Curved surface correction for angle beam on pipe

Flat-plate trigonometry fails on pipe scanned in the circumferential direction. The beam leaves the outside surface at θ to the local radial normal and then travels in a straight line, but the normal rotates continuously around the circumference, so the angle between the beam and the wall keeps changing. The wall curves away from the beam and every depth is reached later, and further round, than a flat plate would suggest.

One geometric quantity makes the problem exact. A straight ray keeps a constant perpendicular distance from the pipe axis, p = R sin θ, so at any radius r the local angle to the radial direction obeys sin φ = R sin θ / r. From that follows the sound path SP = R cos θ − √(r² − R² sin²θ), and the angle subtended at the centre between the entry point and the target, α = asin(R sinθ / r) − θ. Multiplying α by R gives the true surface distance measured around the outside diameter.

The same relation sets a hard coverage limit. Since the ray never gets closer to the axis than p = R sin θ, it cannot reach the bore at all unless R sin θ ≤ r_i, that is θ ≤ asin(ID/OD). On a heavy-wall pipe a 70° or even a 60° probe simply passes over the root and out the far side without ever meeting the inside surface — the root is only covered by dropping the angle or by working from the second leg with a lower angle probe.

Two practical points follow. The wedge must be contoured to the pipe or the entry angle is not what is marked, and the surface distance you chalk on the pipe is an arc, not a chord — the gap between the two grows with both diameter and central angle, from a few tenths of a millimetre on small pipe to several millimetres on large-diameter work, and the difference between either and the flat-plate answer can be tens of millimetres.

R = OD/2,   r = R − depth,   p = R sin θ
sound path  SP = R cos θ − √(r² − p²)
local angle at target  φ = asin(p / r)
central angle  α = φ − θ
arc surface distance = R × α (α in radians)
chord surface distance = 2 R sin(α/2)
maximum angle reaching the bore: θ_max = asin(ID / OD)