ToFD beam spread and wall coverage

The crystal (element) diameter is one of the most consequential choices in a ToFD setup, and it is easy to overlook because it never appears in the depth equation. It acts through beam divergence: the half-angle of the beam is sin γ = k·λ/D, so a small crystal spreads the beam wide and a large one keeps it narrow. ToFD relies on that spread — the technique needs the two beams to insonify as much of the wall as possible from a single pass, which is why ToFD probes use deliberately small, heavily damped crystals (typically 3–6 mm) where a pulse-echo probe of the same frequency might use 10 mm or more.

With the beams aimed at angle θ and crossed on the weld centreline, each edge of the diverging beam meets the centreline at a different depth. The upper beam edge (θ + γ) reaches the centreline at depth S/tan(θ + γ) and the lower edge (θ − γ) at S/tan(θ − γ), with S half the probe centre separation. Everything between those two depths is insonified; everything outside it relies on the weaker skirts of the beam and may not produce usable diffracted signals.

The coverage band shrinks dramatically as the crystal grows. At 5 MHz and 60° on a 25 mm wall, a 6 mm crystal insonifies roughly 13–21 mm of depth — under a third of the wall — while a 3 mm crystal at the same settings covers from 10 mm to the backwall. When the band cannot span the wall, the answer is a smaller crystal, a lower frequency, or splitting the inspection into depth zones with a setup per zone.

The −6 dB edge (k = 0.51) is the conventional working definition of the beam for this check; the −20 dB edge (k = 0.87) is more generous and often defensible for ToFD because diffracted signals remain usable well below the beam axis amplitude.

λ = c / f
sin γ = k · λ / D      (k = 0.51 at −6 dB, 0.87 at −20 dB)
PCS (derived) = (4/3) × T × tan θ,   S = PCS / 2
insonified band on the centreline:  d = S/tan(θ+γ)  to  S/tan(θ−γ)
coverage = (band ∩ wall) / T × 100 %