A certain echocardiogram uses ultrasounds with a frequency f = 6.1130 MHz. The speed of sound in blood is 1570 m/s. The patient is aligned so that blood is flowing through the heart directly towards the transducer (the transducer produces and detects the ultrasound waves).
If blood flows through the artery being measured at a speed of 39.79 cm/s, what frequency signal would be detected by a red blood cell in the artery? Give your answer to four decimal places.
what frequency does the blood hear?
for observer toward source:
f' = f (v+Vobserver)/v
f' =6.113*10^6(1570+.3979)/1570
calculate f'
the blood particle serves as a transponder sending back a signal at that f'
now the source is moving toward the observer
f" = f' v/(v-vs)
f" = f' (1570)/(1570-.3979)
so we want f"
To determine the frequency signal detected by a red blood cell in the artery, we can use the principle of Doppler effect. The Doppler effect states that the frequency of a wave changes when there is relative motion between the source of the wave and the observer.
In this case, the ultrasound waves are being emitted by the transducer and are being reflected back by the red blood cells in the artery. The red blood cells are moving towards the transducer due to the flow of blood in the artery.
The formula for the Doppler effect is given by:
f' = (v + vf) / (v + vs) * f
Where:
- f' is the observed frequency by the red blood cell
- f is the emitted frequency by the transducer
- v is the speed of sound in the blood
- vf is the velocity of the flow of blood
- vs is the velocity of the red blood cell (relative to the flow of blood)
We are given the following values:
- f = 6.1130 MHz = 6.1130 × 10^6 Hz
- v = 1570 m/s
- vf = 39.79 cm/s = 0.3979 m/s
Since the red blood cell is moving in the same direction as the flow of blood, its velocity relative to the flow of blood (vs) is simply equal to the velocity of the flow of blood (vf).
Now, we can plug in the values into the formula to calculate the observed frequency (f'):
f' = (v + vf) / (v + vs) * f
= (1570 + 0.3979) / (1570 + 0.3979) * 6.1130 × 10^6
≈ 6.1130 × 10^6 Hz
Therefore, the frequency signal detected by the red blood cell in the artery is approximately 6.1130 MHz.