Wave Behaviors: Reflection and Refraction
Reflection is the bouncing of a wave off a boundary, obeying the law that the angle of incidence equals the angle of reflection, and refraction is the change in direction of a wave as it passes from one medium into another because its speed changes. In brief: in both, angles are measured from the normal (the line perpendicular to the surface). In refraction, the frequency stays the same, the speed and wavelength change, and the wave bends toward the normal when it slows down. Sound echoes are reflections, and a spoon that looks bent in a glass of water shows refraction of light.
Echoes in a canyon, the image in a mirror and a rainbow all result from reflection and refraction. This guide explains both effects for water waves, sound and light, gives the laws and the relation between angle and speed, compares the two effects in a table, works through calculations and corrects common mistakes.
What is reflection?
When a wave meets a boundary, part or all of it returns into the first medium. The law of reflection: the angle of incidence i equals the angle of reflection r, and the incident ray, the reflected ray and the normal lie in the same plane. The wavelength, frequency and speed do not change on reflection (the wave stays in the same medium). Examples:
- Light reflecting from a mirror (a smooth surface gives specular reflection, while a rough surface gives diffuse reflection, which lets us see objects from any direction).
- Sound echoes from cliffs and walls, used in sonar and ultrasound imaging. Concert halls are designed to control reflections. Soft materials absorb sound and reduce echoes.
- Water waves bouncing off a harbor wall, which can be studied in a ripple tank.
- Waves on a rope reflect from a fixed end inverted, and from a free end without inversion.
What is refraction?
Refraction occurs when a wave crosses a boundary between two media in which it has different speeds. The frequency stays the same, so v = fλ shows that the wavelength changes in proportion to the speed. If the wave hits the boundary at an angle, the part of the wavefront that enters the slower medium first slows down, and the wave changes direction:
- entering a slower medium (denser for light, shallower for water), the wave bends toward the normal;
- entering a faster medium, it bends away from the normal;
- along the normal (angle 0°), there is no change in direction, only in speed and wavelength.
The relation (Snell’s law) is sin i ÷ sin r = v1 ÷ v2, which for light becomes n1 sin i = n2 sin r. Examples: a straw that seems to bend at the water surface, a swimming pool appearing shallower than it is, a lens focusing light, sound bending upward on a warm day (when the air near the ground is warmer, sound speeds up there), and water waves slowing in shallow water, which makes them turn to arrive parallel to the beach.
Reflection and refraction compared
| Feature | Reflection | Refraction |
|---|---|---|
| What happens | Wave returns to the first medium | Wave enters the second medium and changes direction |
| Frequency | Unchanged | Unchanged |
| Speed and wavelength | Unchanged | Both change |
| Law | i = r | sin i ÷ sin r = v1 ÷ v2 |
| Example | Echo, mirror | Lens, bent straw |
Worked example: echo and refraction
- Echo. A person claps and hears the echo from a cliff after 2.0 s. The speed of sound is 343 m/s. The sound travels there and back, so 2.0 × 343 = 686 m is the total path, and the cliff is 686 ÷ 2 = 343 m away.
- Mirror. A ray hits a plane mirror at 35° to the normal. It reflects at 35° to the normal, so the angle between the incident and reflected rays is 70°.
- Water waves. Deep-water waves travel at 2.0 m/s and enter shallow water, where the speed is 1.5 m/s, at an angle of incidence of 40°. Then sin r = sin 40° × (1.5 ÷ 2.0) = 0.643 × 0.75 = 0.482, so r = 28.8°. The wave bends toward the normal. If the frequency is 0.50 Hz, the wavelength changes from 2.0 ÷ 0.50 = 4.0 m to 1.5 ÷ 0.50 = 3.0 m.
- Light. Light travels from air (n = 1.00) into glass (n = 1.50) at 30°. sin r = sin 30° ÷ 1.50 = 0.333, so r = 19.5°. The speed in the glass is 3.00 × 108 ÷ 1.50 = 2.00 × 108 m/s.
How were these laws found?
Euclid and Hero of Alexandria discussed the law of reflection. Ibn al-Haytham (Alhazen) studied reflection and refraction of light experimentally around 1000 CE. Willebrord Snellius found the law of refraction in 1621, and René Descartes published it in 1637. Pierre de Fermat explained it in 1662 using the principle of least time, and Huygens derived it from his wave theory.
Where are they used?
Mirrors, telescopes, periscopes and reflectors on roads use reflection. Eyeglasses, cameras, microscopes and optical fibers rely on refraction. Sonar and medical ultrasound use reflections of sound, and seismologists study the interior of the Earth from the refraction and reflection of earthquake waves.
Which mistakes are common?
Mistake 1: Measuring angles from the surface
Angles of incidence, reflection and refraction are measured from the normal.
Mistake 2: Saying that the frequency changes in refraction
The frequency is determined by the source. Speed and wavelength change.
Mistake 3: Mixing up the direction of bending
A wave slowing down bends toward the normal. A wave speeding up bends away from it.
Mistake 4: Forgetting the two-way path of an echo
The sound travels to the obstacle and back, so the distance is half of the total path.