Snell's law refraction angles

When a compression wave crosses from a wedge into the test material at an angle, it splits. Part of it refracts as a compression wave and part mode-converts to a shear wave, each obeying sin θ₁ / c₁ = sin θ₂ / c₂. Because the shear velocity in steel is much lower than the compression velocity, the shear wave always refracts at a smaller angle than the compression wave from the same incident angle.

This is the whole basis of angle-beam weld inspection. The wedge angle of a 45°, 60° or 70° shear probe is chosen so that the incident angle lies between the first and second critical angles (about 27.6° and 57.4° for perspex on steel). Above the first critical angle the compression wave has been refracted out of the material completely and only the shear wave remains, giving a single, unambiguous beam. Below it you have two beams in the part at once and every indication has two possible depths.

A refracted angle only exists while the calculated sine is 1 or less. When c₂/c₁ · sin θ₁ > 1 that mode has gone past its critical angle and is no longer propagating into the bulk material — the calculator returns no value for it, which is the correct physical answer, not an error.

Wedge velocity matters. Perspex/acrylic is about 2730 m/s, but Rexolite is nearer 2330 m/s and cross-linked polystyrene wedges vary between manufacturers. Wedges also warm up in service and the velocity drops, which is why the actual refracted angle should be verified on a calibration block at working temperature rather than trusted from the label. Enter the wedge temperature below and the calculator de-rates the wedge velocity (about 0.09 % per °C for acrylic) so you can see how far the beam has moved before you verify it.

sin θ₁ / c₁ = sin θ₂ / c₂
c_wedge(T) = c_wedge · [1 − k · (T_wedge − 20)]
sin θ_L = (c_L / c_wedge) · sin θ₁
sin θ_S = (c_S / c_wedge) · sin θ₁
θ = asin(...) exists only while sin θ ≤ 1