13.3 Angle-Beam Techniques, Snell's Law, and Skip Distance

A 0° beam is blind to a tight, vertical weld fusion face. Official UT topic 1 therefore finishes Review of Ultrasonic Techniques with angle-beam work: a plastic wedge, Snell's law, the two critical angles, and the skip geometry that lets a Level II put a shear wave through a V-groove from the plate surface. ASTM E164 and ASME Section V, Article 4 are the industrial weld documents that assume this geometry. They do not publish a secret ASNT wedge angle. If the stem gives velocities or a thickness, compute from those numbers.

This section is still the current outline prior to 5 February 2027. Weld evaluation, DAC, and acceptance belong in later UT chapters. Here you only have to aim the beam and know where it hits.

Snell's law

When a wave crosses an interface at an angle, the ratio of the sine of the angle to the velocity is the same on both sides for each transmitted or reflected mode that exists:

sin θ₁ / v₁ = sin θ₂ / v₂

Angles are measured from the normal (from the perpendicular to the interface), not from the surface. v₁ is the velocity of the incident mode in medium 1. v₂ is the velocity of the refracted mode you are asking about in medium 2. Those two velocities are often different modes. The usual weld-probe story is:

• Medium 1: longitudinal wave in the acrylic (Plexiglas) wedge
• Medium 2: shear wave in the steel

You can also write a Snell line for a refracted longitudinal wave in the steel, using steel's L-wave velocity. Both refracted rays, plus a reflected ray back into the wedge, can exist at once if the angles allow it.

Rearranged for the unknown angle:

θ₂ = arcsin( v₂ sin θ₁ / v₁ )

θ₁ = arcsin( v₁ sin θ₂ / v₂ )

If v₂ sin θ₁ / v₁ is greater than 1, that mode does not refract — it has been totally reflected at the interface. That is the critical-angle story below.

Worked example: incident wedge angle for a 45° shear wave

Typical training velocities (label them typical; a stem table wins):

• Acrylic wedge, longitudinal: v₁ = 2730 m/s (0.107 in/µs)
• Steel, shear: v₂S = 3230 m/s (0.128 in/µs)
• Desired refracted shear: θ₂ = 45°

sin θ₁ / 2730 = sin 45° / 3230

sin θ₁ = 2730 × 0.70711 / 3230 = 1930.4 / 3230 = 0.5976

θ₁ = 36.7°

The machinist cuts the wedge at about 36.7°. The probe is stamped 45° because that is the refracted metal angle the welder and the procedure talk about. A candidate who answers "the wedge is 45° because the weld probe is 45°" has confused incident with refracted.

Same arithmetic for the other two shop angles, same velocities:

If the wedge were polystyrene or a different acrylic lot with v₁ = 2670 m/s, every θ₁ would shift. Compute from the velocities you are given.

A second worked pass in inch units, 60° shear, v₁ = 0.107 in/µs, v₂S = 0.128 in/µs:

sin θ₁ = 0.107 × sin 60° / 0.128 = 0.107 × 0.8660 / 0.128 = 0.724

θ₁ = 46.4°

The 0.6° difference from the SI line is only the rounded typical velocities. The exam will not grade you on a tenth of a degree if you used the stem's table consistently.

First and second critical angles

Raise the incident angle and watch the two refracted rays slide toward the surface.

First critical angle — the incident angle at which the refracted longitudinal ray reaches 90° (it skims the interface and then disappears from the metal as a bulk L-wave).

sin θ_c1 / v₁ = sin 90° / v₂L = 1 / v₂L

sin θ_c1 = v₁ / v₂L

Above θ_c1 there is no bulk longitudinal wave in the metal. Only shear (and a surface-related field) remains. That is the window weld probes live in.

Second critical angle — the incident angle at which the refracted shear ray reaches 90°.

sin θ_c2 = v₁ / v₂S

Above θ_c2 there is no bulk shear wave in the metal either. What remains along the surface is a Rayleigh (surface) wave, plus whatever the wedge still reflects internally.

Worked example: acrylic into steel

Use v₁ = 2730 m/s, v₂L = 5900 m/s, v₂S = 3230 m/s.

First critical: sin θ_c1 = 2730 / 5900 = 0.4627 → θ_c1 = 27.6°

Second critical: sin θ_c2 = 2730 / 3230 = 0.8452 → θ_c2 = 57.7°

Plot the three shop probes on that line:

• A wedge cut at 36.7° (45° metal shear) is above 27.6° and below 57.7° → shear only in the steel.
• 47.0° (60° metal) — still in the shear-only window.
• 52.6° (70° metal) — still just under the second critical.
• An incident angle of 20° is below first critical → both a refracted L-wave and a refracted S-wave in the steel. The A-scan becomes a two-velocity mess. Weld procedures avoid that on purpose.
• An incident angle of 62° is above second critical → no bulk shear. You are in surface-wave territory, not a 70° volumetric weld shot.

