Describing waves in two and three dimensions

When a wave spreads across a surface or through space, a simple displacement–time graph doesn't show enough. We also need to show the shape of the travelling disturbance, and the direction in which energy is being transferred.

A wavefront is a line or surface of constant phase that moves with a wave. In diagrams, neighbouring wavefronts are often one wavelength apart, so their spacing gives the wavelength. With water waves, a crest line is a visible wavefront. With light, we usually can't see the wavefronts directly, but the model still helps.

A ray is a line showing the local direction of energy transfer by a wave. Rays are drawn perpendicular to wavefronts. For circular wavefronts, rays point radially outwards; for plane wavefronts, rays are parallel.

Treat this as a model, not a photograph of what is really there. Ripple tanks, for example, let us see water-wave wavefronts, then use them as an analogue for light or microwaves. The analogy is useful, but it has limits: water needs a medium and has surface effects; electromagnetic waves do not require a material medium.

In sketches, keep the geometry tidy. Draw wavefronts as equally spaced lines or arcs; rays should cross them at right angles. Close wavefronts mean a short wavelength. If the rays spread out, the energy is spreading too.

What happens at a boundary

A boundary is the interface where a wave reaches a different medium or hits an obstacle. When it gets there, the incident wave may reflect, transmit, or do both.

Reflection is a wave behaviour in which a wave returns into the original medium after meeting a boundary. At a plane reflecting boundary, the angle of incidence equals the angle of reflection. A normal is a line drawn perpendicular to a boundary at the point where a ray meets it; all incidence, reflection and refraction angles are measured from the normal, not from the surface.

Transmission is a wave behaviour in which wave energy passes into the second medium. Often, the transmitted wave changes speed, and that’s when refraction is seen.

Refraction is a change in wave direction caused by a change in wave speed as the wave enters a different medium. The source fixes the frequency, so the frequency does not change at the boundary. The wave equation is

If the wave slows down while $f$ stays unchanged, $\lambda$ becomes smaller. That is why water waves entering shallower water have closer wavefronts.

A ripple tank shows this well in practice. A plane dipper can make straight wavefronts; a barrier gives reflection; a shallow region changes speed and produces refraction. When you use ripple-tank evidence, compare what you see with the accepted wave model: unchanged frequency, changed speed, changed wavelength, and a bend at the interface.

For ray diagrams at boundaries in this topic, expect only incident, reflected and transmitted waves. Draw the normal first. That one line prevents most angle errors.

Spreading at edges and gaps

Diffraction is a wave behaviour where a wave spreads into the region beyond an edge, obstacle or aperture. It does not happen because the wave changes medium, so the wave speed, frequency and wavelength stay the same. The direction of parts of the wave changes, and so does the distribution of amplitude.

Diffraction is strongest when the aperture width or obstacle size is close to the wavelength. If the gap is wide compared with the wavelength, the spreading is only slight and mainly near the edges. When the gap is comparable with the wavelength, the emerging wavefront becomes much more curved.

The usual assumption in this topic is normal incidence: the incoming wavefronts are parallel to the slit or grating, so the rays meet it at right angles. Oblique cases do exist, but they are not the cases expected here.

Huygens' construction is a useful way to picture it. Each point on a wavefront can be treated as a source of secondary wavelets, and the new wavefront is the envelope of those wavelets. This model helps explain why wavefronts spread at an aperture, especially at the edges. It is not a perfect historical model of light, but it works very well for sketching diffraction.

You see the same idea outside the lab. Long-wavelength radio waves can diffract around hills and reach places with no direct line of sight to the transmitter. Shorter wavelengths diffract less around the same obstacle, so coverage can disappear behind the hill. Sound travelling around a corner follows the same idea in a familiar setting.

How to read and draw the diagrams

Wavefront-ray diagrams aren’t just decoration; they show the physics. Rays show the direction of travel. The spacing between wavefronts shows the wavelength. Since frequency does not change at a boundary, closer wavefronts mean a lower speed.

For refraction, the ray bends because one side of a wavefront reaches the new medium first, so that side changes speed first. If the transmitted wave is slower, the wavelength decreases and the ray bends towards the normal. If the transmitted wave is faster, the wavelength increases and the ray bends away from the normal.

For diffraction, wavefronts curve after an edge or aperture, but their spacing does not change. That gives you the quick visual difference from refraction: refraction changes wavelength; diffraction does not. Diffraction redistributes amplitude, so the diffracted wave is usually weaker in any one direction because the energy has spread over a larger wavefront.

When you interpret a diagram, check these four things in order:

• Are the rays perpendicular to the wavefronts?
• Is the normal drawn at the boundary where the ray meets it?
• Does wavefront spacing change only when the medium changes?
• Is the wave incident normally on slits or gratings in interference and diffraction pattern questions?

Refractive index and Snell's law

The refractive index of a transparent medium compares the speed of light in a vacuum with the speed of light in that medium. It has no unit:

In IB questions, a larger refractive index means light travels more slowly in that medium.

