Transverse wave

A transverse wave is one in which the disturbance oscillates at right angles to the direction the wave travels. A wave sent along a stretched rope is the familiar example: each piece of rope moves up and down while the pulse runs along it. Light is a transverse electromagnetic wave: in free space its electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ are perpendicular to each other and to the propagation direction. In a longitudinal wave, by contrast, the medium oscillates along the direction of travel, as the compressions and rarefactions of sound in air do. The wavelength of either kind is the crest-to-crest (or compression-to-compression) distance.

Transverse and longitudinal waves compared

A transverse mechanical wave needs a restoring force against sideways displacement. A string supplies it through tension: the speed is $v = \sqrt{T/\mu}$, so a string under 70 N with a linear density of 0.4 g/m carries waves at 418 m/s. Solids resist shear and carry both kinds of wave. Gases and liquids resist compression but not shear, so sound inside them is purely longitudinal. The same fact explains the S-wave shadow in seismology: S-waves do not pass through Earth's liquid outer core, while P-waves do.

Surface waves on water are a mixed case. In deep water each fluid parcel moves in a circle, partly across and partly along the direction of travel, so the surface profile looks transverse while the motion is not purely so. A deep-water wave 10 m long travels at $\sqrt{g\lambda/2\pi}$ = 3.95 m/s.

Light as a transverse wave

Maxwell's equations in free space require $\nabla\cdot\mathbf{E} = 0$, which for a plane wave means $\mathbf{E}$ has no component along the wavevector $\mathbf{k}$. The fields satisfy

$$\mathbf{B} = \frac{1}{\omega}\,\mathbf{k}\times\mathbf{E}, \qquad |\mathbf{E}| = c\,|\mathbf{B}|,$$

and the energy flow, the Poynting vector $\mathbf{S} = \mathbf{E}\times\mathbf{B}/\mu_0$, points along $\mathbf{k}$. For a worked number, sunlight with an irradiance of 1000 W/m² treated as a single plane wave has a peak field

$$E_0 = \sqrt{\frac{2I}{c\,\varepsilon_0}} = 868\ \mathrm{V/m},$$

and a peak magnetic field $E_0/c$ = 2.9 µT, both lying in the plane across the beam.

Consequence: polarization

Because the field of a transverse wave can point in any direction within the plane perpendicular to travel, the wave carries an extra degree of freedom that a longitudinal wave lacks: polarization. The field can oscillate along a fixed line (linear), rotate (circular or elliptical, see polarization states), or vary randomly (unpolarized). A polarizer passes one direction and blocks the orthogonal one. Historically this was the evidence that light is transverse: Fresnel and Arago found (published 1819) that two beams polarized at right angles do not interfere, which a longitudinal wave could not explain, and Young and Fresnel drew the conclusion that the vibration of light lies across the ray. Sound in air has no polarization.

S-waves are polarized in the same sense: seismologists distinguish SH (horizontal) and SV (vertical) components, and their different arrival times in anisotropic rock are the seismic analogue of birefringence.

Where transversality breaks down

The strictly transverse picture belongs to plane waves in uniform, isotropic media. Real fields depart from it in several cases that matter in the laboratory:

• Tight focusing. A lens of high numerical aperture bends rays steeply toward the axis, and the focal field acquires a longitudinal component along the axis; for a radially polarized beam it can dominate at the focus.
• Guided and evanescent waves. Waveguide modes and the evanescent wave at a totally reflecting surface have field components along the propagation direction.
• Anisotropic crystals. In a birefringent crystal the displacement field $\mathbf{D}$ stays transverse to $\mathbf{k}$, but $\mathbf{E}$ can tilt, and the energy walks off from the wavevector.

Common questions

Is light a transverse or longitudinal wave?

Transverse. In free space its electric and magnetic fields oscillate perpendicular to the direction of travel, and the existence of polarization is the direct experimental evidence.

What is the wavelength of a transverse wave?

The distance between two adjacent crests (or troughs), $\lambda = v/f$. For a string wave at 418 m/s and 440 Hz it is 0.95 m; for 632.8 nm light it is 632.8 nm in vacuum.

Are water waves transverse?

Only partly. Surface water waves combine transverse and longitudinal motion, so the water moves in circular or elliptical orbits; ripples driven by surface tension and gravity waves in deep water both behave this way.

Can sound be a transverse wave?

In air or water, no, since fluids do not support shear. In solids, transverse acoustic waves (shear waves) exist alongside longitudinal ones and travel more slowly.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 2, 3 and 8; M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 1; B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).