The math behind NDT Level II: algebra, trig and units

Where the math lives

For UT, the calculations cluster in four places: wavelength, Snell's law, skip distance and decibels. The UT formulas guide states each formula and works one example, so use it as your reference and use this page as your practice ground. MT and PT lean far less on calculation, with magnetizing current the main place a number appears.

Algebra: rearrange before you plug in

Most formulas appear in one shape and get asked in another. Wavelength is velocity divided by frequency, and a question may give you wavelength and velocity and ask for frequency. Rearrange first, then calculate. Three moves cover nearly every case:

• To get a quantity out of a denominator, multiply both sides by it.
• To isolate a variable, undo operations in reverse order.
• To check, plug your answer back into the original formula.

Trigonometry: sine, cosine and tangent

Angle beam work is right triangles. The beam is the hypotenuse, the thickness is one leg and the surface distance is the other. Sine, cosine and tangent connect them. Draw the triangle every time, label the angle and decide which side you know. In UT, angles are measured from the normal, which is the line square to the surface. A drawing catches the common slip of using the wrong reference.

Know your calculator mode. Degrees and radians give wildly different answers for the same keystrokes, so check the mode before you start.

Units: write them down and carry them through

Most unit errors come from one habit: not writing the unit. Keep these pairs in mind:

• Velocity in mm/µs. One mm/µs equals 1,000 m/s.
• Frequency in MHz. Divide by one million only when you move to hertz.
• Thickness and distances in the same unit as the answer you want.

If you keep velocity in mm/µs and frequency in MHz, wavelength comes out in mm with no conversion. Mixing in inches halfway through is where points are lost.

Four original drills

These numbers are invented for practice. Work each one on paper first.

Drill 1: rearrange and units. A shear wave travels at 3.20 mm/µs in a test part. You want a wavelength of 1.5 mm. What frequency do you need?

Rearrange wavelength = velocity / frequency to frequency = velocity / wavelength: 3.20 / 1.5 = 2.13 MHz. Check: 3.20 / 2.13 = 1.50 mm. If velocity were given as 3,200 m/s, you would convert to mm/µs first by dividing by 1,000.

Drill 2: trigonometry and skip. A 70° probe is used on a 12 mm plate. What is the half skip distance, and what is the one-way sound path to the back wall?

Half skip is thickness × tan 70° = 12 × 2.747 = 33.0 mm. Sound path to the back wall is thickness / cos 70° = 12 / 0.342 = 35.1 mm. Check with the right triangle: 12 is the adjacent side, 33.0 is the opposite side and 35.1 is the hypotenuse. Does 12² + 33.0² come to about 35.1²? Yes, 144 + 1,089 = 1,233, and 35.1² is 1,232.

Drill 3: Snell's law. A wedge has a longitudinal velocity of 2.70 mm/µs. The part has a shear velocity of 3.20 mm/µs. What wedge angle gives a 45° shear wave?

sin(incident) = (2.70 / 3.20) × sin 45° = 0.84375 × 0.7071 = 0.5966. The wedge angle is the inverse sine of 0.5966, which is 36.6°. Confirm it on the Snell's law calculator.

Drill 4: decibels. An echo reads 25% of screen height. You want it at 80%. How much gain do you add?

Ratio = 80 / 25 = 3.2. dB = 20 × log₁₀(3.2) = 20 × 0.505 = 10.1 dB. Check with the decibel calculator. Remember the rule of thumb: 6 dB roughly doubles amplitude, and 3.2 sits between a doubling and a tenfold change, so about 10 dB is sensible.

Telling a math miss from a concept miss

After every practice set, ask one question about each miss: if I had the right formula in front of me, would I have gotten it? If yes, it is a math miss, and you need drills. If no, it is a concept miss, and you need to explain the idea in your own words. Tag each miss and count them weekly. Many people discover their misses are mostly math, which is the quickest thing to fix.

The free 10-question check does this sorting for you. It shows exactly which areas to study first, so you know whether your time belongs in math drills or concept reading. Take the free UT check before you plan your week. Then slot the drills into a calendar with the 30, 60, 90 day UT plan, or into short sessions with the full-time study guide.

Calculator habits for exam day

The exam software has a calculator built in, and our exam day guide describes its modes and how to open it. Practice on the scientific layout in the days before the exam so the keystrokes are automatic. Work each problem once, write down the result and move on. Redoing a calculation costs time you might need elsewhere.

FAQ

Do I need calculus for Level II?

No. The UT calculations in our guides use algebra, trigonometry and logarithms for decibels. If you can rearrange a formula, read a sine or tangent and use a log key, you have the tools.

Which formulas matter most?

Wavelength, Snell's law, skip distance and the decibel relationship cover most calculation questions in UT. Learn what each means in a sketch, because the setup is usually harder than the arithmetic. The formulas guide has one worked example for each.

Can I use my own calculator?

Pearson VUE says a calculator is built into the exam software and that you should not bring your own. Our exam day guide covers how to open it and what it offers. Practice on a scientific layout so exam day feels familiar.

How do I stop making unit errors?

Write the unit beside every number, every time. Keep velocity in mm/µs and frequency in MHz so wavelength comes out in mm, and convert only at the start. A final sanity check on size, such as whether a depth exceeds the plate thickness, catches most slips.