Ultrasound reflected from an oncoming bloodstream that is moving at 30 cm/s is mixed with the original frequency of 1.3 MHz to produce beats. What is the beat frequency? (Velocity of sound in blood = 1540 m/s.)
Beat frequency = Hz
HELP: This problem involves two Doppler shifts.
HELP: In the first shift, the source is fixed and the observer is the moving blood. Why?
HELP: In the second shift, the source is the moving blood and the observer is fixed. Why?
HELP: It might be helpful to think of a tiny ultrasound recorder mounted on a blood cell. The recording happens when the ultrasound arrives at the cell, and the echo leaves when the recorded sound is "played back".
HELP: Once you know the frequency of the Doppler-shifted echo, how do you find the beat frequency between the echo and the original frequency?
To find the beat frequency in this problem, we need to understand the two Doppler shifts involved.
In the first shift, the source is fixed, and the observer is the moving blood. This is because the ultrasound is reflected off the moving blood cells, causing a frequency shift. In this case, we can use the formula for the Doppler shift when the observer is in motion:
f' = f * (v + v_obs)/(v + v_src),
where f' is the observed frequency, f is the source frequency, v_obs is the velocity of the observer (in this case, the blood), and v_src is the velocity of the source (in this case, the ultrasound).
In the second shift, the source is the moving blood, and the observer is fixed. This is because the blood cells are now acting as a source of ultrasound waves, and they are moving towards the observer. In this case, we can use the same formula as before, but exchange the roles of the observer and the source:
f'' = f' * (v + v_obs)/(v + v_src).
Now, let's apply these formulas to the given problem. The blood is moving at a velocity of 30 cm/s, which is equal to 0.3 m/s. The velocity of sound in blood is 1540 m/s.
In the first shift, the source is fixed (1.3 MHz), and the observer is the moving blood (velocity = 0.3 m/s). Plugging these values into the first Doppler shift formula, we get:
f' = (1.3 MHz) * (1540 m/s + 0.3 m/s)/(1540 m/s + 0 m/s).
Simplifying this equation, we find the frequency after the first Doppler shift, which is the frequency of the Doppler-shifted echo.
Next, to find the beat frequency between the echo and the original frequency, we subtract the original frequency from the frequency after the first Doppler shift:
beat frequency = f' - f.
This will give us the answer in Hz, as required.
So, to summarize the steps:
1. Use the first Doppler shift formula to find the frequency after the first shift.
2. Subtract the original frequency from the frequency after the first shift to find the beat frequency.
By following these steps, you should be able to calculate the beat frequency in this problem.