The Wave Equation

The Wave Equation

Waves can be transverse or longitudinal. Transverse waves have oscillations perpendicular to the direction of wave travel (like water waves), while longitudinal waves oscillate parallel to the direction of travel (like sound waves). The wave equation applies to both types.

The wave equation links the speed of a wave to its frequency and wavelength. It is written as:

v = f × λ

• v = wave speed (in metres per second, m/s)
• f = frequency (in hertz, Hz)
• λ (lambda) = wavelength (in metres, m)

This equation applies to all types of waves, including sound waves, water waves, and seismic waves. Seismic waves include both transverse and longitudinal waves that travel through the Earth.

Frequency is how many waves pass a point each second. Wavelength is the distance between two identical points on neighbouring waves (e.g., crest to crest or trough to trough).

Diagram: A transverse wave showing wavelength (distance between crests) and frequency (number of waves per second).

For example, if a wave has a frequency of 10 Hz and a wavelength of 2 m, its speed is:

$v = 10 \times 2 = 20 \text{ m/s}$

Wave Speed in Different Media

The speed of a wave depends on the medium it travels through. Waves travel at different speeds in air, liquids, and solids because the particles are arranged differently and interact in different ways.

• Waves generally travel fastest in solids because particles are closely packed and can transfer vibrations quickly.
• They travel slower in liquids where particles are less tightly packed.
• They travel slowest in gases (like air) where particles are far apart.

For example, sound waves travel at about 343 m/s in air at room temperature, but around 1500 m/s in water and even faster in steel (about 5000 m/s).

This difference in speed is important for applications like ultrasound imaging and seismic studies, where waves travel through different materials inside the body or the Earth.

Applications of the Wave Equation

The wave equation is used to calculate wave speed when frequency and wavelength are known, or to find one of these quantities if the other two are given.

It applies to many real-world waves, such as:

• Sound waves travelling through air or other materials
• Seismic waves moving through the Earth during earthquakes
• Water waves on the surface of oceans, lakes, or pools

For example, if you know the frequency of a sound wave and measure its wavelength, you can find its speed and check if it matches the expected speed in that medium.

Understanding wave speed helps in designing musical instruments, sonar systems, and medical imaging devices.

For instance, if a sound wave has a wavelength of 0.5 m and a frequency of 680 Hz, its speed is:

$v = 680 \times 0.5 = 340 \text{ m/s}$