Doppler ultrasound is used to measure the speed of blood flow. If the speed of the red blood cells is v, the speed of sound in blood is u, the ultrasound source emits waves of frequency f, and we assume that the blood cells are moving directly toward the ultrasound source, then the frequency fr of reflected waves detected by the apparatus is given by the formula below.
fr = f
1 + v/u
1 − v/u
The reflected sound interferes with the emitted sound, producing beats. If the speed of red blood cells is 0.12 m/s, the ultrasound frequency used is 4.7 MHz, and the speed of sound in blood is 1570 m/s, what is the beat frequency?
To find the beat frequency, we first need to calculate the reflected frequency fr using the given formula:
fr = f / (1 + v/u) / (1 - v/u)
Given:
v = 0.12 m/s (speed of red blood cells)
f = 4.7 MHz = 4.7 * 10^6 Hz (ultrasound frequency)
u = 1570 m/s (speed of sound in blood)
1. Calculate the value of v/u and store it in a variable:
v_over_u = v / u = 0.12 / 1570 ≈ 7.643312m/s
2. Plug the value of v_over_u into the formula for fr:
fr = f / (1 + v_over_u) / (1 - v_over_u)
fr = 4.7 * 10^6 / (1 + 7.643312) / (1 - 7.643312)
3. Simplify the denominator:
fr = 4.7 * 10^6 / (1 + 7.643312) / (1 - 7.643312)
= 4.7 * 10^6 / (1 + 7.643312) / (-6.643312)
4. Calculate the denominator:
denominator = (1 + 7.643312) * (-6.643312)
≈ (-6.756311) * (-6.643312)
≈ 44.858996
5. Divide the numerator by the denominator to find fr:
fr = 4.7 * 10^6 / 44.858996
≈ 104857.6 Hz
≈ 104.8 KHz
The reflected frequency fr is approximately 104.8 KHz.
Now, to calculate the beat frequency, we need to subtract the original frequency f from the reflected frequency fr:
beat frequency = fr - f
beat frequency = 104.8 KHz - 4.7 MHz
= 104.8 KHz - 4,700 KHz
= -4,595.2 KHz
The beat frequency is approximately -4,595.2 KHz. The negative sign indicates that the beat frequency is lower than the original frequency.