physica.
38 formulas

Waves & Optics

Interference, refraction. Every formula below opens into a live, hands-on simulation.

refraction
n1sin⁡θ1=n2sin⁡θ2n_1 \sin\theta_1 = n_2 \sin\theta_2

Snell's Law

Light bends toward the normal when entering a denser medium.

reflection
θi=θr\theta_i = \theta_r

Law of Reflection

Light bounces off a mirror at the same angle it arrives.

wave mechanics
v=fλv = f\lambda

Wave Speed Equation

Wave speed equals how many wavelengths pass a point each second.

geometric optics
1f=1do+1di\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

Thin Lens Equation

Reciprocals of object and image distances always add up to the lens power.

interference
dsin⁡θ=mλd \sin\theta = m\lambda

Young's Double Slit Interference

Two overlapping wave sources create bright and dark bands.

diffraction
asin⁡θ=mλa \sin\theta = m\lambda

Single Slit Diffraction

A narrow slit spreads light into a pattern of bright and dark bands.

polarization
tan⁡θB=n2n1\tan\theta_B = \frac{n_2}{n_1}

Brewster's Angle

At one special angle, reflected light becomes perfectly polarized.

wave mechanics
f′=fv+vov−vsf' = f \frac{v + v_o}{v - v_s}

Doppler Effect

Moving toward a wave source compresses waves; moving away stretches them.

polarization
I=I0cos⁡2θI = I_0 \cos^2\theta

Malus's Law

Only the component of the electric field aligned with the polarizer axis gets through; the rest is absorbed.

diffraction
θmin=1.22λD\theta_{min} = 1.22 \frac{\lambda}{D}

Rayleigh Criterion

Two point sources are just resolvable when the central maximum of one falls on the first dark ring of the other.

diffraction
dsin⁡θ=mλd \sin\theta = m \lambda

Diffraction Grating Equation

Light constructively interferes whenever the path difference between adjacent slits equals an integer number of wavelengths.

geometric optics
1f=(n−1)(1R1−1R2)\frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)

Lensmaker's Equation

Focal length depends on how strongly light bends at each curved surface, governed by the index step and surface curvature.

geometrical optics
1f=1v+1u\frac{1}{f} = \frac{1}{v} + \frac{1}{u}

Mirror Equation

Object distance, image distance, and focal length lock together — change one and the others must rearrange.

electromagnetic waves
n=cvn = \frac{c}{v}

Refractive Index Definition

The refractive index tells you the factor by which light slows down inside a material.

electromagnetic waves
I=12cϵ0E02I = \frac{1}{2} c \epsilon_0 E_0^2

Intensity of an Electromagnetic Wave

Intensity scales with the square of the electric field amplitude — doubling the field quadruples the power flow.

quantum optics
E=hf=hcλE = h f = \frac{h c}{\lambda}

Photon Energy (Planck-Einstein Relation)

Higher frequency means each light particle carries more punch — color literally equals energy.

wave mechanics
∂2y∂t2=v2∂2y∂x2\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}

Wave Equation

The curvature of the wave in space drives its acceleration in time — sharper bends snap back faster.

wave mechanics
fn=nv2Lf_n = \frac{n v}{2 L}

Standing Wave Frequencies

Only wavelengths that fit a whole number of half-wave humps between the two fixed ends survive — those are the notes.

refraction
θc=arcsin⁡(n2n1)\theta_c = \arcsin\left(\frac{n_2}{n_1}\right)

Critical Angle (Total Internal Reflection)

Past a certain angle, light heading from dense to rare can't refract out — it gets reflected back perfectly.

superposition
fbeat=∣f1−f2∣f_{beat} = |f_1 - f_2|

Beat Frequency

Two waves of nearly equal frequency drift in and out of phase. When they line up they add (loud); when opposed they cancel (silent). The loudness pulses at the difference frequency.

geometric optics
m=−didom = -\frac{d_i}{d_o}

Lateral Magnification

Magnification compares image height to object height. A negative sign means the image is inverted; its magnitude tells you how many times larger or smaller the image is.

interference
2nt=(m+12)λ2 n t = \left(m + \tfrac{1}{2}\right)\lambda

Thin-Film Interference

Light reflecting off the top and bottom surfaces of a film travels different path lengths. A half-wave phase flip at the top surface makes the half-integer condition give bright reflection — and different colors satisfy it at different thicknesses.

diffraction
nλ=2dsin⁡θn\lambda = 2 d \sin\theta

Bragg's Law

Atomic planes act like a stack of partial mirrors. Reflections from successive planes add up only when their extra path length is a whole number of wavelengths — that condition pins the angle for each wavelength.

geometric optics
NA=nsin⁡θmaxNA = n \sin\theta_{max}

Numerical Aperture

Numerical aperture is the sine of the widest cone of light an optical system can gather or emit, scaled by the medium's index. Bigger NA means more light collected and finer resolving power.

