Schlieren Knife-Edge Bench

What is modelled

Air density falls as it warms, and its refractive index n follows the density. Light crossing a warm plume is bent toward the denser, colder side by an angle proportional to the gradient of n. A mirror turns that angle into a tiny shift of the source image in its focal plane, and a knife edge that cuts part of the image turns the shift into more or less light reaching the camera.

Equations

• Gladstone-Dale n − 1 = K ρ with K = 2.26 × 10^-4 m³/kg and ρ = p / (R T).
• Deflection ε = (L / n) ∂n/∂y, focal shift Δa = f ε.
• Brightness change ΔI / I = Δa / (a (1 − c)) for a source image of height a with a fraction c cut.

Checks run in the page

• Gladstone-Dale against the Lorentz-Lorenz relation.
• A Runge-Kutta ray through a thin slice against the deflection formula.
• Rays counted past the knife edge against the transmitted fraction.
• Finite differences on a grid against the analytic peak contrast.

Honest simplifications

A Gaussian temperature plume that is uniform along the light path, an ideal gas at constant pressure, a uniform source image with a sharp knife edge, no diffraction at the edge, no lens aberrations and no camera noise. The rising plume is a cartoon of buoyant spreading. Real schlieren images also show the knife orientation, which this page keeps horizontal.

• Press Candle flame and see the strong dark and bright sides of the plume.
• Press Weak convection and then raise the knife cut to recover the contrast.
• Press Insensitive bench: a short mirror and a tall source image hide the same plume.