Waves: Properties, Wave Equation and Behaviour
What Are Waves?
Waves transfer energy and information without transferring matter. The particles of the medium oscillate about their rest positions — they do not travel with the wave. Evidence for this can be seen with water waves: a floating object bobs up and down but does not travel along with the wave front.
There are two fundamental types. In a transverse wave, the oscillations are perpendicular to the direction of energy transfer. All electromagnetic waves and water waves are transverse. In a longitudinal wave, the oscillations are parallel to the direction of energy transfer. Sound and P-waves (primary seismic waves) are longitudinal.
Longitudinal waves create alternating regions of compression (particles squeezed close) and rarefaction (particles spread out). You can model this with a Slinky spring: push it lengthways to see compressions and rarefactions travel along it.
Transverse waves: oscillations perpendicular to direction of travel (e.g. light, water waves). Longitudinal waves: oscillations parallel to direction of travel (e.g. sound, P-waves). All waves transfer energy without transferring matter.
Wave Properties
Every wave can be described using four properties:
Amplitude is the maximum displacement from the rest position. A larger amplitude means the wave carries more energy.
Wavelength (symbol \(\lambda\)) is the distance from one point on a wave to the identical point on the next wave, e.g. crest to crest.
Frequency (symbol \(f\)) is the number of complete waves passing a point per second, measured in hertz (Hz).
The period (symbol \(T\)) is the time for one complete wave and is linked to frequency by: $$T = \frac{1}{f}$$ If \(f = 50\) Hz, then \(T = \frac{1}{50} = 0.02\) s.
The Wave Equation
Wave speed, frequency and wavelength are related by two equivalent forms of the wave equation that you must be able to recall and apply:
$$v = f \lambda$$
$$v = \frac{x}{t}$$
Where \(v\) is wave speed in metres per second (m/s), \(f\) is frequency in hertz (Hz), \(\lambda\) is wavelength in metres (m), \(x\) is distance in metres and \(t\) is time in seconds.
Both wave equations must be recalled: \(v = f\lambda\) (speed = frequency × wavelength) and \(v = x/t\) (speed = distance ÷ time). Common rearrangement: \(f = v / \lambda\) and \(\lambda = v / f\).
Example 1: A sound wave has a frequency of 440 Hz and a wavelength of 0.75 m. Calculate the wave speed.
\(v = f\lambda = 440 \times 0.75 = 330\) m/s
Example 2: A water wave travels 12 m in 4 s. Calculate the wave speed, then find the wavelength if the frequency is 1.5 Hz.
Speed: \(v = x/t = 12/4 = 3\) m/s. Wavelength: \(\lambda = v/f = 3/1.5 = 2\) m.
Always check: speed in m/s, frequency in Hz, wavelength in m. Do not mix units.
Reflection and Refraction
When waves reach a boundary between two different materials, they may be reflected, refracted, transmitted or absorbed — often a mixture of all four.
Reflection occurs when a wave bounces off a surface. The law of reflection states that the angle of incidence equals the angle of reflection, both measured from the normal (a line perpendicular to the surface at the point of contact).
Refraction occurs when a wave changes speed as it crosses a boundary. If the wave slows down, it bends towards the normal; if it speeds up, it bends away. Frequency does not change during refraction — only speed and wavelength change.
During refraction: speed changes, wavelength changes, frequency stays the same. Direction changes if the wave crosses the boundary at an angle (no direction change if it arrives along the normal).
Core Practical: Wave Speed in a Solid and a Fluid
The core practical requires you to investigate the suitability of equipment to measure wave speed, frequency and wavelength in both a solid and a fluid.
Fluid (ripple tank): A vibrating bar or dipper creates water waves on the surface. A stroboscope can be used to ‘freeze’ the wave pattern. Measure the wavelength directly from the wave pattern projected onto a screen. Measure frequency using the stroboscope setting or a timer. Calculate speed using \(v = f\lambda\). Alternatively, time how long it takes waves to travel a measured distance: \(v = x/t\).
Solid (metal rod): Strike one end of a metal rod with a hammer; the vibration travels as a compression wave along the rod. A microphone at the far end detects when the pulse arrives. With two microphones at a known separation, a data logger records the time between signals. Calculate speed from \(v = x/t\). Evaluate suitability: the rod method is more precise because water waves are affected by surface tension and depth variations.
Seismic Waves, Ultrasound and Infrasound
Sound is only audible to humans between about 20 Hz and 20,000 Hz. Ultrasound (above 20,000 Hz) has uses in foetal scanning, sonar depth-finding and industrial flaw detection. Infrasound (below 20 Hz) is used to study seismic activity and can travel vast distances.
Seismic P-waves (longitudinal) travel through both solids and liquids. S-waves (transverse) can only travel through solids. The detection of S-waves and P-waves by seismographs provides evidence for the internal structure of the Earth.