Wave Behaviour and the Wave Equation
Types of Wave
Waves transfer energy from one place to another without transferring matter. The particles of the medium vibrate about their rest positions but do not travel along with the wave. There are two fundamental categories:
• Transverse waves — the vibrations are perpendicular (at right angles) to the direction of energy transfer. The wave profile forms crests and troughs. All electromagnetic waves, water surface ripples and waves on a string are transverse.
• Longitudinal waves — the vibrations are parallel to the direction of energy transfer. Instead of crests and troughs, these waves have compressions (particles close together) and rarefactions (particles spread apart). Sound is the principal example at GCSE.
A critical distinction: mechanical waves such as sound and water waves require a medium (solid, liquid or gas) to travel through. Electromagnetic waves such as light require no medium and can travel through a vacuum.
Waves transfer energy without transferring matter. Transverse: vibrations perpendicular to travel direction (crests and troughs). Longitudinal: vibrations parallel to travel direction (compressions and rarefactions). Sound is mechanical — it needs a medium.
Wave Properties
All waves — transverse and longitudinal — are described using four measurable properties:
• Amplitude — the maximum displacement from the rest position, measured in metres (m). A larger amplitude means more energy carried: louder sound, brighter light.
• Wavelength (\(\lambda\)) — the distance from one point on a wave to the equivalent point on the next wave (e.g. crest to crest), measured in metres.
• Frequency (\(f\)) — the number of complete waves passing a fixed point per second, measured in hertz (Hz). Higher frequency means higher pitch for sound, or higher energy for light.
• Period (\(T\)) — the time taken for one complete wave to pass a fixed point, measured in seconds. Period and frequency are reciprocals of each other: \(T = \frac{1}{f}\). A wave with a frequency of 200 Hz has a period of \(\frac{1}{200} = 0.005\) s.
The Wave Speed Equation
Wave speed, frequency and wavelength are related by the wave equation:
$$v = f \times \lambda$$
where \(v\) is wave speed in m/s, \(f\) is frequency in Hz, and \(\lambda\) is wavelength in m. Rearranging: \(f = \frac{v}{\lambda}\) and \(\lambda = \frac{v}{f}\). This equation applies to all waves — mechanical and electromagnetic alike.
Question: A sound wave travels through air with a frequency of 440 Hz and a wavelength of 0.77 m. Calculate the wave speed.
Step 1 — identify values: \(f = 440\) Hz, \(\lambda = 0.77\) m.
Step 2 — apply equation: \(v = f \times \lambda = 440 \times 0.77 = 339\) m/s.
Step 3 — state units: wave speed = 339 m/s (close to the speed of sound in air at room temperature — a good sense check).
Reflection, Refraction and Diffraction
Scope note: diffraction is not on this specification. It is included here as background because it explains why waves spread through gaps; exam questions will ask about reflection and refraction instead.
Waves interact with surfaces and boundaries in three important ways:
• Reflection — a wave bounces back from a surface. The angle of incidence equals the angle of reflection (both measured from the normal). Wave speed, frequency and wavelength do not change.
• Refraction — a wave changes speed when it crosses a boundary between two media (e.g. air into glass). If it hits at an angle, it also changes direction. It bends towards the normal when slowing down and away from the normal when speeding up. Frequency stays the same; wavelength changes.
• Diffraction — a wave spreads out after passing through a gap or around the edge of an obstacle. The effect is greatest when the gap width is similar in size to the wavelength. Wider gaps produce less spreading.
Reflection: angle of incidence = angle of reflection. Refraction: wave changes speed (and direction) at a boundary — bends towards the normal when slowing down. Diffraction: wave spreads through a gap most when gap width ≈ wavelength.
In a ripple tank, a vibrating bar produces plane water waves. A light source above projects the wave pattern onto a screen below. To measure wave speed: use a strobe light to “freeze” the pattern, measure several wavelengths and divide by the number of waves to find \(\lambda\). Note the motor frequency for \(f\). Calculate \(v = f\lambda\).
For investigating wave speed in a stretched string: vary the tension by changing the hanging mass, measure the frequency using a signal generator driving the string, and observe the standing wave pattern to determine wavelength from the node spacing. Increasing tension increases wave speed.
Key variables to control: for the string, keep the length and mass per unit length constant when changing tension. For the ripple tank, keep water depth constant — wave speed in water depends on depth.
The speed of sound in a medium depends on the stiffness and density of the material. Sound travels faster in solids than in liquids, and faster in liquids than in gases, because particles in solids are more tightly bonded and can transmit compressions more rapidly. For example, sound travels at approximately 340 m/s in air but around 1,500 m/s in water and over 5,000 m/s in steel.