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

Production and Use of Ultrasound (Cambridge A‑Level Physics 9702 – Syllabus 24.1)

Learning Objective

Recall and use the attenuation relation

\$I = I_{0}\,e^{-\mu x}\$

• I – intensity after travelling a distance x in the material (W m⁻²)
• I₀ – initial intensity (W m⁻²)
• μ – linear attenuation coefficient (m⁻¹), which includes absorption and scattering
• x – path length in the material (m)

1. What is Ultrasound?

Ultrasound = sound waves with frequencies above the upper limit of human hearing (≈20 kHz). In A‑Level examinations the usual range is 0.5 – 10 MHz.

2. Production & Detection of Ultrasound

2.1. Piezo‑electric transducers (core syllabus topic)

• Generation (converse piezoelectric effect) – an alternating voltage makes a crystal repeatedly expand and contract, launching longitudinal sound waves into the surrounding medium.
• Detection (direct piezoelectric effect) – an incoming pressure wave strains the crystal, producing an emf that can be measured.
• Pulse‑echo technique – the same crystal is first driven to emit a short burst (pulse) and then switched to the receiving mode to record the reflected echo.

2.2. Optional/Advanced transducer types (useful enrichment)

• Magnetostrictive transducers – ferromagnetic rods change length in a varying magnetic field (generation) and, conversely, pressure‑induced strain changes magnetic flux (detection).
• Capacitive micromachined ultrasonic transducers (CMUTs) – a thin membrane vibrates under an electric field to emit ultrasound; returning pressure variations alter the membrane capacitance, producing a voltage signal.

These are not required for the AS‑level exam but provide useful context for extended learning.

3. Key Properties of Ultrasound

4. Attenuation of Ultrasound

The intensity of an ultrasonic beam decreases exponentially with distance:

\$I(x)=I_{0}\,e^{-\mu x}\$

• μ – linear attenuation coefficient (units m⁻¹) includes both absorption (conversion to heat) and scattering.
• For most soft tissues μ is roughly proportional to frequency:
  

  \$\mu \approx k\,f\qquad\text{with }k\approx0.5\;\text{dB cm}^{-1}\,\text{MHz}^{-1}\$

• In decibel form:

  \$\text{Loss (dB)} = 10\log{10}\!\left(\frac{I{0}}{I}\right)=8.686\,\mu x\$

Example Calculation

Given

• I₀ = 1.0 W m⁻²
• μ = 0.5 m⁻¹
• x = 3 cm = 0.03 m

Find I after 3 cm:

\$I = 1.0\,e^{-0.5\times0.03}=1.0\,e^{-0.015}\approx0.985\ \text{W m}^{-2}\$

Loss = 8.686 μ x ≈ 0.13 dB – a small attenuation over a short path.

5. Acoustic Impedance and Reflection at Tissue Boundaries

• Specific acoustic impedance:

  \$Z = \rho\,c\$

  where ρ is density (kg m⁻³) and c is the speed of sound (m s⁻¹).

  Medium	Z (kg m⁻² s⁻¹)
  Soft tissue	1.5 × 10⁶
  Fat	1.38 × 10⁶
  Bone	7.8 × 10⁶
  Water	1.48 × 10⁶

• Reflection coefficient for a planar interface between media 1 and 2:

  \$R = \left(\frac{Z{1}-Z{2}}{Z{1}+Z{2}}\right)^{2}\$

  The reflected intensity is \(I{\text{reflected}} = R\,I{\text{incident}}\). Larger impedance mismatches give stronger echoes.

• Diagnostic use

  • Each tissue boundary (e.g., muscle–fat, fluid–organ) produces a characteristic echo.
  • The round‑trip travel time \(t = \dfrac{2x}{c}\) gives the depth \(x\) of the reflector.
  • The echo amplitude (related to R) provides qualitative information about the nature of the interface.

6. Uses of Ultrasound

1. Medical imaging (sonography)

   • Pregnancy scans, abdominal and cardiac imaging, Doppler flow measurement.
   • Relies on the pulse‑echo technique and impedance‑based reflections.

2. Industrial non‑destructive testing (NDT)

   • Detect cracks, voids, and thickness variations in metals, composites, and welds.
   • Both pulse‑echo and through‑transmission modes use the same attenuation and reflection physics.

3. Ultrasonic cleaning

   • High‑frequency cavitation (20–40 kHz) removes contaminants from delicate parts.
   • Lower frequency than imaging to obtain strong cavitation without excessive attenuation.

4. Sonochemistry

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

7. Summary Checklist (for revision)

• Define the linear attenuation coefficient μ (units m⁻¹) and write the exponential attenuation law.
• Re‑arrange \(I = I_{0}e^{-\mu x}\) to solve for any of I, I₀, μ, or x.
• State the converse and direct piezoelectric effects and explain how a single crystal can both generate and detect ultrasound.
• Write the definition of specific acoustic impedance \(Z = \rho c\) and the reflection coefficient \(R = \bigl(\frac{Z{1}-Z{2}}{Z{1}+Z{2}}\bigr)^{2}\).
• Explain qualitatively how impedance mismatches produce the echoes used in diagnostic imaging.
• Identify at least three practical applications of ultrasound and the typical frequency range used for each.