Platform Stats
Wave 26 at a Glance
Three simulations shipped in this wave, spanning three very different disciplines. Electromagnetism has been a long-outstanding gap in the physics category. Galaxy rotation curves revisit Vera Rubin’s landmark discovery from the economics side of cosmology, and IS-LM fills the macroeconomics category with a genuinely interactive policy simulator.
- Electromagnetic Induction — classical electromagnetism, ★★☆
- Galaxy Rotation Curves — astrophysics / cosmology, ★★☆
- IS-LM Model — macroeconomics, ★★☆
New Simulation: Electromagnetic Induction
🧲 Electromagnetic Induction — Faraday’s Law & Lenz’s Law
The most fundamental result in classical electromagnetism: a changing magnetic flux through a conducting loop induces an EMF that drives a current. This simulation lets users move a bar magnet toward and away from a coil, rotate a coil in a uniform field, or study an AC generator, with real-time galvanometer deflection, an EMF history graph, and animated magnetic field lines.
Physics
Faraday’s law in integral form:
Φ = B · A · cos(θ) = B · A · cos(ωt)
ε = N · B · A · ω · sin(ωt) [rotating coil / AC generator mode]
F_Lenz ∝ ε · B_magnet [opposing force via Lenz's law]
Flux from the bar magnet is modelled as
Φ ∝ B₀ / (1 + (d/80)²), a simple dipole-like falloff
with distance d (in canvas pixels). The galvanometer needle
smooth-lerps toward the target angular deflection at each animation
frame, and the EMF trace uses a 200-sample ring buffer for the
history graph.
New Simulation: Galaxy Rotation Curves
🌌 Galaxy Rotation Curves — Dark Matter Evidence
One of the strongest observational arguments for dark matter: the flat rotation curves of spiral galaxies. Newtonian dynamics predicts orbital velocity should fall as v ∝ 1/√r beyond the visible disk, just as planetary velocities decline in a solar system. But galaxy spectra show v ≈ const out to tens of kiloparsecs — an anomaly explained by a massive, dark matter halo surrounding each galaxy.
Physics
M_total(<r) = M_bulge(r) + M_disk(r) + M_NFW(r)
NFW halo density: ρ(r) = ρ₀ / [ (r/rₛ) · (1 + r/rₛ)² ]
NFW enclosed mass: M_NFW(<r) ∝ [ ln(1 + r/rₛ) − r/(r + rₛ) ]
Freeman disk: M_disk(<r) = Mₛ · [ 1 − (1 + r/Rᵈ)·exp(−r/Rᵈ) ]
The left canvas shows an animated face-on view of the galaxy with an NFW halo glow scaling with the dark matter fraction slider. The right panel plots all components (bulge, disk, dark matter) as thin dashed curves, plus the Newtonian total and the full observed rotation curve, against simulated observational data points with error bars.
New Simulation: IS-LM Model
📈 IS-LM Model — Keynesian Macroeconomics
The IS-LM framework, developed by John Hicks in 1937 to formalise Keynes’ General Theory, remains the standard undergraduate introduction to macroeconomic stabilisation policy. The IS curve traces goods-market equilibrium (investment = saving) and the LM curve traces money-market equilibrium (liquidity preference = money supply). Their intersection defines equilibrium national output Y* and interest rate r*.
Equations
where A = C₀ − c₁T + I₀ + G (autonomous spending)
LM curve: r = (k·Y − M/P) / h
where M/P = real money supply, k = income velocity, h = interest sensitivity
Equilibrium Y*: Y = (A + b·M/P/h) / [(1−c₁) + bk/h]
Fiscal multiplier: dY/dG = 1 / [(1−c₁) + bk/h]
Four shock buttons trigger pre-set policy changes (G↑, M↑, T↓, reset) while leaving a ghost of the old curves on screen for easy comparison. Multiplier and crowding-out values update live with every slider change.
Ukrainian Translations
All three Wave 26 simulations ship with complete Ukrainian translations at launch:
- uk/electromagnetic-induction — Електромагнітна Індукція
- uk/galaxy-rotation — Криві Обертання Галактик
- uk/is-lm — Модель IS-LM
Each Ukrainian page shares the identical simulation JavaScript with
its English counterpart (no code duplication) while translating all UI
labels, stat names, info panels, breadcrumbs, and footer text. The
hreflang alternates and canonical URLs are set correctly
on both language variants.
Engineering Notes
Galvanometer animation (electromagnetic-induction)
Galvanometer needles should never snap instantly to the target angle
— real instruments have inertia and damping. The sim does a
simple exponential lerp each frame:
angle += (target − angle) × 0.12, giving a natural
settling behaviour without needing to integrate a full second-order
ODE.
NFW mass integral (galaxy-rotation)
The NFW enclosed mass has an analytical form: M(<r) = M_vir · f(x)
/ f(c), where f(x) = ln(1+x) − x/(1+x) and x = r/r_s, c =
concentration parameter. This avoids numerical integration and
evaluates in O(1) per data point, keeping the 200-point rotation-curve
trace smooth at 60 fps.
IS-LM analytic equilibrium
Rather than iterating to find the IS-LM intersection, the equilibrium is solved analytically in one step by substituting the LM expression for r into the IS equation and solving for Y. This guarantees a unique, exact solution (the model is linear in Y) and means the canvas redraws instantly on every slider input with zero lag.
What’s Next — Wave 27 Preview
Planned topics for Wave 27 include:
- Quantum Tunnelling — 1D time-independent Schrödinger equation, transmission coefficient
- Fluid Vorticity — 2D vortex sheets, Kelvin–Helmholtz instability
- Game Theory — Nash equilibrium finder for 2×2 normal-form games
All Wave 27 simulations will follow the same EN + UK at-launch policy established in Wave 26.