05.08 Wave Phenomena

The big idea Superposition produces interference, wavelength compared with opening size controls diffraction, and the same wave model explains fringes, total internal reflection, polarization, dispersion, and color.

Interference is signed addition

Bright and dark patterns come from what two displacements do at the same point.

Not sure about the basics underneath this lesson?

Superposition and interference

When waves overlap, signed displacements add. Crest with crest gives constructive interference; opposite displacements give partial or complete destructive interference. Ideal waves continue after overlap. A stable pattern needs coherent sources with the same frequency and a constant phase relationship.

The left panel adds signed wave displacements to show constructive and destructive interference. The right panel labels slit separation d, screen distance L, the central bright fringe, and distance x to the first bright fringe in a Young double-slit arrangement. — Signed displacements add locally; in the double-slit geometry, d is slit separation, L is screen distance, and x₁ is measured from the central bright fringe.

Diffraction depends on scale

Diffraction is spreading around an edge or through an opening. It is strongest when the opening or obstacle is comparable with wavelength. Long-wavelength sound bends around a doorway more noticeably than visible light because visible wavelengths are tiny compared with the doorway.

Double-slit pattern

Two coherent slits create alternating bright and dark fringes. Whole-wavelength path differences reinforce; half-step differences cancel. The center is the zero-order maximum, and first-order bright fringes lie on both sides.

Double-slit equation

Bright-fringe position

x_m=\frac{m\lambda L}{d}

m is order, L is screen distance, d is slit separation, and xₘ is measured from the center.

First bright fringe

\frac{\lambda}{d}=\frac{x}{L}

The source lesson uses this m = 1 form.

Conversions

1\,\text{nm}=10^{-9}\,\text{m},\qquad 1\,\text{\AA}=10^{-10}\,\text{m}

Convert before substitution.

Not sure about this bit?

Worked fringe calculations

Find wavelength

Slit separation. $d=4.0\times10^{-3}\,\text{m}$
Screen distance. $L=2.0\,\text{m}$
First fringe. $x=1.8\times10^{-4}\,\text{m}$

Find λ.

Rearrange and substitute.
$\lambda=\frac{xd}{L}=\frac{(1.8\times10^{-4})(4.0\times10^{-3})}{2.0}$

λ = 3.6 × 10⁻⁷ m.

Find first-fringe distance

Wavelength. $\lambda=6.64\times10^{-7}\,\text{m}$
Screen distance. $L=2.75\,\text{m}$
Slit separation. $d=1.20\times10^{-4}\,\text{m}$

Find x.

Use x = λL/d.
$x=\frac{(6.64\times10^{-7})(2.75)}{1.20\times10^{-4}}=1.52\times10^{-2}\,\text{m}$

x = 0.0152 m, or 1.52 cm.

Total internal reflection

It requires light traveling from a higher index to a lower index and an incident angle above the critical angle. At θ_c, the refracted angle is 90°. Above it, ideal light remains in the first medium. Optical fibers use this effect.

Critical angle

\sin\theta_c=\frac{n_2}{n_1}

Use only when n₁ > n₂ and the ray starts in medium 1.

Dispersion, polarization, and color

Dispersion separates white light because refractive index varies with wavelength. It can cause chromatic aberration. Polarization restricts transverse vibration directions, evidence that light is transverse. Screens mix red, green, and blue light additively: all three together at suitable equal intensity appear white.

Wave-phenomena traps

Calling every bend refraction

Refraction happens at a speed-changing boundary; diffraction is spreading at an edge or opening.

Reversing total internal reflection

It starts only in the higher-index medium.

Forgetting fringe order

λ/d = x/L is first order; general position includes m.

Mixing pigment and light

RGB light is additive; paint uses subtractive mixing.

Wave-phenomena practice

Name the phenomenon first, then label and convert every calculation quantity.

Final checklist

Tick these off as you go. Saved in this browser.

I distinguish interference, diffraction, refraction, and reflection.
I identify m, λ, L, d, and xₘ in a slit setup.
I convert nm and Å before using SI values.
I apply both requirements for total internal reflection.
I can explain dispersion, polarization, and RGB light before the 05.08 quiz.