Wheel & Axle Refraction

About

Wave refraction — the bending of a wavefront as it crosses the boundary between two media at different speeds — can be hard to visualise from Snell's law alone. This simulation uses a concrete mechanical analogy: two wheels mounted on a rigid axle. On a uniform surface both wheels roll at the same speed and the axle travels in a straight line. When the axle crosses a surface boundary at an angle, the leading wheel enters the slower surface first, slowing down while the trailing wheel is still moving fast. The speed difference creates a torque that turns the axle toward the slower side — exactly as a wavefront bends toward the normal when it enters a denser medium.

The underlying mathematics is identical to Snell's law for light: v₁ / v₂ = sin θ₁ / sin θ₂, where v₁ and v₂ are the wave speeds in the two media, and θ₁ and θ₂ are the angles of incidence and refraction measured from the normal to the boundary. The simulation displays both the measured exit angle from the running simulation and the theoretical Snell's-law prediction so students can verify the agreement.

Hitting the boundary perpendicularly (0° from normal) causes no bending because both wheels enter the slow surface simultaneously. Swapping which surface is on top reverses the speed gradient, bending the path away from the normal rather than toward it — the analog of a wave speeding up as it crosses into a less dense medium.

Learning Goals

• Explain why a wave bends at a boundary using the wheel-and-axle analogy: the side that slows down first turns the whole front toward that side.
• Predict the direction of bending from the speed ratio: fast → slow bends toward the normal; slow → fast bends away from the normal.
• Verify numerically that the measured exit angle matches the Snell's-law analog: v₁ / v₂ = sin θ₁ / sin θ₂.
• Explain why a wave aimed perpendicularly at a boundary (0° from normal) does not bend, even when the speed changes.
• Connect the mechanical analogy to real-world wave refraction: ocean waves bending toward shore, light slowing in glass, seismic waves curving through the Earth.

How to Use

• Approach angle slider — drag the slider from 0° (straight down, no bending) to 65° (steep approach, strong bending). Watch the θ₁ and θ₂ arcs update on the canvas.
• Readout tiles — compare the Measured exit angle θ₂ (from the live simulation) with the Snell's prediction. They should agree once the axle has fully crossed the boundary.
• Swap surfaces (⇅) — puts the slow surface on top and the fast surface on the bottom. The path now bends away from the normal, just as a wave speeds up crossing into a less-dense medium.
• Watch the tracks — darker tire marks on the slow surface show how the left and right wheels trace different arcs through the transition zone. The tracks become parallel again once both wheels are on the same surface.