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

Production and Use of Ultrasound

Learning Objective

Apply the intensity reflection coefficient formula

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

to analyse how ultrasound behaves at the interface between two media.

1. What is Ultrasound?

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

2. Generation of Ultrasound

1. Piezoelectric transducers – a crystal (e.g., quartz or PZT) deforms when an alternating voltage is applied, producing longitudinal pressure waves.
2. Capacitive micromachined ultrasonic transducers (CMUTs) – a thin membrane vibrates under an electrostatic force.
3. Magnetostrictive materials – change shape in a magnetic field, also used for high‑power applications.

All these devices rely on the rapid conversion of electrical energy into mechanical vibrations, which then propagate as ultrasound.

3. Propagation in Media

The speed of sound \$v\$ in a medium is given by

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

where \$B\$ is the bulk modulus and \$\rho\$ the density. The acoustic impedance \$Z\$ is defined as

\$Z = \rho v\$

Differences in \$Z\$ between adjoining media cause part of the incident wave to be reflected and part to be transmitted.

4. Intensity Reflection Coefficient

When an ultrasound wave of intensity \$I0\$ strikes a planar boundary between medium 1 (impedance \$Z1\$) and medium 2 (impedance \$Z2\$), the reflected intensity \$IR\$ is

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

This expression follows from matching pressure and particle‑velocity boundary conditions and is central to both imaging (e.g., echocardiography) and non‑destructive testing.

5. Practical Use of the Formula

• Predicting echo strength from tissue interfaces in medical ultrasound.
• Designing matching layers for transducers to maximise transmitted energy.
• Estimating signal loss at fluid‑solid boundaries in industrial inspection.

6. Example Calculation

Consider an ultrasound pulse travelling from soft tissue (\$Z{\text{tissue}}\approx1.6\times10^6\ \text{kg m}^{-2}\text{s}^{-1}\$) into bone (\$Z{\text{bone}}\approx7.8\times10^6\ \text{kg m}^{-2}\text{s}^{-1}\$).

1. Calculate the numerator: \$(Z{\text{tissue}}-Z{\text{bone}})^2 = (1.6-7.8)^2\times10^{12}=38.44\times10^{12}\$.
2. Calculate the denominator: \$(Z{\text{tissue}}+Z{\text{bone}})^2 = (1.6+7.8)^2\times10^{12}=88.36\times10^{12}\$.
3. Hence \$\displaystyle \frac{IR}{I0}= \frac{38.44}{88.36}\approx0.44\$.

About 44 % of the incident intensity is reflected at the tissue–bone interface, which explains the bright echo seen in an abdominal scan.

7. Typical Acoustic Impedances

8. Matching Layers

To reduce the large reflection at a high‑impedance–low‑impedance interface, a thin matching layer with impedance \$Zm\approx\sqrt{Z1Z_2}\$ is placed between the transducer and the load. The thickness is usually a quarter of the wavelength in the matching material, giving destructive interference for the reflected wave and constructive interference for the transmitted wave.

9. Applications

• Medical imaging – B‑mode, Doppler, and therapeutic ultrasound.
• Industrial non‑destructive testing – Detecting cracks, measuring thickness, and characterising welds.
• Cleaning – Ultrasonic baths use cavitation generated by high‑intensity waves.
• Sonochemistry – Accelerating chemical reactions via acoustic cavitation.

10. Summary Checklist

1. Know the definition of acoustic impedance \$Z=\rho v\$.
2. Be able to apply \$\displaystyle\frac{IR}{I0}= \frac{(Z1-Z2)^2}{(Z1+Z2)^2}\$ to calculate reflected intensity.
3. Understand why matching layers improve transmission.
4. Identify typical \$Z\$ values for common media.
5. Relate the physics to real‑world ultrasound applications.