Wave Anatomy

A wave is a disturbance that transfers energy through a medium (or through space) without permanently displacing the medium's particles. Every wave is described by four interrelated quantities:

• **Frequency ($f$): The number of complete oscillations (cycles) that pass a fixed point per second. Unit: hertz (Hz)**, where $1\ \text{Hz} = 1\ \text{s}^{-1}$.
• **Period ($T$):** The time required to complete one full oscillation. Unit: seconds (s). Frequency and period are reciprocals:

$$T = \frac{1}{f}$$

• **Wavelength ($\lambda$):** The distance between two successive identical points on a wave — for example, from crest to crest, or trough to trough. Unit: metres (m).
• **Wave speed ($v$): The distance the wave travels per second. Unit: m/s. The wave equation** relates speed, frequency, and wavelength:

$$v = f\lambda$$

Example 1: A wave has a frequency of 200 Hz. Its period is $T = 1/200 = 0.005\ \text{s} = 5\ \text{ms}$.

Example 2: A sound wave in air has $f = 440\ \text{Hz}$ and $\lambda = 0.773\ \text{m}$. Its speed is $v = 440 \times 0.773 \approx 340\ \text{m/s}$.

Longitudinal and Transverse Waves

Waves are classified by the relationship between the direction of particle oscillation and the direction of energy propagation.

Transverse Waves

Particles oscillate perpendicular (at right angles) to the direction of wave travel. The wave displays recognisable crests (high points) and troughs (low points).

Examples: Light and all other electromagnetic waves; waves on a vibrating string or rope; ripples on the surface of water.

Longitudinal Waves

Particles oscillate parallel to (along the same line as) the direction of wave travel. The wave consists of alternating compressions (regions where particles are pushed closer together, creating higher pressure) and rarefactions (regions where particles spread apart, creating lower pressure).

The most important example is sound. Sound is a mechanical longitudinal wave that propagates through a compressible medium (air, water, or a solid). Because it relies on particle compression and expansion, sound cannot travel through a vacuum.

The Doppler Effect

The Doppler effect is the observed change in the frequency of a wave when the source and the observer are in relative motion.

• When the source moves towards the observer, successive wavefronts are compressed together — they arrive more frequently than they would from a stationary source, so the observed frequency is higher (higher pitch for sound).
• When the source moves away, wavefronts are stretched apart and arrive less frequently — the observed frequency is lower (lower pitch).

Everyday example: The rising-then-falling pitch of a passing ambulance siren. The Doppler effect applies to all waves, including light; astronomers use it to determine whether distant stars and galaxies are moving towards or away from Earth.

The Decibel Scale

Sound intensity (power per unit area) is measured in watts per square metre ($\text{W m}^{-2}$). The human ear detects an enormous range of intensities — from the threshold of hearing at $10^{-12}\ \text{W m}^{-2}$ to the pain threshold near $1\ \text{W m}^{-2}$, a ratio of $10^{12}$. The logarithmic decibel (dB) scale compresses this range into manageable numbers.

Representative levels: threshold of hearing (0 dB), whisper (30 dB), normal conversation (60 dB), busy traffic (80 dB), concert speaker close-up (110 dB), pain threshold (≈ 120 dB). Every 10 dB increase represents a tenfold increase in intensity — normal conversation is therefore one million ($10^6$) times more intense than the threshold of hearing.