Acoustic Impedance and Sound Reflection in Ultrasound

The magnitude of this mismatch dictates how much of the sound energy is reflected versus how much is transmitted. If the impedance is identical across an interface, no reflection will occur. However, the larger the difference in $Z$ values, the stronger the reflection will be.

Intensity Reflection and Transmission Coefficients

To quantify exactly how much sound is reflected or transmitted at an interface, we use two specific coefficients:

1. Intensity Reflection Coefficient (IRC): This represents the fraction of the incident intensity that is reflected at the interface. It can be calculated using the $Z$ values of the two media ($Z_1$ and $Z_2$): $$IRC = \left( \frac{Z_2 - Z_1}{Z_2 + Z_1} \right)^2$$
2. Intensity Transmission Coefficient (ITC): This represents the fraction of the incident intensity that continues into the second medium. Since energy must be conserved, the sum of the reflected and transmitted intensity must equal the original intensity: $$ITC = 1 - IRC$$

Exam Tip: A high impedance mismatch (like soft tissue to air) results in a very high IRC, meaning almost all sound is reflected and very little is transmitted. This is why ultrasound gel is required to eliminate air between the transducer and the skin.

Worked Example: Bone vs. Soft Tissue

Let's calculate the reflection and transmission at an interface between bone and soft tissue using the following values:

• Bone ($Z_2$) $= 7.8 \times 10^6$ Rayls
• Soft tissue ($Z_1$) $= 1.63 \times 10^6$ Rayls

Step 1: Calculate IRC

$$\begin{aligned} IRC &= \left( \frac{7.8 \times 10^6 - 1.63 \times 10^6}{7.8 \times 10^6 + 1.63 \times 10^6} \right)^2 \times 100 \\ IRC &= \left( \frac{6.17}{9.43} \right)^2 \times 100 \\ IRC &\approx 43\% \end{aligned}$$

Step 2: Calculate ITC

$$\begin{aligned} ITC &= 1 - 0.43 \\ ITC &= 0.57 \text{ or } 57\% \end{aligned}$$

Reflection and Refraction

When sound hits an interface, it can behave in two distinct ways: it can reflect (bounce back) or refract (bend).

Reflection

Reflection is the sound that returns to the probe to create an image. There are two main types:

• Specular Reflection: Occurs when the sound beam hits a large, smooth surface (greater than one wavelength in diameter), such as the diaphragm. The angle of incidence will always equal the angle of reflection.
• Non-specular (Scatter): Occurs when sound interacts with small or rough surfaces.

Refraction

Refraction refers to the bending of the sound beam as it is transmitted across an interface. Refraction occurs only when two conditions are met:

1. The velocities of the two media are different ($V_1 \neq V_2$).
2. The angle of incidence is non-perpendicular.

Sound follows Snell's Law, which describes how much the beam will bend based on the velocity change:

$$\frac{\sin \theta_i}{\sin \theta_t} = \frac{v_i}{v_t}$$

• If $V_1 > V_2$, the sound bends towards the normal.
• If $V_1 < V_2$, the sound bends away from the normal.
• If the angle reaches a certain "critical value" when $V_1 < V_2$, Total Internal Reflection can occur, where no sound is transmitted at all.

Other Factors in Attenuation

While reflection and refraction are key interactions, they are part of a broader phenomenon called attenuation—the weakening of the sound beam as it travels. Other contributors to attenuation include:

• Absorption: The conversion of sound energy into heat. This is the dominant factor in attenuation, accounting for roughly 80% of the loss.
• Scatter: The result of sound interacting with interfaces smaller than a wavelength. This creates the "grainy" internal texture (parenchyma) of organs seen on ultrasound.
• Wave-front Divergence: The natural spreading of the sound beam as it travels, which increases the area the beam covers and thus decreases its intensity.