For waves going from medium 1 into medium 2, Snell's law is

When light enters glass from air, $n$ increases and the speed falls, so the ray bends towards the normal. When light leaves glass and enters air, $n$ decreases and the speed rises, so the ray bends away from the normal. The frequency still does not change; the wavelength changes with speed.

A semicircular block gives a tidy laboratory method for measuring refractive index. Send a narrow ray through the centre of the flat face, mark the incident and refracted paths, construct the normal, and measure the angles with a protractor. By repeating this for several angles, you can either average values of $\sin i/\sin r$ or plot $\sin i$ against $\sin r$ and use the gradient. Quote an uncertainty from the precision of the angle readings and the scatter in the repeated values.

Critical angle and total internal reflection

The critical angle is the angle of incidence in the higher-refractive-index medium that makes the refracted ray in the lower-refractive-index medium travel along the boundary. At this point, the refracted angle is 90°.

Total internal reflection is a boundary behaviour where all the wave energy reflects back into the original medium because the angle of incidence is greater than the critical angle. It can only occur when the wave is trying to enter a medium where it would travel faster; for light, that means going from higher $n$ to lower $n$.

For light travelling from medium 1 into medium 2, with $n_1 > n_2$, the critical-angle relation is

If $\theta_1 < \theta_c$, some light is refracted and some is reflected. If $\theta_1 = \theta_c$, the refracted ray runs along the boundary. If $\theta_1 > \theta_c$, total internal reflection occurs.

This explains how optical fibres guide light: repeated total internal reflection keeps the light inside the fibre. The same geometry works for non-light waves when the speed conditions allow it; the wave model matters here, not the particular type of wave.

Adding displacements, not energies

The principle of superposition applies when waves of the same type overlap: the resultant displacement at a point is the vector sum of the separate displacements that each wave would have produced there on its own.

Take two pulses on a rope. If both lift the rope upwards at the same place and time, the resultant pulse is larger. If one lifts the rope up while the other pulls it down by the same amount, the rope can be momentarily flat. After that, the pulses keep moving through each other; they haven’t permanently destroyed each other.

Constructive interference is superposition in which overlapping waves produce a resultant amplitude larger than the individual amplitudes at that point. Destructive interference is superposition in which overlapping waves produce a resultant amplitude smaller than the individual amplitudes, possibly zero.

Here, the key word is “displacement”. On a rope, it means the displacement of the rope from equilibrium. In sound, it could mean the displacement of air particles or a pressure variation. For light, the wave model uses the electric and magnetic field values as the quantities that superpose.

A travelling wave transfers energy even when the medium itself has no net transport. In a rope wave, the rope elements oscillate about equilibrium while the disturbance and its energy travel along the rope.

Why coherence is needed

Coherent sources emit waves with a constant phase difference and the same frequency. The word “constant” matters. If the phase relationship drifts randomly, the bright and dark pattern washes out before you can observe it.

Two loudspeakers connected to the same signal generator act as coherent sound sources. Put a microphone along a line in front of them and it detects alternating loud and quiet regions. The loud regions occur where the sound waves arrive in phase; the quiet regions occur where they arrive in antiphase.

For light, coherence is harder to produce because ordinary lamps contain many independent atoms, each emitting short, unrelated wave trains. A laser helps because it is monochromatic and coherent. Young's double-slit experiment can also use one source illuminating two slits; those two slits then behave like coherent secondary sources because the same incoming wavefront reaches both of them.

Overlap is a physical requirement, not just a mathematical one. Two waves can interfere at a point only if they are present at that point at the same time.

Path difference decides the outcome

Path difference is the difference in distance travelled by two waves from their sources to the same point. Let $\Delta x$ be path difference (m), and let $m$ be an integer order number (no unit). For two coherent sources that start in phase, constructive interference occurs when

The waves arrive in phase: crest with crest, trough with trough.

Destructive interference occurs when

$\Delta x = \left(m+\frac{1}{2}\right)\lambda$

Here the waves arrive in antiphase: crest with trough. The result is a minimum, though not always absolute zero unless the two waves have equal amplitudes.

These conditions work for sound, water waves, microwaves and light, as long as the sources are coherent and the waves overlap. In double-source questions, don’t begin with “bright” or “dark”; begin by finding the path difference.

Fringes from two coherent slits

Young's double-slit experiment is an interference experiment where coherent waves from two narrow slits overlap, producing a regular pattern of maxima and minima. For light, the maxima appear as bright fringes; the minima appear as dark fringes.

The standard double-slit formula is

The formula assumes that the screen is far from the slits compared with their separation, and that the waves hit the slits normally.

The formula fits the physics nicely. A larger wavelength gives wider fringe spacing. Move the screen farther away and the pattern spreads out. Increase the slit separation and the fringes move closer together.

You can use this directly to measure wavelength. Shine a laser through a double slit onto a distant screen. Measure across several fringe spacings, rather than only one, then divide to find s. Measure D as well, and use the known slit separation d. Laser safety matters here: keep beams below eye level where possible, avoid reflections, and never look into the direct or reflected beam.

