recall and use I = I0e–μx for the attenuation of ultrasound in matter

Production and Use of Ultrasound

Learning Objective

Recall and use the attenuation relation

\$I = I_0 e^{-\mu x}\$

where I is the intensity after travelling a distance x in a material, I₀ is the initial intensity, and μ is the linear attenuation coefficient (m⁻¹).

1. What is Ultrasound?

Ultrasound refers to sound waves with frequencies above the upper limit of human hearing (≈20 kHz). In most A‑Level contexts the range is 0.5 MHz to 10 MHz.

2. Production of Ultrasound

• Piezoelectric transducers – a crystal (e.g., quartz, PZT) deforms when an alternating voltage is applied, generating longitudinal waves. The same crystal can act as a receiver.
• Magnetostrictive transducers – ferromagnetic rods change length in a varying magnetic field, producing sound.
• Capacitive micromachined ultrasonic transducers (CMUTs) – a thin membrane vibrates under an electric field; used in high‑frequency imaging.

3. Key Properties of Ultrasound

4. Attenuation of Ultrasound

The intensity of an ultrasonic beam decreases exponentially as it propagates through a material:

\$I(x) = I_0 e^{-\mu x}\$

Key points:

• μ (linear attenuation coefficient) incorporates absorption and scattering.
• Attenuation is frequency‑dependent; roughly μ ∝ f for many soft tissues.
• In practice we often use the decibel form:
  

  \$\text{Loss (dB)} = 10 \log{10}\!\left(\frac{I0}{I}\right) = 8.686\,\mu x\$

Example Calculation

Given:

• Initial intensity \$I_0 = 1.0\ \text{W m}^{-2}\$
• Attenuation coefficient \$\mu = 0.5\ \text{m}^{-1}\$
• Travel distance \$x = 3\ \text{cm} = 0.03\ \text{m}\$

Find the intensity after 3 cm.

Solution:

\$I = I_0 e^{-\mu x}=1.0\,e^{-0.5\times0.03}=1.0\,e^{-0.015}\approx 0.985\ \text{W m}^{-2}\$

The loss is small because the distance is short and μ is modest.

5. Uses of Ultrasound

1. Medical imaging (sonography)

   • Pregnancy scans, abdominal organ imaging, Doppler flow measurement.
   • Resolution improves with higher frequency, but penetration depth decreases.

2. Industrial non‑destructive testing (NDT)

   • Detect cracks, voids, and thickness variations in metals and composites.
   • Pulse‑echo and through‑transmission techniques rely on reflected intensity.

3. Ultrasonic cleaning

   • High‑frequency cavitation removes contaminants from delicate parts.
   • Typical frequencies: 20–40 kHz (lower than imaging range).

4. Sonochemistry

   • Acoustic cavitation drives chemical reactions, e.g., synthesis of nanoparticles.

6. Summary Checklist

• Know the definition and units of the attenuation coefficient μ.
• Be able to rearrange \$I = I_0 e^{-\mu x}\$ to solve for any variable.
• Understand how frequency affects both resolution and attenuation.
• Identify at least three practical applications of ultrasound and the typical frequency range used.