use IR / I0 = (Z1 – Z2)2 / (Z1 + Z2)2 for the intensity reflection coefficient of a boundary between two media

Ultrasound – Production, Propagation and Use

Learning Objectives

• Explain how a piezoelectric crystal both generates and detects ultrasound, explicitly linking the applied potential difference (p.d.) to deformation and the resulting e.m.f. to detection.
• Derive and use the intensity‑reflection‑coefficient

  \[

  \frac{IR}{I0}= \frac{(Z1-Z2)^2}{(Z1+Z2)^2}

  \]

  to predict echo strength at an interface.

• Apply the attenuation law \(I = I_0 e^{-\mu x}\) to calculate signal loss in tissue or metal.
• Design an acoustic matching layer using the geometric‑mean impedance and quarter‑wave thickness rule.
• Connect the physics (reflection coefficient, attenuation) to real‑world uses in medical imaging and industrial non‑destructive testing (NDT).

1. What is Ultrasound?

Ultrasound are sound waves with frequencies above the upper limit of human hearing (≈ 20 kHz). In most medical and industrial applications the frequencies lie between 1 MHz and 15 MHz, giving wavelengths in soft tissue of a few millimetres to a few hundred micrometres.

2. Generation & Detection with a Piezoelectric Transducer

2.1. Inverse piezoelectric effect – generation

• When a voltage (potential difference) is applied across a piezoelectric crystal (e.g. quartz, PZT), the electric field produces a mechanical strain – the crystal deforms.
• The rapid deformation launches a longitudinal pressure wave; this is the ultrasound pulse that propagates into the test medium.

2.2. Direct piezoelectric effect – detection

• An incoming acoustic wave compresses and expands the crystal.
• The resulting strain generates an electromotive force (e.m.f.) across the crystal, which is amplified and recorded.
• This is the basis of the pulse‑echo method used in imaging and non‑destructive testing.

2.3. Pulse‑echo principle

1. A short high‑voltage pulse excites the crystal, launching an ultrasound burst.
2. The burst travels, reflects at interfaces, and returns to the same crystal.
3. The returning wave re‑induces an e.m.f.; the measured time‑of‑flight \(t\) gives the depth

   \[

   d = \frac{v\,t}{2},

   \]

   where \(v\) is the speed of sound in the medium.

2.4. Other transducer types (brief)

• Capacitive micromachined ultrasonic transducers (CMUTs) – a thin membrane vibrates under electrostatic force.
• Magnetostrictive transducers – a ferromagnetic material changes shape in a magnetic field, useful for high‑power applications.

3. Propagation in Media

The speed of sound in a homogeneous medium is

\[

v=\sqrt{\frac{B}{\rho}},

\]

where \(B\) is the bulk modulus and \(\rho\) the density.

The acoustic impedance of the medium is

\[

Z = \rho v \qquad\left[\text{kg m}^{-2}\text{s}^{-1}\right].

\]

Differences in \(Z\) at a planar boundary cause part of the incident wave to be reflected and part to be transmitted.

4. Reflection and Transmission at a Planar Boundary

Consider a normally incident plane wave travelling from medium 1 (\(Z1\)) to medium 2 (\(Z2\)). At the interface the acoustic pressure \(p\) and particle velocity \(u\) must be continuous:

\[

p{\text{inc}} + p{\text{ref}} = p_{\text{trans}}, \qquad

\frac{p{\text{inc}}}{Z1} - \frac{p{\text{ref}}}{Z1} = \frac{p{\text{trans}}}{Z2}.

\]

Solving these two equations gives the amplitude reflection coefficient

\[

R = \frac{Z1 - Z2}{Z1 + Z2}.

\]

The intensity of a wave is proportional to the square of its pressure amplitude, so the intensity‑reflection‑coefficient is

\[

\boxed{\frac{IR}{I0}=R^{2}= \frac{(Z1-Z2)^2}{(Z1+Z2)^2}}.

\]

Similarly, the intensity transmission coefficient is \(\displaystyle \frac{IT}{I0}=1-\frac{IR}{I0}\).

5. Using the Intensity‑Reflection‑Coefficient

Example – tissue ↔ bone

Soft tissue: \(Z_t = 1.6\times10^{6}\,\text{kg m}^{-2}\text{s}^{-1}\)

Bone: \(Z_b = 7.8\times10^{6}\,\text{kg m}^{-2}\text{s}^{-1}\)

\[

\frac{IR}{I0}= \frac{(1.6-7.8)^2}{(1.6+7.8)^2}

= \frac{38.44}{88.36}

\approx 0.44 \;(44\%).

\]

Implication for medical imaging: a 44 % reflection produces a bright echo on a B‑mode scan, allowing the tissue‑bone interface to be clearly visualised.

Implication for NDT: the same coefficient predicts that roughly half of the incident acoustic energy is reflected by a high‑impedance defect (e.g. a crack in metal). The size of the reflected signal relative to the background determines the defect‑detectability (signal‑to‑noise ratio).

6. Attenuation of Ultrasound

As the wave propagates, its intensity decreases exponentially:

\[

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

\]

where \(\mu\) is the attenuation coefficient (in m⁻¹) and \(x\) is the travelled distance.

Quantitative example

In soft tissue \(\mu \approx 0.5\;\text{dB cm}^{-1}\) at 5 MHz (≈ 0.057 m⁻¹). For a depth of 5 cm:

\[

I = I0 e^{-0.057\times0.05}\approx I0 \times 0.997 \;(≈ 0.3\% \text{ loss}).

\]

In aluminium \(\mu\) is much larger (≈ 20 dB cm⁻¹), so attenuation dominates the usable inspection depth in metals.

7. Matching Layers – Design Procedure

8. Practical Applications

• Medical diagnostic imaging – B‑mode (2‑D) displays echo brightness proportional to \(\frac{IR}{I0}\); Doppler studies use the same physics to measure blood‑flow velocity.
• Therapeutic ultrasound – High‑intensity focused ultrasound (HIFU) relies on controlled attenuation to deposit energy at a target depth.
• Industrial NDT – Pulse‑echo and through‑transmission techniques detect cracks, weld defects, and wall thinning. The detectable size of a defect is linked to the reflected intensity and the attenuation through the surrounding material.
• Ultrasonic cleaning – Cavitation generated by high‑intensity waves removes contaminants; efficiency depends on the attenuation in the cleaning fluid.
• Sonochemistry – Acoustic cavitation accelerates chemical reactions; again, the intensity reaching the reaction zone follows the attenuation law.

9. Summary Checklist

1. Define acoustic impedance \(Z=\rho v\) and list typical values.
2. State the amplitude reflection coefficient \(R = (Z1-Z2)/(Z1+Z2)\) and the intensity‑reflection‑coefficient \(\displaystyle\frac{IR}{I0}=R^{2}\).
3. Show how the coefficient predicts echo brightness in medical scans and signal‑to‑noise in NDT.
4. Write the attenuation law \(I = I_0 e^{-\mu x}\) and explain how to use it for depth‑dependent loss.
5. Describe the inverse and direct piezoelectric effects, using the terms potential difference (p.d.) and e.m.f..
6. Outline the pulse‑echo principle and the depth calculation \(d = vt/2\).
7. Apply the matching‑layer design steps: compute \(Zm\), choose material, set thickness \(t=\lambdam/4\).
8. Connect the physics to real‑world applications in medicine and industry.