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

Property	Typical value (A‑Level)	Notes
Frequency (f)	0.5 – 10 MHz	Higher f → better spatial resolution, greater attenuation.
Wavelength (λ)	0.15 – 3 mm (in water)	λ = v/f, with v ≈ 1500 m s⁻¹ in soft tissue.
Speed of sound (v)	≈1500 m s⁻¹ in tissue, 340 m s⁻¹ in air	Depends on medium density (ρ) and elasticity.

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.