Attenuation of Ultrasound in Matter (CIE A Level Physics)

• Attenuation of ultrasound is defined as:

The reduction of energy due to the absorption of ultrasound as it travels through a material

• The attenuation coefficient of the ultrasound is expressed in decibels per centimetre lost for every incremental increase in megahertz frequency
  • Generally, 0.5 dB/cm is lost for every 1MHz

• The intensity I of the ultrasound decreases with distance x, according to the equation:

I = I₀ e^−μx

• Where:
  • I₀ = the intensity of the incident beam (W m^-2)
  • I = the intensity of the reflected beam (W m^-2)
  • μ = the absorption coefficient (m^-1)
  • x = distance travelled through the material (m)

• The absorption coefficient μ, will vary from material to material
• Attenuation is not a major problem in ultrasound scanning as the scan relies on the reflection of the ultrasounds at boundaries of materials

When the intensity is expressed in decibels, the amplitudes of the echoes can be seen to decrease linearly

Part (a)

Step 1:           

Write down the equation for intensity reflection coefficient α

Step 2:           

Calculate the intensity reflection coefficient

This means that 0.018 of the intensity is reflected at the interface between fat and muscle. This reflected intensity will move back through the fat towards surface S.

Part (b)

Step 1:           

Write out the known quantities

The intensity of the ultrasound pulse is affected 3 times:

1. 1. Attenuation from the surface S to the fat-muscle boundary
   2. Reflection at the boundary
   3. Attenuation from the boundary back to the surface S

After being transmitted in the fat, the intensity at surface S is given to be 0.012 I.

Therefore, the intensity is 0.018 I at the fat-muscle boundary, and as the ultrasound moves through the fat, it gets attenuated and the new intensity at the surface S is now 0.012 I

Incident intensity, equal to the intensity of the reflected pulse, I₀ = 0.018 I × e^−μx

Transmitted intensity, I = 0.012 I

Absorption coefficient, μ = 48 m^-1

Thickness of fat = x

Step 2:           

Write out the equation for attenuation

I = I₀ e^−μx

Step 3:           

Substitute in values for intensity and simplify

0.012 I = [0.018 I × e^−μx] × e^−μx

0.012 = 0.018 e^−2μx

Step 4:           

Rearrange and take the natural log of both sides

Step 5:           

Rearrange and calculate the thickness x