Acoustic impedance

In a nutshell

Acoustic impedance is the resistance to the propagation of ultrasounds through a tissue. The higher the difference between the acoustic impedance values of two substances, the more an ultrasound beam will be reflected.

Equations

Description	equation
Acoustic impedance	$Z=\rho c$
Intensity reflection coefficient	$\dfrac{I_r}{I_0}=\left( \dfrac{Z_2-Z_1}{Z_2+Z_1} \right)^2$

Variable definitions

quantity name	symbol	derived units	si base units
$acoustic\ impedance$	$Z$	$kg\ m^{-2}\ s^{-1}$	$kg\ m^{-2}\ s^{-1}$
$density$	$\rho$	$kg\ m^{-3}$	$kg\ m^{-3}$
$ultrasound\ speed$	$c$	$m\ s^{-1}$	$m\ s^{-1}$
$intensity$	$I$		$W\ m^{-2}$

Acoustic impedance

When an ultrasound beam is incident at a boundary between two substances, it will be partially reflected and refracted. The amount reflected depends on the acoustic impedance of both substances.

The acoustic impedance $Z$ is defined as the resistance to the propagation of ultrasounds through tissue and has units $kg\ m^{-2}\ s^{-1}$ . It is the product between the density of a substance and the speed of the ultrasound in it:

$Z=\rho c$

The intensity reflection coefficient

When an ultrasound beam gets reflected at the boundary between two substances, the intensity of the reflected ultrasound depends on the acoustic impedance of both substances.

For two substances with acoustic impedance $Z_1$ and $Z_2$ and a beam with an incident angle of $0\degree$ , the ratio of the reflected intensity to the incident intensity is given by the equation:

$\dfrac{I_r}{I_0}=\left( \dfrac{Z_2-Z_1}{Z_2+Z_1} \right)^2$

Where the ratio $\dfrac{I_r}{I_0}$ is known as the intensity reflection coefficient.

The equation shows that the reflection value is higher when the two acoustic impedance values differ more.

Acoustic matching

When a transducer is placed on the skin of a patient, there are always air pockets in between the two. Due two air and skin having very different acoustic impedance values, most of the ultrasound gets reflected before even entering the body.

To get around this problem, a coupling gel is smeared on the skin of the patient and the transducer. This gel fills in the air gaps and has a similar acoustic impedance to that of skin such that the reflection caused by their boundary is negligible.

When two substances have similar acoustic impedance values, it's given the name of acoustic matching.

Example

A beam of ultrasound is incident normally at a boundary between skin and fat. Calculate the intensity reflection coefficient at this boundary as the beam goes from skin to fat. The acoustic impedance of skin and fat are $1.7\times10^{6}\ kg\ m^{-2}\ s^{-1}$ and $1.38\times10^{6}\ kg\ m^{-2}\ s^{-1}$ respectively.

Firstly, write down what you know:

$Z_1=1.7\times10^{6}\ kg\ m^{-2}\ s^{-1} \\Z_2=1.38\times10^{6}\ kg\ m^{-2}\ s^{-1}$ 

Next, write down the equation for the intensity reflection coefficient:

$\dfrac{I_r}{I_0}=\left( \dfrac{Z_2-Z_1}{Z_2+Z_1} \right)^2$ 

Substitute the numbers and calculate the value:

$\dfrac{I_r}{I_0}=\left( \dfrac{1.38\times10^{6}-1.7\times10^{6}}{1.38\times10^{6}+1.7\times10^{6}} \right)^2=0.01$

The intensity reflection coefficient at the boundary between skin and fat is $\underline{0.01}$ .