Water-to-steel immersion refraction uses the same formulas with v₁ ≈ 1480 m/s:

θ_c1 = arcsin(1480/5900) ≈ 14.5°

θ_c2 = arcsin(1480/3230) ≈ 27.3°

A 18° incident water angle is already past first critical and produces a shear wave in the steel. That is how immersion angle-beam is aimed — not by dropping a 45° contact wedge in the tank and hoping.

Incident wedge angle versus refracted metal angle

Say the pair out loud until it is boring:

• Incident (wedge) angle — the angle of the longitudinal ray inside the plastic, from the normal to the wedge–steel face. This is the angle the shop machines. It is not printed in large type on most probe housings.
• Refracted (metal) angle — the angle of the chosen mode in the steel, from the normal to the same face. For a plate, that normal is the plate normal, so a 45° shear wave travels at 45° to the normal and 45° to the surface. A 70° shear wave is only 20° from the plate surface. This is the 45 / 60 / 70 stamp.

Related names you will see on a screen or a calculator:

• Refracted angle, β or θ_r — metal angle.
• Incident angle, α or θ_i — wedge angle.
• Probe angle in a procedure almost always means the metal angle.

A 70° probe on aluminum is not a 70° probe on steel. Aluminum's shear velocity is different, so the same wedge produces a different metal angle. Either use a wedge matched to the material or compute the new θ₂ with Snell's law. High-temperature work has the same problem: steel velocities drop, and a "70°" probe is no longer 70°.

Why weld inspection uses shear at 45°, 60°, and 70°

Four reasons, in the order the general exam cares about:

1. First critical has already happened. A 45/60/70 shear probe is cut so that only shear exists in the steel. You are not interpreting a fast L-wave and a slow S-wave on the same A-scan.
2. Shorter wavelength at the same frequency. v_S is about 0.55 × v_L in steel, so λ_S is about 0.55 × λ_L. A 2.25 MHz shear wave is a tighter ruler than a 2.25 MHz longitudinal wave.
3. Geometry of fusion faces. A single-V weld has bevels typically in the 30°–37.5° from vertical family (procedure-specific). A 60° or 70° shear beam meets that face closer to normal than a 0° beam ever will, so a lack of fusion reflects back toward the probe instead of glancing away. A 45° beam is a better match to some thicker, narrower preps and to mid-volume slag. The procedure names the angle or the set of angles. Production weld books often carry two angles so a face that is specular to one beam is visible to the other.
4. Access from the plate surface. You cannot put a 0° probe on the bevel of a finished weld. Skip geometry (next heading) lets the beam bounce from the scanning face to the root, the opposite fusion face, and the cap.

Why not a 45° longitudinal weld beam as the default? You can build an L-wave angle probe (incident angle below first critical). You then live with a faster wave, a longer λ, a second shear ray that still exists, and a skip distance that does not match the shear calculator on the instrument. Some thick-section and austenitic techniques use refracted L-waves on purpose. They are special procedures, not the carbon-steel 45/60/70 default.

Why those three numbers and not 50° and 65°? They are the conventional set that blocks, calculators, and AWS/ASME training grew up with. A procedure may specify 55° or 65°. The physics does not require the integer trio. The exam will.

Thinner wall favors a larger metal angle (70°) so the half-skip is long enough to stand the probe off the weld cap. Thicker wall can use 45° or 60° so you are not walking a metre of skip to reach the root and so the beam does not glance along the plate. That is a selection heuristic, not a substitute for the technique sheet.

Skip, half-skip, and full-skip

Draw a plate of thickness T. A shear ray leaves the wedge at metal angle θ (from the normal). The ray hits the opposite surface a horizontal distance s from the incident point.

s = T tan θ

That distance is the half-skip (one leg, the first half-V). The beam then reflects (angle of incidence equals angle of reflection for the same mode on a free surface) and returns to the scanning surface a total surface distance

S = 2 T tan θ

from the incident point. That S is the full-skip, full-V, or skip distance in most classroom English. Some shops say "skip" for the half-leg. Read the stem. This guide uses:

The index point is the point on the wedge face where the beam centreline leaves the plastic — the mark you measure from, not the back of the housing.

