Wave phenomena

The **bending** of a wave as it crosses from one medium into another, caused by a **change in its speed** at the boundary.

How much the material **slows light down**: **n = c ÷ v**. A bigger n means slower light and more bending (an optically 'denser' medium).

**n₁ sinθ₁ = n₂ sinθ₂** — the indices and angles (measured from the normal) on each side of a boundary are linked this way. **Given** in the data booklet.

From the **normal** — the line drawn at 90° to the surface — not from the surface itself.

**Toward** the normal — the angle gets smaller.

**Away** from the normal — the angle gets bigger.

When light hitting a boundary is **completely reflected back** into the medium it started in, instead of refracting through. None of it escapes.

The angle of incidence at which the refraction angle is exactly **90°**. Above θc you get total internal reflection.

$\sin\theta_c = \dfrac{n_2}{n_1}$ — derived from Snell's law by setting θ₂ = 90°. n₁ is the denser medium.

1) Light going from a **denser to a less-dense** medium, and 2) an angle of incidence **above the critical angle**.

Rearrange **n = c ÷ v** to **v = c ÷ n**, using c = 3.0 × 10⁸ m s⁻¹.

Measuring the angle from the **surface** instead of from the **normal** — always use the dashed normal line.

The **frequency**. Speed and wavelength change in the same ratio.

**sin θ₁ / sin θ₂ = v₁ / v₂** (given as n₁/n₂ = sin θ₂/sin θ₁ = v₂/v₁).

**55°**: the angle to the normal is 90° − 35°.

Where two (or more) waves overlap, you **add their displacements** at every point to get the resultant.

Waves arrive **in step** (in phase) so their **amplitudes add**, giving a bigger wave (bright / loud).

Waves arrive **half a cycle out of step** (antiphase); equal **amplitudes cancel**, giving zero (dark / quiet).

The **extra distance** one wave travels compared with the other to reach a point, in metres.

**nλ** — a whole number of wavelengths (0, λ, 2λ, …). **Given** in the data booklet.

**(n + ½)λ** — a whole number of wavelengths plus a half. **Given** in the data booklet.

The sources keep a **constant phase difference** (same wavelength, fixed step). Needed for a steady, observable pattern.

So the constructive and destructive points **stay in fixed places**; a drifting phase difference would smear the pattern away.

**Zero.** 1.5λ = (1 + ½)λ is destructive, and equal amplitudes cancel completely.

**2A** — in step, so the amplitudes add.

Only when the two **amplitudes are equal**; otherwise just part of one wave cancels.

Divide by λ: a **whole number** → constructive (nλ); a whole number **+ ½** → destructive ((n+½)λ).

While they overlap the displacements **add**; afterwards each pulse continues **unchanged**.

**2 mm**: the difference of the amplitudes.

**No**: it is redistributed to the bright fringes.

Light of one wavelength through **two close, coherent slits** overlaps on a screen to make a row of **equally spaced bright and dark fringes**.

One of the **bright or dark bands** on the screen in a double-slit (or similar interference) pattern.

The gap from one **bright** fringe to the **next** bright fringe — the same all the way across the screen.

$s = \dfrac{\lambda D}{d}$ — **given** in the data booklet (s spacing, λ wavelength, D slit-to-screen distance, d slit separation).

**s** fringe spacing, **λ** wavelength, **D** slits-to-screen distance, **d** slit separation — all in metres.

s gets **bigger** — d is on the bottom of s = λD/d, so closer slits give wider fringes.

s gets **bigger** — λ is on the top, so a longer wavelength widens the fringes.

They must give light of the **same wavelength** with a **fixed phase relationship**, so the pattern is stable instead of flickering.

About **θ ≈ λ/d** radians, from d sin θ = nλ with sin θ ≈ θ for small angles.

The energy 'missing' at the dark fringes is **redistributed into the bright fringes**; the total energy over the whole screen is unchanged.

**Mixing units** — convert every length to metres (mm = 10⁻³ m, nm = 10⁻⁹ m) before substituting into s = λD/d.

s = λD/d = (6.0×10⁻⁷ × 2.0) / (0.5×10⁻³) = 2.4×10⁻³ m = 2.4 mm.

**1½λ** (dark fringes at ½λ, 1½λ, 2½λ, …).

**2.0 cm**: 6 gaps between 7 fringes.

s **doubles** (s = λD/d).

The **spreading out** of a wave as it passes **through a gap** or **around an edge**.

When the **gap is about the same size as the wavelength** (gap ≈ λ).

Very **little** spreading — the wave carries almost straight on; only the edges curl in.

A **longer** wavelength — it is closer to the gap size, so it spreads more.

A **lower** frequency — lower frequency means a longer wavelength, which spreads more.

**All** of them — water, sound and light (every wave diffracts).

Sound's wavelength (~1 m) is about the size of a doorway (gap ≈ λ → strong diffraction); light's wavelength is far smaller, so it barely spreads.

The length of **one full wave** — for example from one crest to the next.

$v = f\lambda$ (given in the data booklet). Rearranged: $\lambda = \dfrac{v}{f}$.

Thinking a **higher** frequency spreads more — it's the opposite. Higher frequency → shorter λ → **less** diffraction.

**Yes**: they bend into the region behind it, more so for a longer wavelength.

**No**: only its direction spreads; the wavefront spacing stays λ.

**Circular arcs** spreading from the gap, still one wavelength apart.

At $\theta = \lambda/b$, where b is the slit width (small-angle approximation).

**Inversely** — narrower slit (smaller b) ⇒ **wider** central maximum, since $\theta = \lambda/b$.

**Directly** — longer wavelength (red) ⇒ wider spread, since $\theta = \lambda/b$.

$d\sin\theta = n\lambda$, with n = 0, 1, 2, … the order of the maximum.

Take the **reciprocal of the lines per metre**: $d = 1/(\text{lines per metre})$.

500 lines/mm = 500×10³ lines/m, so $d = 1/(500\times10^3) = 2.0\times10^{-6}$ m.

Many slits add in phase only at very precise angles, so each maximum is **narrow and bright**.

The whole number of wavelengths of path difference between **adjacent** slits ($d\sin\theta = n\lambda$).

The **grating** equation has sinθ ($d\sin\theta = n\lambda$); the single-slit small-angle result is just $\theta = \lambda/b$.

sinθ ≤ 1, so $n\lambda \le d$ — orders with $n > d/\lambda$ cannot exist.

**Twice** the first-minimum angle: a full width of about $2\lambda/b$.

Precisely **measuring wavelengths** (spectroscopy), because its sharp maxima pin down θ accurately.

The fringes (λ/d apart) sit **inside a single-slit envelope** with its first zero at λ/b; fringes on a zero are missing.

A **white** maximum: zero path difference for every wavelength.

The whole number below **d / λ(red)** (λ(red) ≈ 700 nm).