Snell's law formula
Let α be the incident angle and θ the refracted angle, both measured from the surface normal. Use c1 and c2 for the selected wave-mode velocities in the incident and receiving media, in the same units.
- sin α / c1 = sin θ / c2.
- Refracted angle: θ = arcsin[(c2 / c1) × sin α].
- Incident angle for a target: α = arcsin[(c1 / c2) × sin θ].
These rearrangements use the ray model in Evident, Ultrasonic Flaw Detection Tutorial: 2.5 Wave Front Dynamics. A sine above 1 has no real angle; a sine of 1 gives a 90° grazing limit. The diagram shows the selected refracted mode only.
First and second critical angles
For longitudinal incidence from a wedge into an isotropic solid, substitute 90° for the refracted angle:
- First critical angle: αc1 = arcsin(cW / cL).
- Second critical angle: αc2 = arcsin(cW / cS).
Here cW is wedge longitudinal velocity; cL and cS are material longitudinal and shear velocities. Each critical angle exists only if its ratio is at most 1. Equality gives a 90° incident limit. Between the two limits the longitudinal bulk ray is unavailable while a refracted shear ray can propagate. At the second limit the shear ray is grazing; this tool does not model surface waves.
Worked example: acrylic wedge into steel 1020
Use the imported reference presets: cW = 2730 m/s, cL = 5890 m/s and cS = 3240 m/s. At α = 30°, the shear angle is arcsin[(3240 / 2730) × sin 30°] = 36.3991°.
The first critical angle is arcsin(2730 / 5890) = 27.6130°; the second is arcsin(2730 / 3240) = 57.4149°. For a target 60° shear ray, α = arcsin[(2730 / 3240) × sin 60°] = 46.8618°. Values are rounded for display.
Second example: acrylic wedge into aluminum
Use cW = 2730 m/s and aluminum with cL = 6320 m/s and cS = 3130 m/s. At α = 35°, the shear angle is arcsin[(3130 / 2730) × sin 35°] = 41.1184°. Because 35° is above the first critical angle of 25.5921° and below the second of 60.7158°, only the shear ray propagates. For a target 70° shear ray, α = arcsin[(2730 / 3130) × sin 70°] = 55.0452°. These numbers are invented for practice. The same rearrangements are on the UT and MT formula sheet.
Reference velocities and limits
The presets reuse the skip-distance calculator's selected published values from Olympus, Ultrasonic Transducers Technical Notes (2019), Table 2, printed p. 11. They cover steel 1020, aluminum and titanium Ti 150A, plus the acrylic longitudinal wedge value. Editing a velocity changes that medium to custom.
This calculation assumes homogeneous, isotropic media and a plane interface. It predicts ray direction, not transmission strength or whether a chosen mode is efficiently excited. Check actual material, temperature, wedge velocity and beam angle by calibration. Curved surfaces and anisotropy need a suitable model.
Continue your UT calculation
Use the skip-distance calculator with your calibrated refracted angle to calculate beam travel in a flat plate. Use the dB and amplitude calculator to compare echo heights.
Snell's law gives the direction of the refracted ray. For how much energy crosses the boundary, use the acoustic impedance calculator.