Snell's law in UT: worked examples and critical angles

The one equation

Snell's law for sound says the ratio of the sine of the angle to the velocity is the same on both sides of the boundary:

sin θ1 / c1 = sin θ2 / c2

Use these labels throughout:

• θ1 is the incident angle in the wedge, measured from the normal (a line square to the surface).
• θ2 is the refracted angle in the part, also from the normal.
• c1 is the sound velocity in the wedge. c2 is the velocity of the wave you are tracking in the part, longitudinal or shear.

Two velocities in the part means two refracted waves. The faster longitudinal wave bends further from the normal than the shear wave does. The shear versus longitudinal guide explains why the shear wave is the one used for angle beam work.

The practice values below are: wedge longitudinal velocity 2.70 mm/µs, steel longitudinal velocity 5.90 mm/µs and steel shear velocity 3.20 mm/µs.

Example 1: refracted shear angle from a wedge angle

A wedge cut at 50° sits on the steel. What is the refracted shear angle?

1. sin θ2 = (c2 / c1) × sin θ1 = (3.20 / 2.70) × sin 50°
2. = 1.1852 × 0.7660 = 0.9079
3. θ2 = inverse sine of 0.9079 = 65.2°

The sine came out below 1, so a shear wave exists. Always check that before you take the inverse sine.

Example 2: the wedge angle for a 60° probe

Now work backward. What wedge angle gives a 60° shear wave in the same steel?

1. sin θ1 = (c1 / c2) × sin θ2 = (2.70 / 3.20) × sin 60°
2. = 0.84375 × 0.8660 = 0.7307
3. θ1 = 46.9°

Same equation, rearranged. Put the velocity of the side you are solving for on top.

Example 3: both critical angles

Set the refracted angle to 90°, where the sine is 1, and solve for the incident angle.

• First critical angle (longitudinal wave grazes the surface): sin θ = 2.70 / 5.90 = 0.4576, so θ = 27.2°.
• Second critical angle (shear wave grazes the surface): sin θ = 2.70 / 3.20 = 0.84375, so θ = 57.5°.

Between 27.2° and 57.5° only the shear wave enters the steel. The 46.9° wedge from Example 2 sits inside that window, which is exactly why it works. The critical angles guide has the three-zone picture.

Example 4: below the first critical angle

A 20° wedge on the same steel. Both waves enter, so work both.

• Longitudinal: sin θ2 = (5.90 / 2.70) × sin 20° = 2.185 × 0.3420 = 0.7474, so θ2 = 48.4°.
• Shear: sin θ2 = (3.20 / 2.70) × 0.3420 = 1.1852 × 0.3420 = 0.4054, so θ2 = 23.9°.

Two beams at two different angles is why a probe in this range gives confusing screens. The angle beam wedge is cut to avoid it.

Example 5: past the second critical angle

A 70° wedge on the same steel.

sin θ2 = (3.20 / 2.70) × sin 70° = 1.1852 × 0.9397 = 1.1137

A sine cannot exceed 1, so there is no refracted shear wave. 70° is past the second critical angle of 57.5°. If a question hands you a wedge angle and the arithmetic returns a number over 1, that is the answer: no shear wave enters.

Where people lose points

• Swapping the velocities. The velocity of the target wave in the part goes on the same side as the angle you want.
• Measuring from the surface. Snell's law uses angles from the normal. If a source quotes an angle from the surface, subtract it from 90° first.
• Using the wrong wave in the part. The longitudinal velocity gives the longitudinal refracted angle and the shear velocity gives the shear angle. Do not mix them in one line.
• Taking the inverse sine of more than 1. That signals the critical angle has been passed. Stop and say no wave enters.
• Calling the probe label the wedge angle. The number printed on an angle beam probe is the refracted shear angle in the test material, not the angle inside the wedge.
• Reusing the numbers on a different material. Aluminum, stainless steel and other alloys have different velocities, so the critical angles move.

Practice it, then check it

Work each example by hand first. Then confirm your answers in the Snell's law calculator, and see the equation placed next to the others in the UT formulas guide. The refracted angle feeds straight into skip distance, so the two skills pay off together. If the trig itself is rusty, start with the math refresher.

Wave modes and refraction sit under the techniques and evaluation headings of the UT General outline for exams before 5 February 2027. What changes after that date shows how the list is regrouped.

See where you stand

Take the free 10-question UT check. It needs no signup, explains every answer and shows which topics to drill first. The course adds timed practice exams, a lesson for each published topic and a study plan built from your exam date, and Echo, the study assistant inside it, can talk through a refraction step you got wrong. See the plans.

FAQ

What is Snell's law in ultrasonic testing?

It says sin θ1 / c1 = sin θ2 / c2 across a boundary, where θ is the angle from the normal and c is the sound velocity of the wave on that side. It gives the refracted angle for a wedge angle and the wedge angle for a wanted refracted angle.

How do you find the first and second critical angles?

Set the refracted angle to 90° and solve for the incident angle. The first critical angle uses the longitudinal velocity in the part, and the second uses the shear velocity. Both need the part to be faster than the wedge for that wave.

Why does the probe label differ from the wedge angle?

The label gives the refracted shear angle in the test material. The wedge is cut at a smaller incident angle that Snell's law converts into the labeled one.

Sources

Original arithmetic and examples, written for this article. The velocities are practice values, not material data. Method background and exam topic headings were read on 26 October 2026.

• Evident, ultrasonic transducer technical notes: Snell's law for sound and critical angles.
• Evident, ultrasonic flaw detection tutorial: angle beam background.
• ASNT NDT Level II Examinations, ASNT Certification Services: UT General topic headings.