Beam spread angle for a circular piezoelectric crystal is calculated from the -6 dB half-angle formula sin(γ) = K·V / (f·D), with K ≈ 0.51 for pulse-echo mode, yielding a full beam spread of 2·γ [S4].
Inputs are frequency f in MHz, crystal diameter D in mm (or inches with unit-matched V), and material longitudinal or shear velocity V in m/s; higher frequency and larger diameter both reduce spread, while higher V increases it [S1][S4].
Working Formula and the -6 dB Constant
The standard approximation for a flat piston source transducer is sin(γ) = K·V / (f·D), where γ is the half-angle of the main lobe measured from the acoustic axis to the -6 dB drop in pulse-echo response [S4][S1]. TWI lists the same form with K as a constant set by the chosen beam edge criterion (full-width, half-maximum, or edge of first null) [S2].
K ≈ 0.51 corresponds to the -6 dB (50% amplitude) pulse-echo beam edge; K ≈ 0.71 corresponds to the edge of the main lobe, and K ≈ 1.22 gives the first-minimum / Fraunhofer edge, the same 1.22 λ/D factor used in optical diffraction [S3][S4]. Picking the wrong K shifts the computed angle by a factor of roughly 1.4 between -6 dB and first-null criteria, which is the single most common error in hand-calculated spread tables.
Worked Example: 2.25 MHz, 12.7 mm, Carbon Steel
For a standard 0.5 inch (12.7 mm), 2.25 MHz longitudinal contact probe on carbon steel at V = 5900 m/s: wavelength λ = V/f = 5900/2.25 ≈ 2.62 mm, ratio K·λ/D = 0.51·2.62/12.7 ≈ 0.105, so γ = arcsin(0.105) ≈ 6.04° half-angle, giving full spread ≈ 12.1° [S4].
The same probe in aluminium (V ≈ 6320 m/s) yields γ ≈ 6.47° half-angle; in Plexiglas/acrylic (V ≈ 2730 m/s) it drops to γ ≈ 2.79°, illustrating that the V term, not f or D, dominates when comparing the same probe across materials [S4]. Engineers who swap a steel calibration block for a composite without re-checking spread routinely over-size flaws at depth.
Comparison: How Spread Varies with f, D, V
Fixing V at 5900 m/s (steel) and a 12.7 mm element, a 1 MHz probe gives γ ≈ 13.7° half-angle, a 2.25 MHz probe 6.0°, a 5 MHz probe 2.7°, and a 10 MHz probe 1.35° [S4]. Holding f = 2.25 MHz and V = 5900 m/s, a 6.35 mm element gives γ ≈ 12.2°, a 12.7 mm element 6.0°, a 19.1 mm element 4.0°, and a 25.4 mm element 3.0° [S4].
Three decision rules follow directly: (1) doubling f halves spread; (2) doubling D halves spread; (3) doubling V (e.g. moving from shear to longitudinal, or plastic to steel) increases spread linearly [S1][S4]. These three levers, not exotic lens attachments, are how procedure writers in PID loop audits and density-compensated level work control effective beam geometry for a given transducer type.
Far-Field Validity and the Near-Field Boundary
The K = 0.51 and K = 1.22 formulas are valid only in the far field (Fraunhofer zone), which begins at N = D²·f / (4·V) for a circular piston; below N, constructive and destructive interference dominates and a single divergence angle is meaningless [S1][S4].
For 2.25 MHz, 12.7 mm, steel, N ≈ 12.7²·2.25/(4·5900) ≈ 15.4 mm, so defect sizing below roughly 15 mm depth requires near-field corrections or hydrophone mapping per ASTM E-1065 Section A6, which standardises pulse-echo ball-target and hydrophone methods for immersion probes in water [S1]. Contact probes on coarse-grained steel, austenitic welds, and cast iron, all classic near-field-heavy applications, are the cases where a hand-calculated γ misleads the operator most.
When the Simple Formula Fails
Rectangular or square elements use the same sin(γ) = K·V/(f·a) form, but with a as the relevant aperture dimension, and the constant K is typically smaller (around 0.5 to 0.6 for -6 dB), so divergence is not isotropic and is wider along the smaller dimension [S3].
Phased-array elements, focused probes, and angle-beam wedges also break the flat-piston model: phased array divergence is set by the aperture-law and pitch, not the disc diameter, and angle-beam spread is added vectorially to the refracted path computed from Snell's law [S4]. Linked transducer selection logic for angle-beam path-tracing parallels the sensor-routing maths used in two-DP density-compensated level measurement, where a single beam assumption gets corrected once the geometry tilts.
Field Procedure: A 4-Step Spread Check
Step 1: read f and D off the probe label; reject any D outside 6.35 to 25.4 mm for general manual UT, the band where γ sits between 2° and 15° in steel [S4]. Step 2: look up V for the test material; use 5900 m/s for carbon-steel longitudinal, 3250 m/s for carbon-steel shear, 6320 m/s for aluminium longitudinal, 2730 m/s for acrylic longitudinal [S4].
Step 3: compute N = D²·f/(4·V) and confirm the inspection depth is at least 1.5·N; if not, treat the near field separately [S1]. Step 4: compute γ = arcsin(0.51·V/(f·D)); verify the projected beam diameter at the back wall (D_beam ≈ D + 2·S·tan(γ), where S is sound path) does not exceed the lateral extent of the test object by more than 20%, otherwise side-wall echoes contaminate the gate [S1][S4]. This same geometry discipline, beam width matched to coupon size, is what determines whether construction machinery and equipment welds, in-service pipe, and lifting-boom NDT reports are defensible in an audit.
Trackable signals for the next quarter: ASTM E-1065 revision activity on immersion beam-profile measurement (Section A6 is the only published standard test method for full 3-D sound-field mapping cited here [S1]) and the continued migration of procedure-writing tools to online calculators that auto-flag near-field violations, two visible proxies for whether the -6 dB K = 0.51 approximation is being re-validated or quietly overwritten by phased-array toolchains [S4].
The underlying component specifications are covered under angle grinder, and proximity probe.