Beat frequency from Doppler-shifted ultrasound echo in blood flow

[j2dye]

Ultrasound reflected from an oncoming bloodstream that is moving at 31 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.)

I know that there are two doppler shifts; one where the source is fixed and the other where the observer is fixed.

So I'm supposed to use the doppler shift equation to solve for this:

f1=f((V + Vo) / (V - Vs))

but I don't know what numbers to plug into which variables.

After I find the frequency of the doppler shifted echo all I have to do is subtract it from the original frequency of 1.3 MHz, right?

 

[Andrew Mason]

So I'm supposed to use the doppler shift equation to solve for this:

f1=f((V + Vo) / (V - Vs))

but I don't know what numbers to plug into which variables.

You have used the doppler expression where the source and observer are both moving relative to the medium. You also have to keep in mind that there is a reflection so there are two doppler shifts.

Use the moving observer form:
$$f_{blood} = f_{source}(\frac{v_{sound}+ v_{blood}}{v_{sound}})$$

to find the apparent frequency that the blood receives. The blood then acts as a moving source and reflects this back to the observer so used the moving source form:

$$f_{observer} = f_{blood}(\frac{v_{sound}}{v_{sound} - v_{blood}})$$

After I find the frequency of the doppler shifted echo all I have to do is subtract it from the original frequency of 1.3 MHz, right?

You are correct that the beat frequency is the difference between the original and the reflected sound.

AM

 

[wais]

The beat frequency can be calculated by subtracting the two frequencies, as you mentioned. However, in order to find the frequency of the doppler shifted echo, we first need to determine the velocity of the ultrasound wave in the blood. This can be done using the velocity of sound in blood (1540 m/s) and the frequency of the original ultrasound (1.3 MHz).

Using the formula for velocity (V = fλ), we can rearrange it to solve for the wavelength (λ) of the ultrasound wave in the blood. This gives us a wavelength of approximately 0.118 m.

Now, we can use this wavelength in the doppler shift equation, where f1 represents the frequency of the doppler shifted echo, f is the original frequency (1.3 MHz), V is the velocity of sound in blood (1540 m/s), and Vs is the velocity of the bloodstream (31 cm/s or 0.31 m/s).

Plugging these values into the equation, we get:

f1 = 1.3 MHz * ((1540 m/s + 0.31 m/s) / (1540 m/s - 0.31 m/s))

Simplifying, we get f1 = 1.3 MHz * 1.0002, which gives us a frequency of approximately 1.3 MHz.

Subtracting this frequency from the original frequency of 1.3 MHz, we get a beat frequency of 0.0002 MHz or 200 Hz.

In summary, the beat frequency produced by the ultrasound reflected from an oncoming bloodstream moving at 31 cm/s is 200 Hz.