Wavelength

Wavelength is an important idea in physics and mathematics that helps us understand waves and repeating patterns. It describes the distance between two high points, called crests, in a wave. This measurement applies to many types of waves.

Wavelength is usually represented by the Greek letter lambda (λ). It is closely related to the frequency of a wave. The wavelength changes depending on what the wave is traveling through, such as air or water.

We see and experience waves in many parts of everyday life. Sound waves travel through air. Light and other forms of electromagnetic radiation involve changes in electric and magnetic fields. Water waves are changes in the height of water.

The range of possible wavelengths for different kinds of waves is called a spectrum. This idea started with the colors we see in visible light but now includes all types of electromagnetic waves, sound waves, and even vibrations.

Sinusoidal waves

In simple materials, waves can be thought of as made up of smaller repeating parts called sinusoidal waves. The wavelength of a wave is the distance over which the wave’s shape repeats. For example, it is the distance between two high points (crests) or two low points (troughs) of the wave.

Wavelength depends on the speed of the wave and its frequency. For light waves traveling through empty space, the speed is constant. Sound waves in air travel more slowly, so their wavelengths are much longer.

Standing waves

A standing wave is a wave that stays in one place. It has points that do not move, called nodes, and the wavelength is twice the distance between these nodes. Standing waves can be seen in a box where the walls force the wave to have nodes at the edges.

The speed of a wave changes depending on the material it travels through. When light enters a material, its speed decreases, which shortens its wavelength. This change in speed also causes the wave to change direction.

More general waveforms

The idea of wavelength is used with smooth, repeating waves. These waves keep their shape as they move. The wavelength tells us about the wave in space and is linked to how often the wave repeats, which we call frequency. Smooth, repeating waves are the simplest kind of traveling waves. More complicated waves can be made by combining many of these simple waves together.

In some special situations, waves that are not smooth can also travel without changing their shape. For example, ocean waves in shallow water can have sharper peaks and flatter valleys than smooth waves.

If a traveling wave looks the same over and over again in space or time, we call it a repeating wave. Even if these waves aren’t perfectly smooth, we can still talk about their wavelength. We measure wavelength by looking at the distance between two matching points on the wave, like from one peak to the next peak.

Wave packets

Main article: Wave packet

Sometimes, waves come in short bursts called wave packets. These bursts travel as a single unit, and the distance between the peaks inside the burst is sometimes called a local wavelength. The whole burst moves at a different speed than the small waves inside it.

We can break wave packets into many smooth waves of different sizes using a math tool called Fourier analysis.

Louis de Broglie suggested that tiny particles, like electrons, also act like waves with a special wavelength. For example, the electrons in a TV screen have a very tiny wavelength. To keep these wave-like particles in one place, de Broglie suggested using wave packets. The size of the wave packet and the different sizes of the smooth waves inside it relate to how exactly where the particle is and how it moves, which is described by a rule called the Heisenberg uncertainty principle.

Interference and diffraction

Double-slit interference

Main article: Interference (wave propagation)

When waves meet, they can make light brighter or darker depending on how they line up. This idea is used in tools called interferometers. A famous example is an experiment where light passes through two small openings and shines on a screen. The light’s path to each spot on the screen is different for the two openings.

If we know the wavelength of the light, we can figure out how far apart the openings are by looking at the pattern of bright and dark spots on the screen. For many openings, the pattern becomes more complex, but the total amount of light stays the same.

Single-slit diffraction

Main articles: Diffraction and Diffraction formalism

When light passes through a single narrow opening and hits a screen, it spreads out into a wider shape. This spreading is called diffraction. There are two kinds of diffraction: one that happens when the light source and screen are far apart (Fraunhofer diffraction) and one when they are close together (Fresnel diffraction).

In the far-field case, the light’s intensity on the screen creates a pattern that depends on the wavelength of the light and the width of the opening.

Diffraction-limited resolution

Main articles: Angular resolution and Diffraction-limited system

Diffraction sets the limit on how sharply optical tools, like telescopes and microscopes, can see. For a circular opening, the smallest spot of light that can be made is called an Airy disk. The size of this spot depends on the wavelength of the light and the size of the opening. Shorter wavelengths allow for clearer and more detailed images.

Subwavelength

The word subwavelength means something is smaller than the length of a wave it touches. For example, a subwavelength-diameter optical fibre is a thin tube that carries light, and it is narrower than the distance the light travels in one full wave.

A subwavelength particle is very small compared to the wave of light around it (see Rayleigh scattering). Subwavelength apertures are tiny openings, smaller than the light wave going through them. These special shapes help in extraordinary optical transmission and zero-mode waveguides, which are important in the study of light, known as photonics.

Subwavelength can also describe effects made by these small objects, like subwavelength imaging.