Ultrasonic testing formulas for UT Level II, worked examples

The equations on this page come from Evident's published UT material, linked at each formula. Where two sources write the same idea differently, both are shown. Every worked example uses made-up numbers for an imaginary probe and test material, so do not reuse them as real material data. For what the exam covers, read the UT Level II General exam guide.

Units and conventions used on this page

• Velocity in mm/µs (1 mm/µs is 1 km/s, or 1,000 m/s). Frequency in MHz. With those two, wavelength and near field come out in mm with no extra conversion.
• Angles are measured from the normal (a line square to the surface), as on Evident's angle beam diagrams.
• Flat, parallel surfaces and an isotropic material. Curved parts, wedge delay and real beam width are ignored.
• Acoustic impedance in units of 10⁶ kg/(m²·s), the units in Evident's materials table. Density in g/cm³ times velocity in km/s gives the same units.
• Answers are rounded at the end, not between steps.

Wavelength

λ = c / f, where c is the sound velocity in the material and f is the frequency. Evident gives this as Equation 2 of its transducer technical notes (p. 2). The velocity must match the wave mode. A shear wave in the same part has a shorter wavelength than a longitudinal wave, because shear velocity is lower.

Worked example. A 4 MHz shear wave travels in a test alloy at 3.10 mm/µs.

λ = 3.10 / 4 = 0.775 mm

Halve the frequency and the wavelength doubles.

Near field length

Evident's beam characteristics tutorial (section 2.4) gives the near field of a circular element as N = D²f / 4c, or the same thing written as N = D² / 4λ, where D is the element diameter. The near field ends at the last on-axis maximum. Amplitude-based evaluation is harder inside it, because the sound pressure there swings between maxima and minima.

Two versions in Evident's own material. The technical notes (p. 10) say the near field table was calculated from a fuller expression, N = (D² / 4λ) × [1 − (λ / D)²], and that the short form was derived from it. When D is many wavelengths across, the correction term is tiny. If a practice source gives an answer slightly lower than yours, it may be using the fuller form, so check which one it uses before marking yourself wrong.

Worked example. A 12 mm diameter, 4 MHz element on a material with a longitudinal velocity of 6.00 mm/µs.

1. λ = 6.00 / 4 = 1.50 mm
2. Short form: N = 12² × 4 / (4 × 6.00) = 576 / 24 = 24.0 mm
3. Fuller form: N = (144 / 6.00) × [1 − (1.50 / 12)²] = 24.0 × 0.984 = 23.6 mm

Either way, a reflector at 10 mm depth sits inside the near field.

Beam spread

Evident's flaw detection tutorial PDF (section 2.5, p. 16) gives the −6 dB half-angle of an unfocused transducer as sin(α) = 0.514 c / (f D). The technical notes (Equation 11, p. 5) write the same relation as sin(α/2) and define α/2 as the half-angle between the −6 dB points, so both describe the angle from the beam axis to the −6 dB edge. Beam spread shrinks as frequency or element diameter goes up.

The constant depends on where you call the edge of the beam. TWI's FAQ on beam spread in the far zone uses a constant of 1.08 for the edge where energy is down to one tenth (−20 dB), and 1.22 for the extreme edge of the beam. A larger constant gives a wider angle for the same probe. Use the constant your question or procedure names, and never mix two conventions in one answer.

Worked example. The same 12 mm, 4 MHz probe on the 6.00 mm/µs material, using Evident's −6 dB constant.

sin(α) = 0.514 × 6.00 / (4 × 12) = 3.084 / 48 = 0.0643

α = 3.7° from the beam axis to the −6 dB edge

Check your own numbers with the UT beam calculator, which shows the constant and dB edge it used.

Snell's law and the critical angles

Snell's law links the sines of the angles to the velocities on each side of the boundary. Evident writes it as sin θi / ci = sin θrl / crl = sin θrs / crs, with θi the incident angle in the wedge, θrl and θrs the refracted longitudinal and shear angles, and c the matching velocities (technical notes, Equation 12, p. 6; wave front tutorial, section 2.5).

The two critical angles follow from setting a refracted angle to 90°:

• First critical angle: the incident angle that refracts the longitudinal wave to 90°. Beyond it, only a shear wave enters the part. sin θ₁ = c(wedge) / c(longitudinal in part).
• Second critical angle: the incident angle that refracts the shear wave to 90°, where Evident says a surface wave forms. sin θ₂ = c(wedge) / c(shear in part).

A critical angle only exists when the second material is faster than the wedge for that wave mode.

Worked example. A plastic wedge with a longitudinal velocity of 2.60 mm/µs on a test alloy with longitudinal 6.00 mm/µs and shear 3.10 mm/µs. What wedge angle gives a 60° shear wave, and where are the critical angles?

1. sin θi = (2.60 / 3.10) × sin 60° = 0.8387 × 0.8660 = 0.7264, so θi = 46.6°
2. First critical: sin θ₁ = 2.60 / 6.00 = 0.4333, so θ₁ = 25.7°
3. Second critical: sin θ₂ = 2.60 / 3.10 = 0.8387, so θ₂ = 57.0°

46.6° sits between the two critical angles, so only the shear wave enters the alloy. Check your own numbers with the Snell's law calculator.

Skip distance, sound path, surface distance and depth

Evident's angle beam inspection section (section 5.2, p. 36) and the technical notes (p. 7) give the flat-plate geometry, with T the thickness and θ the refracted angle:

• Leg (sound path to the back wall) = T / cos θ
• V path (full skip sound path) = 2T / cos θ
• Skip distance = 2T × tan θ (half skip is T × tan θ)
• Surface distance = sound path × sin θ
• Depth, first leg = sound path × cos θ
• Depth, second leg = 2T − (sound path × cos θ)

Skip distance is measured on the surface from the beam index point. Sound path here is the one-way path in the part, not the wedge.

