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

Property	Typical \cdot alue (A‑Level)	Notes
Frequency (f)	0.5 – 10 MHz	Higher f → better 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 and elasticity

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:
  $$\text{Loss (dB)} = 10 \log_{10}\!\left(\frac{I_0}{I}\right) = 8.686\,\mu x$$

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.