Specific Acoustic Impedance (Cambridge International A-Level Physics)

Spec point — 24.1.4

Specific acoustic impedance

• The acoustic impedance, Z, of a medium is defined as: The product of the speed of the ultrasound in the medium and the density of the medium
• This quantity describes how much resistance an ultrasound beam encounters as it passes through a tissue
• Acoustic impedance can be calculated using the equation:

$Z=ρc$

• Where:

  • Z = acoustic impedance (kg m^-2 s^-1)
  • ρ = the density of the material (kg m^-3)
  • c = the speed of sound in the material (m s^-1)

• This equation tells us:

  • The higher the density of a tissue, the greater the acoustic impedance
  • The faster the ultrasound travels through the material, the greater the acoustic impedance also

• This is because sound travels faster in denser materials

  • Sound is fastest in solids and slowest in gases
  • This is because the closer the particles in the material, the faster the vibrations can move through the material

• At the boundary between media of different acoustic impedances, some of the wave energy is reflected and some is transmitted

• The greater the difference in acoustic impedance between the two media, the greater the reflection and the smaller the transmission

  • Two materials with the same acoustic impedance would give no reflection
  • Two materials with a large difference in values would give much larger reflections

• Air has an acoustic impedance of Z_air = 400 kg m^-2 s^-1

• Skin has an acoustic impedance of Z_skin = 1.7 × 10⁶ kg m^-2 s^-1

  • The large difference means ultrasound would be significantly reflected, hence a coupling gel is necessary
  • The coupling gel used has a similar Z value to the skin, meaning that very little ultrasound is reflected

Light travelling through two materials with different acoustic impedance

Refraction and reflection of ultrasound waves at a boundary between two materials with different acoustic impedances (in this case, Z₁< Z₂ )

Worked Example The table shows the speed of sound acoustic impedance in four different materials.

medium	speed of ultrasound / m s–1	acoustic impedance / kg m–2s–1
air	330	4.3 × 102
gel	1500	1.5 × 106
soft tisuse	1600	1.6 × 106
bone	4100	7.0 × 106

Use this information to calculate the value for the density of bone.

Answer:

Step 1: Write down known quantities

• Acoustic impedance of bone, Z = 7.0 × 10⁶ kg m^-2 s^-1
• Speed of ultrasound in bone, c = 4100 m s^-1

Step 2: Write out the equation for acoustic impedance

$Z=ρc$

Step 3: Rearrange for density and calculate

$ρ=\frac{Z}{c}=\frac{7.0\times10}{4100}=1700\mathrm{kg}m^{-3}$

Exam Hint A common mistake is to confuse the c in the acoustic impedance equation for the speed of light - don’t do this!