, often denoted by c. This is the maximum possible speed in the Universe.
The Wave Equation
The frequency of a wave is proportional to the speed of the wave when all other factors are constant. For a wave moving at a particular speed, the longer its wavelength, the fewer complete waves will pass a point each second.
The Wave Equation:v = fλ
speed of a wave = frequency × wavelength
where:
- v = wave speed (m s⁻¹)
- f = frequency (Hz or s⁻¹)
- λ = wavelength (m)
Relationship Between Period and Frequency
f = (1/T) and T = (1/f)For example, if the frequency is 10 Hz (10 complete oscillations per second), then one oscillation takes 1/10 s to pass a fixed point.
ImportantThe speed of electromagnetic waves and sound in air (at a particular temperature and density) is the same for all frequencies. When waves change speed in different media, their frequency remains constant (since this depends on the wave source), so their wavelengths will change according to the medium.
The Ripple Tank
The ripple tank is a transparent glass or plastic tray filled with water, used to show wave behaviour of surface water ripples. Images of the water waves can be projected onto a screen for viewing.
- A round 'dipper' dips into the water and vibrates up and down to produce circular water waves
- A rectangular 'paddle' generates plane (straight) waves
- The vibrations are generated using an electric motor
- Can demonstrate reflection, refraction, and diffraction
Phase
Points on a wave that have the same displacement and are moving in the same direction are said to be oscillating in phase. Instead of using arbitrary position numbers, we can use an angle from 0° to 360° to refer to a point on a wave. This is called the phase angle.
A phase angle of 360° is equivalent to:- A distance of one wavelength
- A time interval of one period
Phase Difference
Phase difference is the difference in phase angle between two waves at the same time.
| Phase Difference (degrees) | Phase Difference (radians) | Description |
|---|---|---|
| 0° | 0 | In phase |
| 90° | π/2 | Quarter cycle difference |
| 180° | π | In antiphase |
| 270° | 3π/2 | Three-quarter cycle difference |
| 360° | 2π | In phase (one complete cycle) |
Show two sinusoidal waves on the same axes demonstrating different phase relationships: (a) waves in phase (0° difference), (b) waves in antiphase (180° difference), and (c) waves with 90° phase difference. Label the phase difference and show how points on the waves correspond.
Key Points- 360° = 2π radians
- If two waves have a phase difference of 0 or 2π (or 4π, ...) they are in phase
- If two waves have a phase difference of π (or 3π, ...) they are in antiphase
Wave Intensity
Intensity is a measure of the rate of transfer of energy by a wave to unit area that is perpendicular to the direction of travel of the wave.
I = (P/A)intensity (W m⁻²) = power (W) ÷ area (m²)
Intensity and Amplitude
When a wave passes through a medium, it is the kinetic energy of the oscillating particles that is transferred. As kinetic energy is given by ½mv², the kinetic energy is proportional to the square of speed.
The maximum speed of oscillating particles is proportional to their amplitude of oscillation (if frequency doesn't change). If the amplitude doubles, particles travel twice as far in the same time, so speed must double.
I ∝ A^2intensity ∝ (amplitude)² Example
If amplitude doubles (×2), intensity quadruples (×4).
If amplitude triples (×3), intensity increases by factor of 9 (×9).
Worked Examples
Worked Example 1: Wavelength of LightQuestion: The frequency of yellow light is 530 THz. Calculate the wavelength of this light in air. Give your answer in nm. (Speed of light in air = 3.00 × 10⁸ m s⁻¹)
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