Describing waves in two and three dimensions

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

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

A ray is a line that shows the local direction of energy transfer by a wave. Draw rays perpendicular to wavefronts. With circular wavefronts, rays point radially outwards; with plane wavefronts, the rays are parallel.

Treat this as a model, not a photograph of reality. Ripple tanks, for example, let us see water-wave wavefronts and use them as an analogue for light or microwaves. That’s useful, but the analogy has limits: water needs a medium and has surface effects; electromagnetic waves do not require a material medium.

In sketches, keep the geometry clean. Draw wavefronts as equally spaced lines or arcs, with rays crossing 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 place where a wave meets another medium or an obstacle. When a wave reaches a boundary, some of it may reflect, some may transmit, or both may happen.

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, which is when refraction occurs.

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 it does not change at the boundary. The wave equation is

[v=f\lambda]

where v is wave speed (m s⁻¹), f is frequency (Hz) and λ is wavelength (m). If the wave slows down while f stays the same, λ becomes smaller. That is why water waves entering shallower water have closer wavefronts.

A ripple tank shows this neatly. A plane dipper can make straight wavefronts; a barrier gives reflection; a shallow region changes speed and gives refraction. When using 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 doesn't happen because the medium changes, so the wave speed, frequency and wavelength stay the same. What changes is the direction of parts of the wave, along with how the amplitude is distributed.

Diffraction is strongest when the aperture width or obstacle size is close to the wavelength. If the gap is much wider than the wavelength, the wave only spreads a little near the edges. If the gap is comparable with the wavelength, the emerging wavefront becomes much more curved.

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

Huygens' construction gives a useful model. Treat each point on a wavefront as a source of secondary wavelets; the new wavefront is the envelope of those wavelets. This explains why wavefronts spread at an aperture, especially near the edges. It is not a perfect historical model of light, but it works very well for sketching diffraction.

The effect shows up outside the lab too. 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 carry the physics. Rays show the direction of travel. The spacing between wavefronts shows the wavelength. Since frequency stays unchanged at a boundary, closer wavefronts mean a smaller speed.

In refraction, the ray bends because one side of a wavefront reaches the new medium first and 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.

With diffraction, wavefronts curve after an edge or aperture, but their spacing does not change. That is the fast visual check against refraction: refraction changes wavelength; diffraction does not. Diffraction spreads out amplitude, so the diffracted wave is usually weaker in any one direction because the energy has spread over a larger wavefront.

When interpreting 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:

[n=\frac{c}{v}]

where n is refractive index (no unit) and c is the speed of light in a vacuum (m s⁻¹). In IB questions, a larger refractive index tells you that light travels more slowly in that medium.

Snell's law for waves travelling from medium 1 into medium 2 is

[\frac{n_1}{n_2}=\frac{\sin * heta_2}{\sin * heta_1}=\frac{v_2}{v_1}]

where n₁ is the refractive index of medium 1 (no unit), n₂ is the refractive index of medium 2 (no unit), θ₁ is the angle between the incident ray and the normal (rad or °), θ₂ is the angle between the refracted ray and the normal (rad or °), v₁ is the wave speed in medium 1 (m s⁻¹) and v₂ is the wave speed in medium 2 (m s⁻¹).

When light enters glass from air, n increases and speed decreases, so the ray bends towards the normal. When light leaves glass and enters air, n decreases and speed increases, 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, then mark the incident and refracted paths. After that, construct the normal and measure the angles with a protractor. Repeating the method for several angles lets you average values of (\sin i/\sin r), or plot (\sin i) against (\sin r) and use the gradient. Quote an uncertainty from the angle-reading precision and the scatter in 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 happens at a boundary when all the wave energy reflects back into the original medium because the angle of incidence is greater than the critical angle. It can happen only when the wave is trying to enter a medium where it would travel faster; for light, that means moving from higher n to lower n.

For light travelling from medium 1 into medium 2, with n₁ > n₂, the critical-angle relation is

[\sin * heta*_c=\frac{n_2}{n_1}]

where θ꜀ is the critical angle (rad or °). If θ₁ < θ꜀, some light is refracted and some is reflected. If θ₁ = θ꜀, the refracted ray runs along the boundary. If θ₁ > θ꜀, total internal reflection occurs.

This explains how optical fibres guide light: repeated total internal reflection keeps the light inside the fibre. The same geometry can apply to 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 individual displacements each wave would have produced there on its own.

Take two pulses on a rope. If both push the rope upwards at the same place and time, the resultant pulse is larger. If one pushes it upwards and the other downwards by the same amount, the rope can be momentarily flat. After that, the pulses carry on through each other; they haven’t permanently destroyed each other.

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

The key word is “displacement”. On a rope, that means the displacement of the rope from equilibrium. In sound, it could be the displacement of air particles or pressure variation. In 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 the same frequency and a constant phase difference. “Constant” matters here. If the phase relationship drifts randomly, the bright and dark pattern washes out before you have time to observe it.

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

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

The overlap condition is physical, not just mathematical. 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 means the difference between the distances travelled by two waves from their sources to the same point. Let Δx 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

[\Delta x=m\lambda]

where Δx is path difference (m) and m may be 0, 1, 2, 3, … . The waves reach the point 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. That gives a minimum, though not always absolute zero unless the two waves have equal amplitudes.

The same 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 regularly spaced maxima and minima. For light, the maxima appear as bright fringes and the minima as dark fringes.

The standard double-slit formula is

[s=\frac{\lambda D}{d}]

Here, s is the separation between neighbouring fringes (m), D is the distance from the slits to the screen (m), and d is the separation between neighbouring slits or lines in an aperture array (m). The formula assumes that the screen is far from the slits compared with their separation, and that the waves hit the slits normally.

The physics is quite direct. A larger wavelength gives wider fringe spacing. Put the screen farther away, and the whole pattern spreads out. Increase the slit separation, and the fringes move closer together.

You can measure wavelength this way in the lab. Shine a laser through a double slit onto a distant screen, measure across several fringe spacings instead of only one, then divide to find s. Measure D, use the known slit separation d, and substitute into the formula. Laser safety still matters: 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. It’s 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 with very low light intensity showed that, even when detections arrive one at a time, the accumulated pattern is still an interference pattern. That is why modern physics doesn’t describe light as “just a wave” or “just a particle”; the model chosen depends on the observation being explained.

The pattern from one rectangular slit

A single-slit diffraction pattern is the intensity pattern you get 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 part of the pattern, and also the widest. On both sides, 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 look faint in practice.

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

[* heta*=\frac{\lambda}{b}]

where θ is the angular position of the first minimum measured from the central axis (rad) and b is the slit width (m). 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 gives the model behind 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 has a matching source in the bottom half, and its wave arrives half a wavelength out of phase. The waves cancel pair by pair.

Slit width has a clear qualitative effect: 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.

The intensity changes too. A narrower slit lets through less wave energy and spreads that energy over a wider angular range, so the pattern becomes dimmer and broader. In intensity-against-angle sketches, draw a tall central maximum, first minima placed symmetrically, and much smaller side maxima.

To observe the pattern 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 use the diffraction equation. Keep the laser beam and 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 finite width, so each one also produces a single-slit diffraction envelope. What you observe is the regular 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, 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 narrows.