08 Ultrasound

Snell's law (refraction)

sin θ₂ / sin θ₁ = c₂ / c₁. Critical angle when θ₂ = 90°.

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Simulation

Snell's law (refraction) — Change the numbers; the scene follows.

Where it works

Ultrasound

Tissue path

Along the beam in tissue — impedance mismatch, Doppler shift, MI/TI at the focus.

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Formula

\frac{\sin\theta_2}{\sin\theta_1}=\frac{c_2}{c_1},\quad \theta_c=\arcsin(c_1/c_2)

Variables

θ₁ · Incidence angle°c₁ · Speed medium 1m/sc₂ · Speed medium 2m/s

Results

θ₂
Refraction angle
18.79°
θ_c
Critical angle
—°

Explanation

\frac{\sin\theta_2}{\sin\theta_1}=\frac{c_2}{c_1},\quad \theta_c=\arcsin(c_1/c_2)

What it means

Refraction obeys Snell: sinθ₂/sinθ₁ = c₂/c₁. If the second medium is faster and θ₁ exceeds the critical angle, the wave totally internally reflects — an edge shadow beside a cyst or gallbladder is often this plus defocusing. This is a working relation in Ultrasound.

Where it is used

Clinically it sits on the Ultrasound — Tissue path. Along the beam in tissue — impedance mismatch, Doppler shift, MI/TI at the focus. 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.

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How to use it

Soft tissue 1540 vs fat 1450 vs fluid 1480 vs bone 4000 m/s. Critical angle exists only when c₂ > c₁. Distorted needle paths under refraction are a safety topic. Change one input and watch the curve and the simulation follow.

Symbols

θ₁Incidence angle20 °
c₁Speed medium 11,540 m/s
c₂Speed medium 21,450 m/s

Worked example

A typical case from the default values: θ₁ = 20 ° (Incidence angle); c₁ = 1,540 m/s (Speed medium 1); c₂ = 1,450 m/s (Speed medium 2). Substituting into the relation gives θ₂ = 18.79 °; θ_c = — °. These are teaching numbers — align them with your machine.

Typical values give

θ₂ = 18.79°
θ_c = —°

Where it comes from

The displayed formula is the working relation. sin θ₂ / sin θ₁ = c₂ / c₁. Critical angle when θ₂ = 90°. Usual reference: Bushberg. Derive it in the specialty lesson, then return here to pin the numbers.

Reference: Bushberg

Assumptions & limits

Geometric acoustics, planar interface, no beam width. Real interfaces are curved; rays focus or defocus (edge shadow).

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. Geometric acoustics, planar interface, no beam width. Real interfaces are curved; rays focus or defocus (edge shadow).

Keep this

Gel, angle, and assumed speed of sound — get those three right before you trust a centimetre. Geometric acoustics, planar interface, no beam width. Real interfaces are curved; rays focus or defocus (edge shadow).

In this specialty

Ultrasound