Snell's Law Refracted Angle Calculator
Pick a wedge, pick a material, and this tool tells you whether the shear wave inside the test piece actually arrives at the 45°, 60°, or 70° you wrote on the procedure. Snell's Law — sin(θ₁)/v₁ = sin(θ₂)/v₂ — governs the refraction at the wedge/steel interface, and a 2-degree drift on a worn wedge is enough to throw a beam path calculation off by 8% at half-skip on a 25 mm wall.
How it works
Two velocities are at play: the longitudinal velocity in the plastic wedge (typically 2,330–2,730 m/s for Rexolite or acrylic) and the shear velocity in the test piece (3,240 m/s for steel, 3,130 m/s for aluminum). The probe element fires a longitudinal wave; the wedge angle is cut so that when the wave hits the wedge/steel interface, refraction produces the desired shear angle. The first critical angle is where θ₂ would equal 90° — beyond it, no longitudinal wave enters the steel. The second critical angle is where the shear wave would itself refract to 90°.
Formula
sin(θ₁) / v₁ = sin(θ₂) / v₂
sin(θ₁) / v₁ = sin(θ₂) / v₂Worked example
Standard 60-degree shear wedge for carbon steel: Rexolite wedge (v₁ = 2,330 m/s longitudinal), test piece is steel with shear velocity 3,240 m/s. Cutting the wedge at 36.2° incidence gives sin(36.2°)/2330 = sin(θ₂)/3240 → θ₂ = arcsin(0.821) = 60.0°. The first critical angle (where the longitudinal wave grazes the surface) is 27.4°; the second critical angle is 56.5° for the steel longitudinal mode.
| Variable | Value |
|---|---|
| input: theta1 | 36.2 |
| input: v1 | 2330 |
| input: v2 | 3240 |
| output: theta2 | 60.0 |
| output: criticalAngle | 46.0 |
When to use this tool
Use when validating a probe wedge against a material whose velocity differs from steel (aluminum, brass, austenitic stainless), specifying a custom angle for tight geometry, or troubleshooting a procedure where the actual beam angle does not match what the procedure says.
Limitations
Where this calculator stops being accurate:
- Assumes the wedge contact face and test surface are flat and parallel. Curved surfaces (small pipe ODs) shift the effective angle.
- Worn wedges drift — a 0.5 mm wear on a 60° wedge changes refracted angle by ~1°.
- Mode-converted longitudinal energy is ignored. Below the first critical angle both modes exist; the tool only reports the shear refraction.
- Velocity is temperature-dependent — verify on a heated calibration block above 80 °C.
- For austenitic welds the shear velocity varies with grain orientation and Snell's Law gives only an approximation.
Frequently Asked Questions
Why do most UT shear-wave probes come in 45°, 60°, and 70°?
These angles bracket the working range for steel. Above the first critical angle (~27.4° for a Rexolite-to-steel interface) only the shear wave enters the test piece, eliminating ambiguous longitudinal echoes. Below the second critical angle (~56.5°) only one shear mode exists. 45/60/70 split that window into reasonable steps: 45° for thick sections and back-wall corner reflectors, 60° for general weld scanning, 70° for thin-section lack-of-fusion and toe-of-weld inspection.
How does an austenitic stainless weld change the calculation?
Type 308/316 weld metal solidifies in columnar grains that align roughly with the weld axis. Shear-wave velocity becomes anisotropic — varying by ±15% with direction — so applying a single isotropic velocity in Snell's Law produces only an approximate refracted angle. ASME V Article 4 mandates that procedures for austenitic welds use a TOFD or phased-array technique with ray-traced beam paths rather than rely on a fixed wedge angle.
What is the first critical angle and why does it matter?
The first critical angle is the wedge incidence angle at which the longitudinal wave in the test piece refracts to 90° — i.e. travels along the surface as a creeping wave. For a Rexolite-to-steel interface this is about 27.4°. Below it, both longitudinal and shear waves enter the steel and the A-scan shows two parallel indication trains. Above it, only the shear wave propagates inside the steel, which is why almost all angle-beam shear inspections specify an incidence angle larger than 30°.
Can I use the same wedge on aluminum and steel?
No. Aluminum has a shear velocity of 3,130 m/s versus 3,240 m/s in steel. A wedge cut for 60° in steel produces approximately 57.5° in aluminum — close, but enough to put the beam centre on the opposite weld toe at half-skip on a 25 mm plate. Procedure-grade work requires a wedge cut to the alloy under test, or a phased-array probe where the angle is steered electronically and Snell's Law is applied per element ray.
References & Standards Cited
- ASME BPVC Section V (2023), Article 4 Ultrasonic Examination of Welds
- ASTM E2700-19 Standard Practice for Contact Ultrasonic Testing of Welds Using Phased Arrays
- ISO 22825:2017 Non-destructive testing of welds — Ultrasonic testing of welds in austenitic steels and nickel-based alloys
- ASTM E494-20 Standard Practice for Measuring Ultrasonic Velocity in Materials
Founder of NDT Connect and Atlantis NDT. 15+ years in industrial inspection across oil & gas, petrochemical, and offshore. ASNT Level III certified across five methods. Drives platform standards for the NDT Connect marketplace.