What is the refraction of a wave

• Refraction is the change of direction that a wave undergoes because of a change in its propagation speed.
• In an ideal homogeneous medium waves propagate in a straight line. For example, light in vacuum, or sound in a homogeneous solid at equilibrium. Refraction is a deviation from straight-line propagation, and can be caused by the fact that the properties of the medium change from place to place.
• A situation in which the phenomenon is particularly easy to visualize is one in which the properties of the medium change suddenly and sharply. This typically happens at the surface separating two different media.

Refraction at the boundary between two media

When a wave strikes the separation surface (interface) between two different media, it is partly reflected and partly transmitted to the second medium. In passing from the first to the second medium, the direction of propagation of the wave can change. This phenomenon, known as refraction, is a general feature of wave propagation, that is, it occurs for all kinds of waves.

Reasoning in terms of rays, the situation is well illustrated by the picture alongside, where the incident ray coming from medium 1 is partly reflected (note that the angle of reflection is equal to the angle of incidence) and partly refracted. θi = angle of incidence θr = angle of refraction Note, however, that it is possible to reason in terms of rays only under suitable conditions, and in particular when the size of the separation surface and its radius of curvature are much larger than the wavelength of the waves striking it. This condition is easily met when visible light strikes macroscopic objects, but not in the case of audible sound waves. In the latter case one cannot resort to the geometric simplification of rays, but has to solve the sound wave equation completely, which also takes into account diffraction phenomena.

Which way does the wave bend?

Think of a wavefront arriving obliquely at the boundary surface. The part of the front that reaches the surface first enters the second medium and changes speed before the others, like a row of people marching side by side who pass obliquely from asphalt onto a muddy field: whoever enters the mud first slows down first, and the row "turns". Consequently:

• if the wave is slower in the second medium (for example light passing from air into water or glass) the refracted ray moves closer to the normal, that is, to the perpendicular to the surface;
• if the wave is faster in the second medium (for example sound passing from air into water) the refracted ray moves away from the normal.

Quantitatively, the ratio of the sines of the angles equals the ratio of the speeds: ${\displaystyle \sin {\theta }_i/\sin {\theta }_r=v_1/v_2}$ (Snell's law, see refractive index). In the passage the frequency of the wave does not change, while the wavelength changes in proportion to the speed. If the wave passes into a medium where it is faster and strikes very obliquely, beyond a certain angle (the critical angle) it can no longer get out and is completely reflected: this is total internal reflection, on which optical fibres are based.

Importance of refraction

The phenomenon of refraction is important because

1. it underlies the explanation of a large number of natural phenomena. Refraction makes it possible to explain:
   • the formation of rainbows;
   • the formation of mirages and other strange optical phenomena (fata Morgana,...);
   • the shift of the apparent position of an object (think of a coin at the bottom of a swimming pool, or a stick partly immersed in a transparent liquid, which looks "broken")
   • the ability of certain materials to carry and confine light, as if they were channels (optical fibres, waveguides...);
   • the ability of certain materials, suitably shaped, to bend light (lenses)
2. by investigating the change of direction of the refracted ray (governed by a law relating the angles of incidence and refraction, called Snell's law) it is possible:
   • to derive physical properties of the medium in which the refraction occurs. These properties are summarized by a number called the refractive index;
   • to derive information on the nature of the wave being refracted. Sometimes the refractive index depends on the wavelength of the incident wave (in these cases the medium is said to be dispersive). If an incident wave that is not "monochromatic" (that is, made up of the superposition of "component" waves of different wavelengths) enters a dispersive medium, the individual components will be deflected at different angles. Very well known is the experiment in which Newton showed, by passing sunlight through a glass prism, that sunlight is made up of light of different wavelengths (that is, of different colours);
   • to calculate the efficiency with which a wave undergoing refraction transfers energy from one medium to another. This problem, dealt with in the section on impedance matching, is fundamental for understanding how antennas and musical instruments work.
3. historically, the explanation of refraction was a serious "test bench" for the corpuscular and wave theories of light. Indeed, had the experimental means of the late 17th century allowed it, refraction would have provided an experimentum crucis to settle the question between the corpuscular theory and the wave theory well before Young's experiment. On this subject, visit the section refractive index

Check your understanding

Virtual laboratory: experiments on refraction

If you are interested in visualizing refraction phenomena, click on 2D Waves virtual laboratory. On the pages listed below you will find a series of guided experiments, but you are free to try new configurations and new experiments. The app is very intuitive; to understand correctly what you observe, keep in mind that the coloured areas represent the amplitude of the oscillation. The brighter the colour, the larger the amplitude of the oscillations (green for crests, red for troughs). Remember also that what we have called the wave ray is always perpendicular to the wavefronts shown in the animation.

Guided paths in the virtual laboratory

• experiments on refraction
• experiments on total reflection
• experiments on total reflection from a circular prism
• experiments on mirages and other strange phenomena

Further reading and links

• If you want to see refraction at work, visit the section rifrazione in una dimensione (in Italian)
• To see how refraction explains many of the phenomena described, visit the section on the refractive index
• Since it is a wave phenomenon, refraction also affects sound waves. On the dedicated page you will find examples of sound mirages.
• The phenomenon of dispersion has profound consequences in many areas of wave theory. Visit the section devoted to it!!
• The rainbow is a spectacular natural phenomenon due to the combined phenomena of refraction and dispersion. In the section waves and rainbows you can find out how rainbows form, and what their features are.
• If you are interested in finding out more about optical fibres, visit the section waveguides.
• Related questions and answers:
  • Frequency and wavelength are linked: why then, going from air to water, does the wavelength change but not the frequency?