6.2 Wave properties and the wave equation

Describing a wave

• The amplitude (A) is measured from the rest position (the middle line) to a crest or a trough. A bigger amplitude means a wave carrying more energy. For a sound wave, a bigger amplitude means a louder sound; for light, a brighter light.
• The wavelength (\lambda, the Greek letter "lambda") is the length of one complete wave. It is measured in metres (m).
• The frequency (f) is the number of complete waves that pass a point each second. 1 hertz (1 Hz) is 1 wave per second. Frequencies are often large: 1 kHz = 1000 Hz and 1 MHz = 1 000 000 Hz. For sound, a higher frequency is a higher pitch.
• The period (T) is the time for one complete wave.

Frequency and period

Frequency and period are linked by:

T = \frac{1}{f}

A wave with a frequency of 2 Hz has a period of \frac{1}{2} = 0.5 \text{ s}.

On a displacement-time graph, the period is the time between two equivalent points, such as two peaks. In the graph, the period is 0.5 s, so the frequency is \frac{1}{0.5} = 2 \text{ Hz}.

The wave equation

The speed of a wave depends on its frequency and wavelength. In one second, f waves pass a point, each of length \lambda, so the wave travels a distance of f \times \lambda in one second. This gives the wave equation:

v = f \lambda

• v is wave speed in metres per second (m/s)
• f is frequency in hertz (Hz)
• \lambda is wavelength in metres (m)

Wave speed can also be found using v = \frac{\text{distance}}{\text{time}}, the same as for any moving object.

Worked example 1. A wave has a frequency of 50 Hz and a wavelength of 6.0 m. Calculate its speed.

v = f \lambda = 50 \times 6.0 = 300 \text{ m/s}

Worked example 2. A sound wave of frequency 440 Hz travels at 330 m/s in air. Calculate its wavelength.

\lambda = \frac{v}{f} = \frac{330}{440} = 0.75 \text{ m}

Worked example 3. Waves on the sea have a wavelength of 20 m and a period of 8.0 s. Calculate the wave speed.

f = \frac{1}{T} = \frac{1}{8.0} = 0.125 \text{ Hz}, so v = f \lambda = 0.125 \times 20 = 2.5 \text{ m/s}

Worked example 4. A radio station broadcasts at 100 MHz. Radio waves travel at 3.0 \times 10^8 \text{ m/s}. Calculate the wavelength.

f = 100 \times 10^6 = 1.0 \times 10^8 \text{ Hz}, so \lambda = \frac{3.0 \times 10^8}{1.0 \times 10^8} = 3.0 \text{ m}

What decides the wave speed?

• The frequency is set by the source: the wave has the same frequency as the thing that makes it.
• The speed is set by the medium. For example, sound travels at about 330 m/s in air, 1500 m/s in water and 5000 m/s in steel.
• When a wave moves from one medium to another, its frequency stays the same but its speed and wavelength change. If the speed is lower, the wavelength is shorter (since v = f \lambda, and f is constant).
• All electromagnetic waves travel at the same speed in a vacuum, 3.0 \times 10^8 \text{ m/s}.