Revise Wave phenomena for IB Physics (C.3)

When a wave meets a boundary, some of it is always reflected. The law of reflection states that the angle of incidence equals the angle of reflection, with both angles measured from the normal — the line drawn perpendicular to the surface at the point where the ray strikes. The incident ray, the reflected ray and the normal all lie in the same plane. Measuring from the normal rather than from the surface is essential: it is the convention used everywhere in this topic, including in Snell's law.

When light crosses from one transparent medium into another it changes speed, and unless it strikes along the normal this change of speed bends the ray. The refractive index of a medium, $n = c/v$, measures how much light slows down in it: $c$ is the speed of light in a vacuum and $v$ its speed in the medium. Because light never travels faster than $c$, $n \ge 1$; a larger $n$ means slower light and an optically denser medium. Light slowing down (entering a denser medium) bends towards the normal; light speeding up (entering a less dense medium) bends away from the normal.

Refractive index: ; n = \dfrac{c}{v} \[4pt] Snell's law: $\; n_1\sin\theta_1 = n_2\sin\theta_2$

In Snell's law, $\theta_1$ is the angle in the medium of index $n_1$ and $\theta_2$ is the angle in the medium of index $n_2$, both measured from the normal. The single rule that prevents most errors: each refractive index must be paired with the angle in that same medium. Keep the subscripts consistent from the first line of working and the algebra takes care of itself.

When light travels from a denser medium towards a less dense one, it bends away from the normal, so the refracted angle is larger than the angle of incidence. Increase the angle of incidence and eventually the refracted ray reaches 90°, grazing along the boundary. The angle of incidence that produces this is the critical angle, $\theta_c$. Beyond it there is no refracted ray at all: the light is entirely reflected back into the denser medium. This is total internal reflection.

At the critical angle the refracted angle is $90°$, so Snell's law $n_1\sin\theta_c = n_2\sin 90°$ gives: $\sin\theta_c = \dfrac{n_2}{n_1} \quad (n_1 > n_2)$

Diffraction is the spreading of a wave as it passes through a gap or bends around an obstacle. Crucially, the amount of spreading depends on the ratio of the gap width $b$ to the wavelength $\lambda$. When the gap is much wider than the wavelength ($b \gg \lambda$) the wave passes almost straight through with only slight fraying at the edges. As the gap narrows towards the wavelength ($b \approx \lambda$) the spreading becomes dramatic, with the wave fanning out in wide arcs on the far side. This is why sound (long wavelength) bends round a doorway readily, while light (very short wavelength) needs an extremely fine slit before its diffraction is visible.

When two or more waves meet, the principle of superposition states that the resultant displacement at any instant is the vector sum of the individual displacements. Where two crests coincide the wave is reinforced (constructive interference); where a crest meets a trough they cancel (destructive interference). For a fixed, observable pattern the sources must be coherent — same frequency and a constant phase difference. Which type of interference occurs at a given point is decided by the path difference: the extra distance one wave travels compared with the other to reach that point.

For two coherent, in-phase sources: $\text{Constructive: path difference} = n\lambda$ $\text{Destructive: path difference} = \left(n + \tfrac{1}{2}\right)\lambda$ with $n = 0, 1, 2, \ldots$

Thomas Young passed monochromatic light through two very narrow, closely spaced slits. Light diffracts at each slit, so the two slits behave as coherent sources whose waves overlap and interfere. On a distant screen this produces a pattern of equally spaced bright and dark fringes: bright where the path difference is a whole number of wavelengths, dark where it is an odd number of half-wavelengths. The experiment is one of the strongest pieces of evidence that light is a wave.

The spacing between adjacent bright fringes is: $s = \dfrac{\lambda D}{d}$ where $\lambda$ is the wavelength, $D$ the perpendicular slit-to-screen distance and $d$ the slit separation. The relation holds when $D \gg d$.

This section is HL only. Passing monochromatic light through one narrow slit of width $b$ produces a diffraction pattern: a broad, bright central maximum flanked by dimmer secondary maxima separated by dark minima. The first minimum sits at an angle $\theta = \lambda/b$ (in radians) from the centre, which makes the central maximum twice as wide as the secondary ones. Narrowing the slit or using a longer wavelength widens the whole pattern — the same gap-to-wavelength dependence seen in diffraction generally.

Single-slit first minimum: $\; \theta = \dfrac{\lambda}{b}$ \[4pt] Diffraction grating maxima: $\; d\sin\theta = n\lambda$

A diffraction grating replaces two slits with thousands of equally spaced ones. Every slit contributes light that interferes, and the many-slit interference makes the bright maxima far sharper, brighter and more widely separated than a double slit. Their positions obey $d\sin\theta = n\lambda$, where $d$ is the spacing between adjacent slits, $\theta$ the angle of the $n$-th order maximum from the straight-through direction, and $n = 0, 1, 2, \ldots$ Gratings are often quoted as $N$ lines per metre, in which case $d = 1/N$; convert to a spacing in metres before substituting.

In Paper 2 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an answer mark (A) — and error-carried-forward (ECF) means a wrong number early on does not have to cost you the marks that follow. But that protection only exists if your method is written down, and in an 'explain' part the reasoning marks are separate from the calculation marks. Study how each mark below is earned by a specific line.

These wave phenomena reappear throughout physics. Total internal reflection underpins optical fibres and modern communications; interference and path difference return in thin films, standing waves and, at HL, in the resolution of optical instruments. The habit built here — measure angles from the normal, pair each index with its own angle, and let the path difference decide constructive from destructive — carries directly into every later wave and optics problem. Master the method, show every line, and the rest of wave physics becomes variations on rules you already own.