NDT math drill: trig, logs, ratios and units

How the drill works

Pick a drill type or leave it on all types. A mixed set covers all 6 types before it repeats any, and a single-type set gives you 10 problems of that type. Type a number, check it and read the steps. The key accepts a small rounding difference, so 36.6 and 36.58 both count, and a trailing unit is fine.

You score one point per problem. At the end you see your score by type, weakest first, your longest streak and a link to the calculator or guide that fixes your weakest type. Hit New set for fresh numbers, and quote the set number if you report an error.

What each type drills

• Wavelength and frequency. λ = c ÷ f, and the same relation turned around to find a frequency. With velocity in mm/µs and frequency in MHz, the answer comes out in mm.
• Snell's law and wedge angles. sin θ(incident) ÷ c(wedge) = sin θ(refracted) ÷ c(part). You find a refracted shear angle or the wedge angle that gives a target angle.
• Near field length. N = D² ÷ 4λ for a round element. The key also accepts the fuller form that carries a (1 − (λ ÷ D)²) factor.
• Decibels and amplitude ratios. dB = 20 log₁₀ of the amplitude ratio, and ratio = 10^(dB ÷ 20). You find a gain, a ratio or a new screen height.
• Skip distance, sound path and depth. Leg = T ÷ cos θ, skip = 2T × tan θ, surface distance = sound path × sin θ, and depth is path × cos θ on the first leg or 2T − path × cos θ on the second.
• Attenuation loss. Loss in dB = coefficient in dB/mm × path in mm. A back wall echo travels down and back, so the path is twice the thickness. Beam spread and reflection losses are separate.

These are public engineering formulas in our own words, as in Evident's published UT material. Where a source writes one differently, the UT formulas guide shows both and says which convention to watch.

Worked examples

Example 1: near field length

A 13 mm diameter, 5 MHz round element sends a longitudinal wave into a material at 5.9 mm/µs. Step 1: wavelength. λ = 5.9 ÷ 5 = 1.18 mm. Step 2: near field. N = 13² ÷ (4 × 1.18) = 169 ÷ 4.72 = 35.8 mm. Step 3: sanity check. The element is 11 wavelengths across, so the fuller form, 35.8 × [1 − (1.18 ÷ 13)²] = 35.5 mm, lands within 1%. Either answer passes the drill. Check any element in the UT beam calculator.

Example 2: which leg and how deep

A 60° shear wave probe is on a flat plate 30 mm thick, and an indication peaks at a sound path of 75 mm. Step 1: one leg. 30 ÷ cos 60° = 60.0 mm. Step 2: compare. 75 mm is longer than one leg, so the beam has bounced off the back wall and the reflector is on the second leg. Step 3: depth. 2T − path × cos θ = 60 − 75 × 0.5 = 22.5 mm. Step 4: surface distance. 75 × sin 60° = 65.0 mm from the beam index point. The first-leg formula would have said 37.5 mm, deeper than the plate, which is the tell. The skip distance calculator runs the same geometry.

Calculator tips for exam day

• Check degrees mode before the first sine, cosine or tangent. Radians turn a clean problem into nonsense.
• Use the inverse sine key (sin⁻¹) as the last step of a Snell's law problem. Solving for the sine is only the middle of the problem.
• Keep full precision on the screen and round once at the end. Rounding λ early shifts a near field answer.
• Write the unit beside every number. Velocity in mm/µs and frequency in MHz give wavelength in mm.
• Finish with a size check: depth cannot exceed the plate thickness, and 6 dB is close to a doubling.

The exam software has its own calculator, so practice on a scientific layout. The exam day guide covers how to open it.

How the problems are made

Each problem is generated from one formula with numbers drawn from short lists of made-up probes, materials and plate thicknesses. None is an exam item or a recalled question, and no text or values come from ASME BPVC Section V or any other standard. The key is calculated by the same functions that power the calculators linked above, and the same numbers always give the same set, so a set number is enough to report a problem.

NDT math FAQ