Acoustic Impedance (OCR A Level Physics)

• 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 shows:
  • 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
  • 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 skin, meaning that very little ultrasound is reflected

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

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

• The intensity reflection coefficient α is defined as:

The ratio of the intensity of the reflected wave relative to the incident (transmitted) wave

• This can be calculated using the fraction:

• Where:
  • α = intensity reflection coefficient
  • I_R = intensity of the reflected wave (W m^-2)
  • I₀ = intensity of the incident wave (W m^-2)
  • Z₁ = acoustic impedance of one material (kg m^-2 s^-1)
  • Z₂ = acoustic impedance of a second material (kg m^-2 s^-1)

• This ratio shows:
  • If there is a large difference between the impedance of the two materials, then most of the energy will be reflected
  • If the impedance is the same, then there will be no reflection

Coupling Medium

• When ultrasound is used in medical imaging, a coupler is needed between the transducer and the body
  • This is because the soft tissues of the body are much denser than air
• If air is present between the transducer and the body, then almost all the ultrasound energy will be reflected
• To counter this, a coupling gel is placed between the transducer and the body
  • This is because skin and coupling gel have a similar density, so little ultrasound is reflected
• This is an example of impedance matching, which is defined as when:

Two media have a similar acoustic impedance, resulting in little to no reflection of the ultrasound wave

• In terms of intensity reflection coefficient, α, between the two media:
  • At lower values of α, the media are impedance matched, so less reflection occurs
  • At higher values of α, the media are not impedance matched, so more reflection occurs

Part (a)

Step 1: List the known quantities

• • Intensity of incident wave = I
  • Intensity of the transmitted wave = I_T
  • Intensity of the reflected wave = I_R

Step 2: Relate the quantities:

• • The incident intensity is equal to the sum of the transmitted and reflected intensities:

Incident intensity = Transmitted intensity + Reflected intensity

I = I_T + I_R

Part (b)(i)

Step 1: List the known quantities

• • Acoustic impedance of gel, Z₁ = 1.5 × 10⁶ kg m^-2 s^-1
  • Acoustic impedance of soft tissue, Z₂ = 1.6 × 10⁶ kg m^-2 s^-1

Step 2: Write down the equation for intensity reflection coefficient α

Step 3: Calculate the intensity reflection coefficient

• • This result means that only 0.1% of the incident intensity will be reflected, with the remaining being transmitted

Part (b)(ii)

Step 1: List the known quantities

• • Air, Z₁ = 4.3 × 10² kg m^-2 s^-1
  • Soft tissue, Z₂ = 1.6 × 10⁶ kg m^-2 s^-1

Step 2: Calculate the intensity reflection coefficient

• • This result means that 100% of the incident intensity will be reflected, with none being transmitted

Part (c)

Why gel is usually put on the skin during medical diagnosis using ultrasound

• • At the air-soft tissue boundary, the intensity reflection coefficient is α ≈ 1
    • Therefore, without gel, there is almost complete reflection - no ultrasound is transmitted through the skin

  • At the gel-soft tissue boundary, the intensity reflection coefficient is α = 0.001
    • Therefore, the gel enables almost complete transmission of the ultrasound through the skin, with very little reflection