Learning outcomes
- Define amplitude, wavelength, frequency and wave speed.
- Identify wave quantities from diagrams and oscilloscope traces.
- Recall and use v = fλ with consistent units.
- Explain how frequency, period and wavelength are related for a wave of fixed speed.
2.1 Mean position, crest and trough
The mean or equilibrium position is the position a particle occupies when no wave disturbance is present. A crest is a point of maximum positive displacement and a trough is a point of maximum negative displacement. These labels apply to a transverse displacement graph; a longitudinal wave is better represented using compressions, rarefactions or a pressure–distance graph.
A graph of displacement against distance is a snapshot of the wave at one instant. It shows how different particles are displaced at that moment. A graph of displacement against time follows one particle as the wave passes. Both may look sinusoidal, but the horizontal axes represent different quantities and must not be confused.
2.2 Amplitude and wavelength
Amplitude is the maximum displacement from the mean position. It is not the vertical distance from a crest to a trough; that distance is twice the amplitude. Depending on the wave, amplitude may be measured as a displacement, pressure variation, electric-field strength or another physical quantity.
Wavelength, symbol λ, is the shortest distance between two consecutive points in the same phase. On a transverse wave it can be measured from crest to crest, trough to trough, or between any matching points on successive cycles. For a longitudinal wave, it is the distance from the centre of one compression to the centre of the next compression, or from rarefaction to rarefaction.
2.3 Frequency and period
Frequency, symbol f, is the number of complete waves or oscillations passing a point per unit time. Its unit is the hertz, where 1 Hz means one cycle per second. A source vibrating 50 times each second produces a frequency of 50 Hz. Frequency is set by the source and normally remains unchanged when a wave enters another medium.
The period T is the time for one complete oscillation. Although the syllabus emphasises frequency, period is useful in practical work. Frequency and period are reciprocals: f = 1/T. Timing many cycles and dividing gives a more reliable period than timing one cycle.
2.4 Wave speed
Wave speed is the rate at which a particular phase point, such as a crest, travels. It is not the speed of a vibrating particle. Wave speed depends mainly on the properties of the medium: tension and mass per unit length for a string, depth for shallow water waves, and elasticity and density for sound.
The wave equation is v = fλ. It follows because one wavelength passes a point during one period. Since f = 1/T, the same relationship may be written v = λ/T. Use SI units unless a question clearly permits others: speed in m/s, frequency in Hz and wavelength in m.
2.5 Rearrangement and proportional reasoning
The equation can be rearranged to f = v/λ and λ = v/f. When the wave speed is constant, increasing frequency decreases wavelength in inverse proportion. Doubling the frequency halves the wavelength. When a wave crosses into another medium, frequency stays constant but speed may change, so wavelength changes in the same ratio as speed.
Unit conversion is a common source of lost marks. Convert centimetres or nanometres to metres before using a speed in m/s. Convert kilohertz to hertz by multiplying by 1000. State the final unit and use a sensible number of significant figures based on the data.
Worked examples
Water-wave calculation
A wave has frequency 4.0 Hz and wavelength 0.30 m. Its speed is v = fλ = 4.0 × 0.30 = 1.2 m/s.
Finding wavelength
Radio waves travel at 3.0 × 10⁸ m/s and have frequency 100 MHz. Convert 100 MHz to 1.00 × 10⁸ Hz. Then λ = v/f = 3.0 × 10⁸ ÷ 1.00 × 10⁸ = 3.0 m.
Reading amplitude
A displacement graph has a crest at +6 mm and a trough at −6 mm. The amplitude is 6 mm, not 12 mm.
Practical focus
Investigation
Use a ripple tank with a stroboscope. Measure the length occupied by five wavelengths and divide by five. Read the vibrator frequency and calculate v = fλ. Measuring several wavelengths reduces the percentage uncertainty.
Examination guidance
- Frequency is the number of complete cycles per second, not the number of crests only.
- A crest-to-trough distance is half a wavelength horizontally but twice the amplitude vertically; inspect the direction being measured.
- Always convert MHz, kHz, cm and nm before substitution.
Check your understanding
- A wave travels at 12 m/s with frequency 8.0 Hz. Find its wavelength.
- Five complete waves occupy 2.5 m. What is the wavelength?
- A wave enters a slower medium without changing frequency. What happens to its wavelength?
Answers
- λ = 12/8.0 = 1.5 m.
- λ = 2.5/5 = 0.50 m.
- The wavelength decreases in the same ratio as the speed.
