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Learn7408Paper 3B · Option: Medical physicsNon-ionising imagingUltrasound
Non-ionising imaging · Subtopic 1 of 1

Ultrasound

Ultrasound (above 2020 kHz; medical uses 11–1515 MHz) is generated and detected by a piezoelectric crystal, which vibrates when an alternating p.d. is applied and produces an e.m.f. when squeezed by returning echoes. Pulses reflect at boundaries between tissues. The fraction reflected depends on the acoustic impedances Z=ρcZ=\rho c: IRI0=(Z1−Z2)2(Z1+Z2)2\dfrac{I_R}{I_0}=\dfrac{(Z_1-Z_2)^2}{(Z_1+Z_2)^2}. A big mismatch reflects almost everything, so gel is used to couple the transducer to the skin. Ultrasound is attenuated exponentially: I=I0e−μxI=I_0e^{-\mu x}.

Cambridge 9702 Paper 4AQA 7408 Paper 3B≈ 40 minRevision 5 · updated 2026-10-27
Section 1 of 14

Learning objectives

  • Explain how a piezoelectric transducer produces and detects ultrasound.
  • Use Z=ρcZ=\rho c and the intensity reflection coefficient.
  • Explain impedance matching and the use of coupling gel.
  • Use I=I0e−μxI=I_0e^{-\mu x} and find depths from echo times.
Pause and think

Before an ultrasound scan, the doctor spreads cold gel on your skin. Why can't they just press the probe on directly?

Section 2 of 14

Core idea

Start hereSeeing a baby with sound

Ultrasound scans build pictures from echoes. Sound bounces off each boundary between tissues, and timing the echoes tells how deep each boundary is. How strongly it bounces depends on how different the two materials are.

The big idea

Z=ρcZ=\rho c. Reflected fraction IRI0=(Z2−Z1)2(Z2+Z1)2\frac{I_R}{I_0}=\frac{(Z_2-Z_1)^2}{(Z_2+Z_1)^2}. Depth =ct2=\frac{ct}2.

1. Producing ultrasound

  • Piezoelectric crystal: vibrates with an alternating p.d.
  • Produces a p.d. when echoes hit it: one transducer sends and receives.

2. Reflection

A-scan: echoes from tissue boundaries
probefatmusclebonetime of echo → depth
Time between echoes × speed ÷ 2 gives the depth of each boundary.
IRI0=(Z2−Z1Z2+Z1)2\frac{I_R}{I_0}=\left(\frac{Z_2-Z_1}{Z_2+Z_1}\right)^2
where
ZZ
acoustic impedance ρc\rho c (kg m⁻² s⁻¹).
Worked exampleTissue to bone

Z1=1.63×106Z_1=1.63\times10^6, Z2=7.8×106Z_2=7.8\times10^6.

Goal
Reflected fraction.
Strategy
Formula.
  1. 7.8−1.637.8+1.63=6.179.43\frac{7.8-1.63}{7.8+1.63}=\frac{6.17}{9.43}
    Powers of 10 cancel.
  2. 0.6542=0.430.654^2=0.43
    Square.
Answer43% reflected.
✓ CheckAir (Z≈400Z\approx400) to skin: almost 100%, so gel is used ✓.

3. Depth from echoes

Fraction reflected against impedance ratio
2468100.20.40.60.810Z₂ / Z₁fraction reflected((Z₂ − Z₁)/(Z₂ + Z₁))²matched: no reflection
The bigger the mismatch, the stronger the echo. Soft tissue/bone gives strong echoes.
Worked exampleEcho timing

Echo after 65 μs; c=1540c=1540 m s−1^{-1}.

Goal
Depth.
Strategy
Halve the round trip.
  1. 1540×65×10−6=0.100 m1540\times65\times10^{-6}=0.100\text{ m}
    There and back.
  2. 0.050 m0.050\text{ m}
    Halve.
Answer5.05.0 cm.
✓ CheckTypical organ depth ✓.
Key takeawayFrequency trade-off Higher frequency: better resolution, but more absorption (less depth).

Summary

  • Z=ρcZ=\rho c; reflection formula.
  • Depth =ct/2=ct/2.
Key words
Acoustic impedance
Density × speed of sound.
HistoryFrom submarines to babies. Ultrasound echo-ranging (sonar) was developed to detect submarines in World War I. In 19581958 Ian Donald, a Glasgow obstetrician who'd served in the RAF, published the first ultrasound images of unborn babies, transforming pregnancy care.
Fun factBats and dolphins. Bats emit ultrasound up to about 200200 kHz and can detect insects as thin as a human hair from the echoes. Dolphins use clicks up to 150150 kHz to "see" inside other animals, sensing the air-filled lungs and swim bladders that reflect strongly.
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