Look at a shop window on a bright day and you see something strange: the mannequins inside the shop, and a faint ghost of the street behind you, both at once. The same sheet of glass is doing two jobs. Some of the light coming from the street passes through it into the shop, and some of it bounces back off the surface into your eyes. That is the whole story of this page: what happens when a wave arrives at a boundary — the join between one material and another.

Whenever a wave — light, sound, water ripples — meets a boundary, its energy is split three ways. Part of it can be

• reflected — it bounces off and stays in the first material;
• refracted (transmitted) — it crosses into the new material, and if it hit at an angle it bends;
• absorbed — its energy soaks into the material and warms it up.

Usually all three happen together, in some mixture. A mirror is nearly all reflection; a clean pane of glass is mostly refraction with a little reflection; a black jumper is mostly absorption. This page zooms in on the first two — the neat bounce, and the sideways bend.

Reflection: bounce off at the same angle

Reflection is the tidy bounce. To describe it precisely we need one helper line: the normal. The normal is an imaginary line drawn at exactly 90^\circ to the surface, right at the point where the ray lands. Every angle in this topic is measured from the normal — never from the surface itself. Get that one habit right and the rest is easy.

Call the angle between the incoming ray and the normal the angle of incidence i, and the angle between the bounced ray and the normal the angle of reflection r. The rule the bounce always obeys is beautifully simple:

So a ray striking a mirror at 30^\circ to the normal leaves at 30^\circ the other side; one arriving at 50^\circ leaves at 50^\circ. Lean the incoming ray in more, and the reflected ray leans out by exactly the same amount.

This is why a smooth surface makes a clear picture but a rough one does not. On a flat mirror every tiny patch faces the same way, so parallel rays bounce off still parallel — the image survives. We call this specular reflection. On a rough wall each bump faces a slightly different way, so the same rays scatter off in all directions — the image is shredded. That is diffuse reflection. Both obey i = r at every point; the surfaces just point their normals in different directions.

Refraction: change speed, and you bend

Refraction is the sideways bend, and its cause is one thing only: a wave changes speed when it crosses into a new material. Light travels fastest in empty space and air; it goes slower in water, and slower still in glass, because the material gets in its way. That change of speed is what bends the ray.

Picture a toy car rolling off smooth pavement onto grass at an angle. The wheel that reaches the grass first slows down while the other wheel is still racing on the pavement — so the car slews round and changes direction. A wavefront does exactly the same at a boundary. The rule that comes out of it:

• Entering a slower material (air \to glass or water): the ray bends towards the normal.
• Entering a faster material (glass or water \to air): the ray bends away from the normal.

There is one special case worth remembering: a ray travelling straight along the normal (hitting the surface head-on, angle of incidence 0^\circ) still changes speed, but it does not bend — both wheels of the car hit the grass at once, so there is nothing to slew it round. It slows, but carries straight on.

What changes, and what doesn't

Here is the part that trips people up. A wave's speed v, frequency f and wavelength \lambda are tied together by

When light crosses into glass it slows down, so v drops. But the frequency stays the same — the source is still wiggling the wave the same number of times each second, and the boundary can't add or lose wiggles. If f is fixed and v falls, then \lambda = v / f must fall too: the wavelength shrinks.

Worked example. Green light of frequency f = 5.0\times10^{14}\ \text{Hz} travels through air at v = 3.0\times10^{8}\ \text{m/s}. Its wavelength there is

Now it enters glass, where it slows to v = 2.0\times10^{8}\ \text{m/s}. The frequency is unchanged at 5.0\times10^{14}\ \text{Hz}, so the new wavelength is

The speed fell to \tfrac{2}{3}, so the wavelength fell to \tfrac{2}{3} as well — from 600 to 400\ \text{nm} — while the frequency (and so the colour) never moved.

See it happen

Below, a single ray strikes a boundary at the middle. Tilt it with the angle slider and watch two things at once: the reflected ray always mirrors it at an equal angle (i = r), while the refracted ray crosses into the new material and bends. Flip the switch to send it the other way — from air into glass (slowing down, bending towards the normal) or from glass into air (speeding up, bending away). Notice the reflected ray never cares which material lies beyond; only the refracted one bends.

Everyday refraction

Once you can see the bend, the world fills up with it. A few favourites, all the same physics:

• The bent straw. Stand a straw in a glass of water and it looks snapped in two at the surface. The straw is fine — the light leaving the underwater part speeds up as it exits the water into the air and bends away from the normal, so your eye traces it back to the wrong place.
• The shallow pool. A swimming pool always looks shallower than it really is. Light from the bottom bends away from the normal as it leaves the water, so the rays reach your eye as if the bottom were higher up. Every year people misjudge the depth because of it.
• Spectacles, cameras, telescopes. A lens is just a cleverly curved piece of glass that refracts every ray by the right amount to bring a whole picture to a focus.