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.

Compression or shear

Worked example

Wave mode	compression
Material velocity (override)	0 m/s
Probe frequency	5 MHz
Element diameter	10 mm
Beam edge definition	0.51
Velocity used	5900 m/s
Wavelength λ	1.18 mm
Divergence half-angle θ	3.45 deg
Full beam angle 2θ	6.9 deg
Near field length N	21.19 mm

10 mm, 5 MHz compression probe on steel. λ = 1.18 mm; sin θ = 0.51 × 1.18 / 10 = 0.06018, so θ = 3.45° and the full −6 dB beam angle is 6.90°. At k = 1.22 the same probe gives a first-null half-angle of 8.28°.

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