Question

A Doppler ultrasound is used to measure blood flow speeds, for instance, in a patient's artery. Assume the probe emits ultrasound waves at a frequency of 5 x 1O^6 Hz and the speed of sound in human tissue is 1540 m/s. (a) Why is the Doppler effect the basis of the mechanism of an ultrasound machine? (b) What is the frequency, f" received by the probe if the beat frequency, that is, the frequency shift following reflection, is 400 Hz and the blood cell is approaching the probe? (c) What is the frequency, f" received by the probe if the beat frequency, that is, the frequency shift following reflection, is 400 Hz and the blood cell is moving away from the probe? What is the speed of blood in the artery?

Answer 1

a.) The mechanism of ultrasound employs the Doppler effect to give the frequency shift. Initially, the probe acts as the source and the blood cell acts as the object and the sound wave is reflected back from the blood cell and is again received by the probe. In the second case, where the wave is reflected from the blood cell and received by the probe, the blood cell acts as the source and the probe as the receiver. Hence there are two doppler shifts taking place in this mechanism.

b.) Since the blood cell is approaching the probe, there will be a rise in the frequency. So,

the perceived final frequency = initial frequency + frequency shift

f'' = f + 400 = 5 x 10⁶  + 400 = 5000400 Hz

c.) Since the blood cell is moving away from the probe, there will be a drop in the frequncy. So,

perceived frequency = initial frequency - frequency shift

f'' = f - 400 = 5 x 10⁶  - 400 = 4999600 Hz

d.) When the blood cell is moving towards the probe,

the frequency perceived by the blood cell will be f' = f [ ( v_wave + v_blood ) / v_wave ]

when the wave is, in turn, reflected from the blood cell, the blood cell acts as the source now and hence emits a wave of frequency f' (perceived by the blood cell)

So, the final frequency received by the probe is f'' = f' [ v_wave / ( v_wave - v_blood ) ]

f'' = f' [ v_wave / ( v_wave - v_blood ) ] = f [ ( v_wave + v_blood ) / v_wave ][ v_wave / ( v_wave - v_blood ) ]

f'' = f [ ( v_wave + v_blood ) / ( v_wave - v_blood ) ]

putting f'' = 5000400 Hz , f = 5000000 Hz , and v_wave = 1540 m/s

we get,

5000400 = 5000000 [ (1540 + v_blood ) / (1540 - v_blood ) ]

7700616000 - 5000400 v_blood = 7700000000 + 5000000 v_blood

616000 = 10000400 v_blood

v_blood = 0.0615975361 m/s ~ 0.061 m/s

(Note: if you do the same procedure for the case where blood cell moves away from the probe you'll get a slightly different answer,which is 0.0616024614 m/s. However, this is because the questioner gave the frequency shift for both the cases to be 400 Hz but in reality, they will not be equal. But even then the answer rounded off to three decimals is sufficient for all practical purposes.)