Worked example. A 20 mm plate scanned with a 60° probe. An indication peaks at a sound path of 55 mm.

1. Leg = 20 / cos 60° = 40.0 mm, so V path = 80.0 mm
2. Skip distance = 2 × 20 × tan 60° = 69.3 mm (half skip 34.6 mm)
3. 55 mm is longer than one leg, so the reflector is on the second leg
4. Surface distance = 55 × sin 60° = 47.6 mm from the beam index point
5. Depth = 40 − (55 × cos 60°) = 40 − 27.5 = 12.5 mm

A common slip is to use the first-leg depth formula on a second-leg signal, which here would give 27.5 mm, deeper than the plate. If depth comes out larger than T, recheck the leg. The skip distance calculator works the same geometry for legs 1 and 2.

Decibel and amplitude ratio

Evident defines the decibel difference between two amplitudes as dB = 20 log₁₀ (amplitude ratio). Its documents put the ratio different ways round. The tutorial glossary (p. 72) uses 20 log (V2 / V1), while the technical notes (Equation 5, p. 3) use 20 log (A1 / A2). The size of the answer is the same; only the sign changes. Our decibel calculator reports a positive number when the second signal is higher than the first.

Going the other way, ratio = 10^(dB / 20). Each 6 dB is close to a doubling of amplitude, and 20 dB is a factor of 10.

Worked example. A reference echo is set to 80% full screen height. A second echo reads 32%, with the gain unchanged.

1. dB = 20 log₁₀ (32 / 80) = 20 log₁₀ (0.4) = −8.0 dB
2. To bring the second echo up to the reference height, add about 8.0 dB of gain.
3. Reverse check: a 9 dB difference is an amplitude ratio of 10^(9/20) = 2.82

These calculations assume the screen reading is linear and not clipped above 100%.

Acoustic impedance and reflection coefficient

Acoustic impedance is Z = ρc, density times velocity (technical notes, Equation 6, p. 4). When sound meets a boundary square to the beam, the impedance mismatch decides how much is reflected. Evident's tutorial (section 2.5, p. 17) states that similar impedances reflect little, identical ones reflect nothing in theory, and a very large mismatch such as steel to air reflects close to 100%.

Amplitude versus energy. The tutorial gives R = (Z₂ − Z₁) / (Z₂ + Z₁) and labels R a percentage of sound energy. The technical notes (Equations 7a and 7b, p. 4) square the impedance terms when they work out energy loss: reflected energy uses (Z₂ − Z₁)² / (Z₁ + Z₂)², and transmitted energy uses 4Z₁Z₂ / (Z₁ + Z₂)². Our reading is that the unsquared ratio is the sound pressure (amplitude) ratio and the squared form is the energy fraction. The two energy fractions add up to 1, which is a quick check. Read the question carefully for which one it wants.

Worked example. Sound in a solid with density 7.0 g/cm³ and velocity 5.0 mm/µs meets a liquid with density 1.2 g/cm³ and velocity 1.6 mm/µs.

1. Z₁ = 7.0 × 5.0 = 35.0; Z₂ = 1.2 × 1.6 = 1.92 (both × 10⁶ kg/(m²·s))
2. Amplitude ratio = (1.92 − 35.0) / (1.92 + 35.0) = −33.08 / 36.92 = −0.896. The minus sign means the echo is phase inverted, which Evident notes happens when Z₂ is lower than Z₁.
3. Reflected energy = 0.896² = 0.803, about 80%
4. Transmitted energy = 4 × 35.0 × 1.92 / 36.92² = 0.197, about 20%
5. Check: 0.803 + 0.197 = 1.000

How to practice these

• Draw the beam path for every angle beam question. A sketch catches the wrong-leg error quickly.
• Work each example by hand, then confirm it with the matching calculator.
• When a source gives a different answer, find the convention first: the beam edge constant, the near field form, the dB ratio direction, or amplitude versus energy.

ASNT's examinations page says no one source should be the only basis for exam preparation, and that applies to this page too. To see whether UT math or another topic is costing you the most, take the free 10-question check. A practice score does not predict your exam result. If your exam is after 5 February 2027, read the UT outline change guide as well.

Sources

Show the list

• ASNT NDT Level II Examinations, ASNT Certification Services. Retrieved 20 Oct 2026.
• ASNT 9712 Level II Examinations: Information and Scheduling, ASNT Certification Services. Retrieved 20 Oct 2026.
• Ultrasonic Flaw Detection Tutorial, 2.4 Beam Characteristics, Evident. Retrieved 20 Oct 2026.
• Ultrasonic Flaw Detection Tutorial, 2.5 Wave Front Dynamics, Evident. Retrieved 20 Oct 2026.
• Ultrasonic Flaw Detection Tutorial (PDF, 2021), Olympus, hosted by Evident: sections 2.4, 2.5 and 5.2, glossary p. 72. Retrieved 20 Oct 2026.
• Ultrasonic Transducers Technical Notes (PDF, 2019), Olympus, hosted by Evident: Equations 2, 5, 6, 7a, 7b, 8, 8a, 11 and 12, and pp. 7 and 10. Retrieved 20 Oct 2026.
• What factors influence the beam spread of an ultrasonic probe in the far zone?, TWI. Retrieved 20 Oct 2026.