define the specific acoustic impedance of a medium as Z = ρc, where c is the speed of sound in the medium

Production and Use of Ultrasound

Learning Objective

Define the specific acoustic impedance of a medium as

\$Z = \rho c\$

where ρ is the density of the medium and c is the speed of sound in that medium.

1. What is Ultrasound?

Ultrasound refers to sound waves with frequencies above the upper limit of human hearing (typically > 20 kHz). In medical and industrial applications the frequencies are usually in the range 1 MHz – 20 MHz, giving wavelengths of a few millimetres to a few centimetres.

2. Production of Ultrasound

• Piezoelectric transducers – a crystal (e.g., quartz, PZT) deforms when an alternating voltage is applied, generating longitudinal sound waves. The same crystal can act as a receiver.
• Magnetostrictive transducers – ferromagnetic materials change length in a magnetic field, producing sound when the field varies.
• Capacitive micromachined ultrasonic transducers (CMUTs) – a thin membrane vibrates under an electrostatic force, useful for high‑frequency arrays.

3. Uses of Ultrasound

1. Medical imaging (sonography) – visualising soft tissues, fetal monitoring, Doppler flow measurement.
2. Therapeutic ultrasound – physiotherapy, lithotripsy, targeted drug delivery.
3. Industrial non‑destructive testing (NDT) – detecting cracks, measuring thickness, characterising material properties.
4. Cleaning and sonochemistry – high‑intensity fields generate cavitation for cleaning delicate parts.

4. Acoustic Impedance

The specific acoustic impedance Z of a medium quantifies how much resistance the medium offers to the passage of a sound wave. It is defined as

\$Z = \rho c\$

where:

• ρ – density of the medium (kg m⁻³)
• c – speed of sound in the medium (m s⁻¹)

5. Why Impedance Matters

When an ultrasonic wave encounters a boundary between two media with different impedances, part of the wave is reflected and part is transmitted. The reflection coefficient for intensity is

\$R = \left(\frac{Z2 - Z1}{Z2 + Z1}\right)^2\$

and the transmission coefficient is T = 1 - R. Matching impedances (e.g., using a coupling gel) maximises transmitted energy, which is crucial for clear images.

6. Typical Acoustic Impedances

7. Example Calculation

Calculate the impedance of soft tissue with ρ = 1050 kg m⁻³ and c = 1540 m s⁻¹.

\$Z = (1050\;\text{kg m}^{-3})(1540\;\text{m s}^{-1}) = 1.617 \times 10^{6}\;\text{Rayl}\$

If the wave travels from soft tissue into bone (Z₁ = 1.62 × 10⁶ Rayl, Z₂ = 7.75 × 10⁶ Rayl), the intensity reflection coefficient is

\$R = \left(\frac{7.75-1.62}{7.75+1.62}\right)^2 \approx 0.46\$

Thus about 46 % of the incident intensity is reflected at the tissue‑bone interface.

8. Practical Tips for Using Ultrasound

• Apply a coupling medium (gel) to eliminate the large impedance mismatch between the transducer (usually quartz, Z ≈ 15 × 10⁶ Rayl) and skin.
• Choose a frequency that balances resolution (higher frequency) against penetration depth (lower frequency).
• For quantitative measurements, calibrate the system using a reference material of known impedance.

9. Summary

Understanding the specific acoustic impedance \$Z = \rho c\$ is fundamental to both the generation and effective use of ultrasound. Impedance mismatches dictate how much of the ultrasonic energy is reflected or transmitted at interfaces, influencing image quality in medical diagnostics and the sensitivity of industrial inspections.