A wave is a disturbance that travels through a medium transporting energy and information without the bulk transport of matter, as seen when a pebble disturbs still water or when sound moves through air. In a transverse wave the medium particles vibrate perpendicular to the direction of propagation, while in a longitudinal wave they vibrate along it; sound in air is longitudinal. A travelling harmonic wave is described by \(y(x,t)=A\sin(kx-\omega t+\phi)\), where \(A\) is amplitude, \(k=2\pi/\lambda\) the angular wavenumber and \(\omega=2\pi\nu\) the angular frequency; the wave speed is \(v=\nu\lambda=\omega/k\). The speed of a transverse wave on a stretched string is \(\sqrt{T/\mu}\), and the speed of sound in a medium depends on its elasticity and density (Newton's formula corrected by Laplace gives \(\sqrt{\gamma P/\rho}\) for a gas). When two waves overlap, the principle of superposition gives a resultant displacement equal to the sum; this leads to interference, to standing waves with nodes and antinodes on strings and in air columns producing harmonics, and to beats when two slightly different frequencies combine. The Doppler effect describes the change in observed frequency due to relative motion of source and observer.
Transverse and longitudinal wavesDisplacement relation for a progressive waveSpeed of a travelling waveSpeed of waves on a string and of soundPrinciple of superposition of wavesStanding waves and normal modesBeatsThe Doppler effect
Key terms
Wave
A disturbance that propagates through a medium carrying energy and information without net transport of matter.
Transverse wave
A wave in which the medium particles oscillate perpendicular to the direction of wave propagation.
Longitudinal wave
A wave in which the medium particles oscillate along the direction of propagation, such as sound in air.
Wavelength
The distance between two consecutive points in the same phase of a wave, related to wavenumber by \(k=2\pi/\lambda\).
Standing wave
A wave pattern formed by superposition of two identical waves travelling in opposite directions, having fixed nodes and antinodes.
Beats
The periodic variation in loudness produced when two waves of slightly different frequencies superpose.
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In a transverse wave, the particles of the medium vibrate:
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