The same idea works with microwaves, just on a larger scale. A transmitter, a pair of metal-sheet slits, and a receiver moved sideways can show maxima and minima on a meter. This is a useful reminder that the wave model is not “a light trick”; it is a wave principle.

Young's result matters historically because a stable interference pattern is very hard to explain if light is treated only as a stream of classical particles. It gives strong evidence that light has wave-like behaviour. Later quantum experiments, using very low light intensity, showed that even when detections arrive one at a time, the accumulated pattern is still an interference pattern. That is one reason modern physics does not say “light is just a wave” or “light is just a particle”; the model chosen depends on the observation being explained.

The pattern from one rectangular slit

A single-slit diffraction pattern forms when monochromatic waves pass through one narrow rectangular aperture and spread out. For this topic, take the light to be monochromatic and normally incident on the slit.

The central maximum is the brightest and widest part of the pattern. On both sides, much weaker secondary maxima appear, with dark minima between them. The first secondary maximum has only a small fraction of the central intensity, so the side maxima can be hard to see in real life.

For the first diffraction minimum, the guide gives the small-angle condition as

The central maximum runs from the first minimum on one side to the first minimum on the other, so its angular width is about twice this value.

Superposition explains the formula. Treat the slit as many tiny Huygens sources. In the direction of the first minimum, each source in the top half of the slit pairs with one in the bottom half whose wave arrives half a wavelength out of phase. Pair by pair, the waves cancel.

Slit width matters in a clear qualitative way: a narrower slit spreads the diffraction pattern out, while a wider slit makes the pattern narrower. A longer wavelength spreads the pattern more as well. Red light diffracts more than blue light through the same slit.

Slit width also affects intensity. A narrower slit lets in less wave energy and spreads that energy over a wider angular range, so the pattern becomes dimmer and broader. In sketches of intensity against angle, draw a tall central maximum, first minima placed symmetrically, and much smaller side maxima.

To observe this safely, use a low-power laser, a single slit or adjustable calipers, and a screen several metres away. Measure the separation of the first minima, use the screen distance to find the small angle, and then apply the diffraction equation. Keep the laser beam and any reflections out of eyes throughout.

Real double slits are also single slits

The simple double-slit model treats each slit as extremely narrow. Real slits have a finite width, so each slit also makes its own single-slit diffraction envelope. What you see is the regular double-slit interference fringe pattern sitting inside that broader diffraction envelope.

Modulation is a variation of one pattern by another pattern that controls its overall amplitude. In this case, the single-slit diffraction pattern modulates the double-slit interference pattern. The fringes are brightest near the centre, then fade as the diffraction envelope becomes small.

At angles where the single-slit pattern has a minimum, the double-slit fringes are suppressed. That happens even if the double-slit condition alone would predict a bright fringe there. These are often called missing orders or missing fringes.

Two different quantities control the pattern. Slit separation controls fringe spacing through the double-slit equation. Slit width controls the width of the diffraction envelope through the single-slit condition. Increase the slit separation and more interference fringes fit inside the same diffraction envelope. Increase the slit width and the envelope itself becomes narrower.

From two slits to many

A multiple-slit interference pattern is an interference pattern produced by more than two equally spaced coherent slits. Increase the number of illuminated slits and the principal maxima stay at the same angular positions, but they become sharper and more intense.

If $N$ is the number of illuminated slits (no unit), the intensity of a principal maximum is proportional to $N^2$, assuming the slits are illuminated equally. Here’s the useful reason: field amplitude adds roughly in proportion to $N$, while intensity is proportional to amplitude squared.

For three or more slits, subsidiary maxima can appear between the principal maxima. The key effect, though, is better separation of wavelengths. The whole pattern is still limited by the single-slit diffraction envelope caused by the finite width of each slit.

Diffraction gratings

A diffraction grating is an optical component made of many equally spaced parallel slits or lines that diffract and interfere light to produce sharp maxima. Gratings are often described by the number of lines per millimetre; the grating spacing $d$ is the reciprocal of that line density, converted into metres for calculations.

For a grating at normal incidence, the maxima satisfy

The syllabus may write the same relation as $n\lambda=d\sin\theta$, but I prefer $m$ here so that it is not confused with refractive index.

The central maximum is the zero order. All wavelengths have zero path difference there, so white light remains white. Away from the centre, each wavelength satisfies the grating equation at a different angle. Red light, with the longer wavelength, appears at a larger angle than blue light for the same order. Monochromatic light gives sharp spots or lines; white light gives spectra on both sides of the central maximum.

There is also a maximum possible order because $\sin\theta_m$ cannot exceed 1. If a calculation requires a sine greater than 1, that order simply does not exist.

A good grating experiment measures wavelength by plotting $\sin\theta_m$ against order $m$. Measure the screen distance and the displacement of each order from the central maximum, convert the geometry to an angle, then use the gradient with the grating equation. Include uncertainties from distance measurements; at larger angles, uncertainty in position can noticeably affect $\sin\theta_m$.