wave propagation
vg=dωdkv_g = \frac{d\omega}{dk}

Group Velocity

A localized wave packet is a sum of many frequencies. The crests move at the phase velocity, but the packet's envelope — where the energy and information ride — moves at the group velocity, the slope of the dispersion curve.

interferometry
ΔN=2 Δdλ\Delta N = \frac{2\,\Delta d}{\lambda}

Michelson Interferometer Fringe Shift

Light in one arm makes a round trip, so moving the mirror by Delta d changes the path by 2*Delta d. Each whole wavelength of extra path slides the fringe pattern by exactly one fringe — turning displacement into a count.

wave propagation
v=rt⇒r=v tv = \frac{r}{t} \quad\Rightarrow\quad r = v\,t

Huygens' Principle

Every point a wave reaches becomes the launch point of a new little wave. Add all those wavelets up and their leading edge is where the wave goes next. Where speed changes, the wavelets shrink and the front tilts — that bend is refraction.

wave kinematics
vp=ωk=fλv_p = \frac{\omega}{k} = f\lambda

Phase Velocity

Pick one crest and follow it. In one period T it advances exactly one wavelength lambda, so its speed is lambda/T = f*lambda. Equivalently, the point of constant phase moves at omega/k.

geometric optics
OPL=n d\text{OPL} = n\,d

Optical Path Length

Inside glass light crawls slower, so a 1 mm slab packs in more wave crests than 1 mm of vacuum. Multiplying the real thickness d by the refractive index n converts physical distance into 'how much phase the wave accumulated' — that is what interference cares about.

acoustics
L=10 log⁡10 ⁣(II0)L = 10\,\log_{10}\!\left(\frac{I}{I_0}\right)

Sound Intensity Level (Decibels)

Ears respond to ratios, not differences. The decibel measures how many factors of ten your sound sits above the faintest audible intensity I0 = 1e-12 W/m^2. Ten times the power is +10 dB; a hundred times is +20 dB.

acoustics
Z=ρ cZ = \rho\,c

Acoustic Impedance

Impedance measures how hard a medium pushes back when sound tries to move it: dense, stiff media (high rho, high c) resist more. When a wave meets a different impedance, part of it reflects — the bigger the mismatch, the more bounces back.

dispersion
n(λ)=A+Bλ2n(\lambda) = A + \frac{B}{\lambda^2}

Optical Dispersion

A material's electrons resonate at ultraviolet frequencies. Visible light pushes those electrons, and the closer its frequency is to resonance (toward blue), the more strongly the medium responds — raising the index. Shorter wavelength means larger n, so blue bends more.

interference
rm=m λ Rr_m = \sqrt{m\,\lambda\,R}

Newton's Rings

The air gap under a curved lens thickens with distance from the contact point. Wherever the round-trip through the gap equals a whole number of wavelengths you get a dark ring (the half-wave reflection flip makes the center dark). Because the gap grows quadratically with radius, the rings crowd together outward.

scattering
I∝1λ4I \propto \frac{1}{\lambda^4}

Why the Sky Is Blue (Rayleigh)

A passing light wave jiggles the electrons in a tiny air molecule, which then re-radiate in all directions. Higher-frequency (blue) light jiggles them harder, and the re-radiated power scales as frequency to the fourth — so blue scatters about 5-10 times more than red, painting the whole sky blue.

geometric optics
θ=4 arcsin⁡ ⁣(sin⁡θin)−2θi\theta = 4\,\arcsin\!\left(\frac{\sin\theta_i}{n}\right) - 2\theta_i

How Rainbows Form

Each ray that enters a spherical drop bends in, bounces off the far wall, and bends out. Rays piling up at the angle of maximum deviation make a bright cone. Because the drop refracts red and blue by slightly different amounts, the cone's edge splits into colors — and you see an arc.

geometric optics
n(y) sin⁡θ(y)=constn(y)\,\sin\theta(y) = \text{const}

Why Mirages Appear on Hot Roads

Refractive index drops as the air gets hotter near the ground. A nearly horizontal ray heading down keeps hitting ever-less-dense air, bending a little more each time until it turns and heads back up — total internal reflection spread out smoothly. Your brain projects it straight back, placing sky on the ground.

reflection and refraction
Rs=∣n1cos⁡θi−n2cos⁡θtn1cos⁡θi+n2cos⁡θt∣2R_s = \left|\frac{n_1\cos\theta_i - n_2\cos\theta_t}{n_1\cos\theta_i + n_2\cos\theta_t}\right|^2

Fresnel Equations

At a boundary the wave's electric field must stay continuous. Solving that boundary condition for the two polarizations (s = perpendicular, p = parallel) gives how much reflects. The p-component vanishes entirely at Brewster's angle; near grazing both shoot up to total reflection.

variational optics
δ∫ABn(s) ds=0\delta \int_A^B n(s)\,ds = 0

Fermat's Principle of Least Time

Among all conceivable routes from A to B, light takes the one where small wiggles do not change the travel time to first order. Neighboring paths then arrive nearly in phase and reinforce; all others interfere away. Snell's and the reflection law are just this condition written out.