Worked example: skip distance

Plate T = 25 mm, refracted shear 70°. tan 70° = 2.747.

Half-skip: 25 × 2.747 = 68.7 mm (about 69 mm).

Full-skip: 2 × 25 × 2.747 = 137.4 mm (about 137 mm).

Same plate, 45° (tan 45° = 1):

Half-skip = 25 mm. Full-skip = 50 mm.

Inch version. T = 1.000 in, 70°:

Half-skip = 1.000 × 2.747 = 2.75 in. Full-skip = 5.49 in.

T = 0.500 in, 60° (tan 60° = 1.732):

Half-skip = 0.866 in. Full-skip = 1.73 in.

A root indication on a single-V weld, scanned from the plate next to the cap, is often a half-skip problem: the beam hits the opposite-side root after one leg. A cap or near-side fusion indication may sit at a full-skip. The Level II measures the surface distance from the index to the probe mark when the echo peaks, then compares it with T tan θ and 2T tan θ (and with the weld width). Depth along the beam is v_S × Δt / 2; vertical depth in the plate is that sound-path times cos θ. Depth formulas are evaluation-chapter tools; the topic 1 item is usually the surface triangle.

Full-skip uses 2T, not 2θ. Candidates who write 2T tan(2θ) or T / tan θ are inventing a different triangle.

Curved pipe uses the same Snell law at the wedge but a different skip formula (OD, ID, and whether you are on the outside). If the stem is a flat plate, use T tan θ. If it is a pipe and gives a formula or a calculator output, use that. Do not force plate skip onto a 50 mm pipe and call it good.

Putting Snell and skip on the same setup

A 25 mm carbon-steel V-weld, procedure 70° shear, acrylic wedge, typical velocities.

1. Confirm the wedge is in the shear-only window: incident ≈ 52.6°, which sits between 27.6° and 57.7°.
2. Half-skip ≈ 69 mm. Stand the index about 69 mm from the centreline to put the beam on the opposite-side root (plus or minus the weld width and the index-to-exit offset the probe card gives).
3. Full-skip ≈ 137 mm. A same-side cap or near-face indication that peaks near that surface distance is on the second leg.
4. If the cap is so wide that 69 mm puts the wedge on top of weld reinforcement, the procedure may switch to 60° (half-skip 43 mm) or require grinding — it will not ask you to use a 20° incident L+S beam because the geometry is awkward.

Sound path for one half-skip leg is T / cos θ. At 70° and 25 mm that is 25 / 0.3420 = 73.1 mm. Time of flight is 2 × 73.1 mm / v_S. That clock is how the instrument draws a true-depth or sound-path number. If the stem asks for surface distance, still use T tan θ, not the sound path.

Realistic exam scenarios

A stem gives v₁ = 2730 m/s, v₂S = 3230 m/s, and asks for the incident angle that produces 45° shear. Only 36.7° matches sin θ₁/2730 = sin 45/3230. 45° is the metal angle. 27.6° is first critical. 57.7° is second critical.

A candidate aims a 20° acrylic wedge at a steel weld "to get more energy in." First critical is 27.6°. The part now carries both a refracted L-wave and a refracted S-wave. Peaks arrive at two velocities. That is not a standard 45/60/70 technique.

A 70° probe is moved from steel to aluminum without a new calculation. Aluminum shear is slower than steel shear in some tables and faster in others — either way it is not 3230 m/s. The metal angle is not 70° until you recompute.

A 25 mm plate, 70° shear, and a surface distance of about 137 mm to a peaked echo: that is a full-skip (2T tan 70°), not a half-skip. Calling it 25 mm deep because "the wall is 25 mm" ignores the V path.

A Level II reports skip distance as 25 × 70 = 1750 mm. Angle in degrees is not a length. tan θ is the factor.

An immersion manipulator sets a 12° water incident angle on steel and the operator expects a 70° shear wave. First critical in water–steel is about 14.5°. At 12° a longitudinal refracted ray still exists. Compute θ₂ from Snell's law; do not import the contact-wedge stamp into the tank.

What angle-beam items are really testing

If the stem gives two velocities and one angle, write sin θ₁ / v₁ = sin θ₂ / v₂ and decide which angle is the wedge and which is the metal. If it names a critical angle, ask which ray is at 90° — L at the first, S at the second. If it names 45/60/70, those are refracted shear metal angles in the window after the first critical. If it names skip, draw the triangle: half-skip = T tan θ, full-skip = 2T tan θ, measure from the index, and do not feed the metal angle into the sine formula when the question asked for surface distance. Those four moves are the angle-beam third of official UT topic 1.