08 Ultrasound

Near-field (Fresnel) length

N = D² / (4 λ) = D² f / (4 c). The last axial maximum of an unfocused circular piston.

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Simulation

Near-field (Fresnel) length — Change the numbers; the scene follows.

Where it works

Ultrasound

Transducer

At the transducer face — wavelength, pulse length, and the near/far field of the beam.

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Formula

N=\frac{D^2}{4\lambda}=\frac{D^2 f}{4c}

Variables

D · Aperture diametermmf · FrequencyMHzc · Speed of soundm/s

Results

λ
Wavelength
0.308mm
N
Near-field length
81.1688mm
N
Near-field length
8.1169cm
θ_div
Far-field half-angle
2.1535°

Explanation

N=\frac{D^2}{4\lambda}=\frac{D^2 f}{4c}

What it means

A circular piston has a messy near field (Fresnel zone) of interference maxima and minima, then a smoothly diverging far field (Fraunhofer). The transition sits at N = D²/4λ. A 10 mm, 5 MHz probe in soft tissue (c=1540 m/s, λ=0.31 mm) has N ≈ 8 cm — which is why arrays are focused (electronically) rather than left as pistons. Lateral resolution is best near the focus / around N for an unfocused disc. This is a working relation in Ultrasound.

Where it is used

Clinically it sits on the Ultrasound — Transducer. At the transducer face — wavelength, pulse length, and the near/far field of the beam. Ultrasound equations sit on the probe face and along the beam: impedance, Snell, Doppler, MI and TI. They explain why gel matters, why aliasing appears, and why a mechanical index is on the screen.

Ultrasound · Open this machine

How to use it

Enter aperture D (mm) and frequency (MHz); speed defaults to 1540 m/s. Read λ, N, and the far-field divergence half-angle sin⁻¹(1.22 λ/D). Change one input and watch the curve and the simulation follow.

Symbols

DAperture diameter10 mm
fFrequency5 MHz
cSpeed of sound1,540 m/s

Worked example

A typical case from the default values: D = 10 mm (Aperture diameter); f = 5 MHz (Frequency); c = 1,540 m/s (Speed of sound). Substituting into the relation gives λ = 0.308 mm; N = 81.1688 mm; N = 8.1169 cm; θ_div = 2.1535 °. These are teaching numbers — align them with your machine.

Typical values give

λ = 0.308mm
N = 81.1688mm
N = 8.1169cm
θ_div = 2.1535°

Where it comes from

The displayed formula is the working relation. N = D² / (4 λ) = D² f / (4 c). The last axial maximum of an unfocused circular piston. Usual reference: Kremkau / Edelman. Derive it in the specialty lesson, then return here to pin the numbers.

Reference: Kremkau / Edelman

Assumptions & limits

Continuous-wave circular piston in a homogeneous medium. Linear arrays have a rectangular Fresnel formula (N ≈ D²/4λ still as a scale). Focusing pulls the last maximum closer than N.

Pitfalls

Soft-tissue 1540 m/s is an assumption — not a measurement in that patient. Doppler angle 90° gives no shift. MI and TI are on-screen estimates, not absorbed dose. Continuous-wave circular piston in a homogeneous medium. Linear arrays have a rectangular Fresnel formula (N ≈ D²/4λ still as a scale). Focusing pulls the last maximum closer than N.

Keep this

Gel, angle, and assumed speed of sound — get those three right before you trust a centimetre. Continuous-wave circular piston in a homogeneous medium. Linear arrays have a rectangular Fresnel formula (N ≈ D²/4λ still as a scale). Focusing pulls the last maximum closer than N.

In this specialty

Ultrasound