Beam divergence angle

Beyond the near field the beam spreads as a cone. The half-angle is given by sin θ = k · λ / D, where D is the element diameter and k depends on which beam edge you are defining. The three values in normal use are k = 0.51 for the −6 dB (half-amplitude) edge, k = 0.87 for the −20 dB edge, and k = 1.22 for the first pressure null, which is the outer limit of the main lobe.

Which one to use depends on the job. The −6 dB edge defines the working beam for scan-plan coverage and index spacing — it is the width over which sensitivity is within 6 dB of the axis. The −20 dB edge is the one to use when you need to be confident a defect could not have been missed at the edge of a pass, and the first-null angle tells you where the main lobe stops and the side lobes begin.

Divergence is set by the ratio λ/D. A big, high-frequency element gives a tight beam with a long near field; a small, low-frequency element gives a wide beam with a short near field. There is no way to have both, which is the fundamental trade-off in probe selection. Working in shear roughly halves the wavelength for the same probe, so a shear beam is about half as divergent as a compression beam from the same crystal.

The formula describes an unfocused, circular, single-element probe radiating into a homogeneous medium. It does not describe a focused probe, a phased-array beam that is being steered, or a beam in coarse-grained austenitic material where scattering and anisotropy dominate.

sin θ = k · λ / D
k = 0.51 → −6 dB edge;  0.87 → −20 dB edge;  1.22 → first null
Full beam angle = 2 θ