Waves

A wave is a propagating disturbance that transfers energy and momentum without transporting the medium as a whole from source to destination. The same basic language—amplitude, wavelength, frequency, phase and speed—works across water waves, sound, strings and electromagnetic waves.

Describe a repeating wave

Amplitude describes the size of the disturbance; wavelength describes its spatial repetition. Frequency tells how many cycles pass a point each second.

v = fλ and T = 1/f

Frequency f is measured in hertz (cycles per second), period T in seconds, wavelength λ in metres, and wave speed v in metres per second. Frequency is set by the source. In a nondispersive medium, wave speed is set mainly by the medium and wave type, so wavelength adjusts through v = fλ.

Wave relation explorer

Set any two of speed, frequency and wavelength through v = fλ.

Frequency f (Hz) Wave speed v (m/s)

Transverse and longitudinal motion

Wave type refers to how the medium moves relative to propagation. A string wave is transverse; sound in air is predominantly longitudinal.

In a transverse wave, the disturbance is perpendicular to the direction of propagation. In a longitudinal wave, oscillations are parallel to propagation, producing compressions and rarefactions.

The particles of the medium usually oscillate around equilibrium rather than travelling with the wave over long distances.

Superposition and interference

When waves overlap in a linear medium, their displacements add. If crest meets crest, the result is constructive interference; if a crest overlaps a trough, cancellation can be partial or complete.

After the overlap, the waves continue. Interference is not usually a collision that destroys the waves.

Phase tells where one oscillation is within its cycle relative to another. Equal-frequency waves in phase reinforce; waves 180° out of phase cancel most strongly when their amplitudes are equal. Intermediate phase differences produce partial interference.

Reflection and standing waves

Boundaries can reflect waves. A reflected wave can overlap the incoming wave and form a standing wave with nodes that remain at zero displacement and antinodes with maximum oscillation.

For a string fixed at both ends, allowed standing-wave wavelengths satisfy λ_n = 2L/n, giving frequencies f_n = nv/(2L).

Boundary conditions decide which patterns are allowed. A fixed end must be a displacement node; a free end can be an antinode. This is why a string or air column supports a discrete family of normal modes rather than every possible standing-wave frequency.

Amplitude and energy are not the same as speed

Larger amplitude generally means greater wave energy, but it does not automatically mean the wave travels faster. For example, the speed of small-amplitude waves on a given stretched string is set primarily by tension and linear mass density, not by amplitude.

Some media are dispersive: different frequencies travel at different phase speeds. Then a pulse can change shape as it travels. The simple relation v = fλ still holds for each sinusoidal component, but v itself may depend on frequency.

When a wave crosses into a new medium, its frequency remains tied to the source. If the wave speed changes, the wavelength changes with it. This distinction becomes essential in refraction and in comparing sound or light across different media.

Worked examples

1. Sound wavelength

A 440 Hz sound travels at 343 m s⁻¹. Find its wavelength.

Solution

λ = v/f = 343/440 = 0.780 m.

2. Period from frequency

A wave has frequency 25 Hz. What is its period?

Solution

T = 1/f = 1/25 = 0.040 s.

3. Superposition

At one point, two pulses produce displacements +3.0 cm and −1.2 cm at the same instant. What is the resulting displacement?

Solution

Superposition adds displacements algebraically: 3.0 + (−1.2) = +1.8 cm.