Doppler Effect in Ultrasound (OCR A Level Physics)

• The Doppler effect is defined as:

The frequency change of a wave due to the relative motion between a source and an observer

• When the source starts to move towards the observer, the wavelength of the waves is shortened
  • The sound therefore appears at a higher frequency to the observer

• The frequency is increased when the source is moving towards the observer
• The frequency is decreased when the source is moving away from the observer
  • This applies to ultrasound in the exact same way as it does to sound waves

Doppler effect of a moving source and stationary observer

• For a wave of frequency, f₀, that reflects at a moving boundary, the new frequency, f_r, of the reflected pulse is given by:

• Where:
  • f₀ = frequency of original wave (Hz)
  • f_r = frequency of reflected wave (Hz)
  • v = velocity of the moving boundary (m s⁻¹)
  • c = velocity of the wave (m s⁻¹)

• The factor of 2 appears due to the wave travelling to the boundary and back again

• In medicine, Doppler imaging can be used as a non-invasive technique to measure the speed of blood flow in the heart or in an artery
  • This is effective because blood contains iron which is very reflective

• The journey of the ultrasound is as follows:
  • Pulses of ultrasound are emitted from a transducer into a blood vessel
  • The ultrasound pulses are reflected by moving blood cells
  • The moving blood causes a shift in the frequency of the ultrasound
  • The transducer detects an increase in frequency if the blood is moving towards the transducer and a decrease if it is moving away

• This frequency shift, Δf, can be calculated using the equation:

• Where:
  • f₀ = frequency of ultrasound emitted by the transducer (Hz)
  • f_r = frequency of ultrasound received by the transducer (Hz)
  • Δf = difference between emitted and received frequencies (Hz)
  • v = velocity of the blood (m s⁻¹)
  • c = velocity of ultrasound in blood (m s⁻¹)
  • θ = angle between the transducer and the blood vessel (°)

• This can be rearranged to give:

Part (a)

Step 1: List the known quantities

• • Transmitted frequency of the ultrasound, f₀ = 8.5 MHz
  • Reflected frequency of the ultrasound, f_r = 8.495 MHz
  • Change in frequency, Δf = 8.5 − 8.495 = 0.005 MHz
  • Velocity of sound in blood, c = 1570 m s⁻¹
  • Angle between the transducer and skin, θ = 15°

Step 2: Write out the equation for Doppler shift of ultrasound

Step 3: Rearrange for the speed, v, and calculate

= 0.478

v = 0.48 m s⁻¹

Part (b)

Step 1: List the known quantities

• • Radius of the aorta, r = 15 mm = 1.5 cm
  • Blood flow rate, v = 0.48 m s⁻¹ = 48 cm s⁻¹

Step 2: Calculate the cross-sectional area of the aorta

• • Cross-sectional area = πr² = π(1.5)² = 7.1 cm²

Step 3: Determine the volume of blood passing through the aorta each second

• • Volume flow rate = cross-sectional area (cm²) × blood flow rate (cm s⁻¹)
  • Volume flow rate = 7.1 × 48 = 341 cm³ s⁻¹

Step 4: State the volume of blood in 1 s

• • The volume of blood that flows in 1 s = 341 cm³