Formulas and who prints them
c is the sound velocity of the chosen wave mode in the medium the beam travels through, f is the frequency and D is the active element diameter. Lengths come out in the unit you enter D in. The model is an unfocused circular (disc) element radiating into one isotropic medium.
- Wavelength: λ = c / f. Olympus/Evident Technical Notes, Equation 2.
- Near-field length: N = D² / 4λ, the same as D²f / 4c. Technical Notes, Equations 8 and 8a, and Evident's tutorial section 2.4.
- Fuller near-field form: N = (D² / 4λ) × [1 − (λ / D)²]. Technical Notes p. 10, which says Equations 8 and 8a were derived from it.
- Beam-spread half angle: sin α = K × λ / D = K × c / (f × D). Technical Notes Equation 11 prints K = 0.514 for the −6 dB pulse-echo half angle; TWI writes sin(θ/2) = Kλ / D with K = 1.08 at the −20 dB edge and 1.22 at the extreme edge.
Multiplying out the fuller form gives N − λ/4, so the two near-field values differ by a quarter wavelength. That gap is small when the element is many wavelengths across, which is why both forms appear in study material. If a practice answer is slightly lower than yours, check which form it used. N = r² / λ with r the element radius is the same short form written another way; if an answer is exactly four times yours, check for a radius and diameter mix-up.
Which beam-edge constant to use
The half angle depends on where you call the edge of the beam. A larger K puts the edge farther out, so the angle is wider for the same probe. The 0.514 constant describes the −6 dB points of the pulse-echo beam, as Evident defines it. TWI's 1.08 marks the edge where energy is one tenth of the beam center (−20 dB), and 1.22 marks the extreme edge. Use the constant your question, procedure or transducer data sheet names, and never mix two conventions in one answer. The calculator shows all three side by side so you can see the spread.
Worked example: 5 MHz, 10 mm element on steel
Use a longitudinal velocity of 5890 m/s (typical for steel 1020), f = 5 MHz and D = 10 mm.
- λ = 5890 / (5 × 1000) = 1.178 mm.
- N = 10² / (4 × 1.178) = 100 / 4.712 = 21.2224 mm. The fuller form gives 21.2224 − 1.178 / 4 = 20.9279 mm.
- −6 dB: sin α = 0.514 × 1.178 / 10 = 0.060549, so α = 3.4713°.
- −20 dB: sin α = 1.08 × 1.178 / 10 = 0.127224, so α = 7.3092°.
- Extreme edge: sin α = 1.22 × 1.178 / 10 = 0.143716, so α = 8.2629°.
A reflector 15 mm deep sits inside this near field, where on-axis amplitude swings between maxima and minima. Doubling the frequency to 10 MHz halves λ, doubles N to about 42.4 mm and roughly halves each angle.
Assumptions and use limits
These are textbook far-field approximations for a flat, unfocused circular element. They ignore wedges, delay lines, curved surfaces, focusing, attenuation, rectangular or phased-array apertures and the real (often slightly smaller) element size. For angle-beam probes the beam is formed partly in the wedge, so use the manufacturer's beam data rather than these figures. Material presets are typical published values, not a calibration: velocity changes with alloy, heat treatment and temperature, so measure it on your own part or a reference block where the result matters.
Practice the related math in the UT Level II formulas guide, and use the Snell's law calculator for refracted and critical angles.