Question

For this problem, assume that high frequency sound waves travel through human tissue at 1500 m/s (about the speed of sound in water).

a) Measurements of the blood flow in the ascending aorta shows a maximum speed of 92.00 cm/sec. Assuming the ultrasound frequency used is 2.000000 MHz (the paper uses 2.25 MHz), calculate the frequency as seen by the blood that is traveling toward the ultrasound transducer at 92.00 cm/s. The transducer, which both transmits and receives the sound waves, is not moving. It’s fine to use lots of significant figures here, so you can see the difference in frequency.

b)Calculate the shift in frequency for the waves received by the transducer after the waves have reflected off the flowing blood back to the transducer. The shift in frequency is the difference in frequency between the incoming and outgoing waves.

c)The transducer is then re-positioned so that the blood is flowing away from the transducer. Again, calculate the shift in frequency for the waves received by the transducer after the waves have reflected off the flowing blood back to the transducer. [

d)For (c), calculate the magnitude of the percentage difference between the incoming and outgoing frequencies. (It’s essentially the same magnitude for part (b), too, because the speed of the blood is so much smaller than the speed of sound.)

Answer 1

a) Speed of sound in human tissue = 1500 m/s

Speed of blood in aorta = 92 cm/sec= 0.92m/s

Frequency of of ultrasound = 2 x 10⁶ Hz

From Doppler effect,

c- velocity of the waves with respect to the medium

v_r - Velocity of the receiver with respect to medium;positive if the receiver is moving towards the source (and negative in the other direction)

v_s - velocity of fourcce with respect to medium; positive if the source is moving away from the receiver (and negative in the other direction).

Since the blood is flowing towards the transducer and the frquency as seen by the blood needs to be measured,

v_r = 0.92, v_s = 0 as receiver is sattionary w.r.t. the tissue

f = 2.0012266666666666666666666666667 x 10⁶ Hz

b) For the reflected wave, the source is the moving blood (v_s = - 0.92), and receiver is the transducer (stationary v_r = 0)

The frequency of the reflected wave as seen by the transducer will be

f = 2.0024548389679003121914774394962 x 10⁶ Hz

The frequency shift will be  2.0024548389679003121914774394962 x 10⁶ Hz - 2.0012266666666666666666666666667 x 10⁶ = 0.00122817230123364552481077282946 MHz

= 1228.17230123364552481077282946 Hz

c)

Since the blood is flowing away from the transducer (v_r = -0.92) and the frquency as seen by the blood needs to be measured, v_s = 0

f = 1.9987733333333333333333333333333 x 10⁶

For the reflected wave, the source is the moving blood (v_s = 0.92), and receiver is the transducer (stationary v_r = 0)

The frequency of the reflected wave as seen by the transducer will be

f = 1.9975481704554539882205580577246 x 10⁶

Frequency Shift = 1.9987733333333333333333333333333 - 1.9975481704554539882205580577246 MHz

= 0.0012251628778793451127752756087 MHz = 1225.1628778793451127752756087Hz

d) percentage differencce between incoming and outgoing frequencies =

=

= 0.061258143893967255638763780435%