# Ultrasound Intensity Reflection and Attenuation

## Intensity Reflection Coefficient

When a beam of ultrasound is directed into a body, a certain proportion of the initial intensity would be reflected at the boundary. The ratio of the reflected intensity to the initial intensity can be obtained using the formula

\begin{aligned}\frac{I_r}{I_0}=\frac{(Z_2-Z_1)^2}{(Z_2+Z_1)^2}\end{aligned}

## Air-Tissue Boundary

Using the formula from above, we can calculate that the intensity reflection coefficient of an air-tissue boundary is more than 99%.

\begin{aligned}\frac{I_r}{I_0}=\frac{(1.63 - 0.0004)^2}{(1.63 + 0.0004)^2}=0.999\end{aligned}

Note that the squaring of both the numerator and the denominator means that it does not matter which medium is assigned as $Z_1$ and $Z_2$.

This is the reason why impedance matching need to be done when ultrasound is performed on a patient.

## Intensity Attenuation

When ultrasound travels across a tissue, its intensity attenuates. This is because the wave loses energy to the tissue as it travels. The intensity of the wave at any particular depth of the tissue is determined by the formula

$I=I_0 e^{-mu x}$

where x is the depth of the point at which the intensity is determined, $latex μ$ is the absorption coefficient of the material and $I_0$ is the original intensity before the ultrasound wave enters the particular medium.

To find the intensity of the ultrasound that is received at the edge of medium 1, we can employ the following sequence of calculations:

1. $I_0$ is the initial intensity.
2. When the wave reaches the boundary of medium 1 and medium 2, it would have been attenuated by the amount $I=I_0 e^{-\mu x}$ where $x$ is the depth of medium 1.
3. This wave is then reflected at the boundary, where \begin{aligned} I_2 = \frac{(Z_2-Z_1)^2}{Z_2+Z_1)^2}\end{aligned}.
4. Lastly, when the wave reaches the edge of medium 1, its final intensity $I_3=I_2 e^{-\mu x}$

## Summary

1. When you are calculating the reflection of ultrasound at a boundary, use $\frac{I_r}{I_0}=\frac{(Z_2-Z_1)^2}{(Z_2+Z_1)^2}$
2. When you are calculating the intensity attenuation over a distance, use $I=I_0 e^{-\mu x}$