The Doppler effect when waves reflect off an object and retu

The Doppler effect when waves reflect off an object and return to their source is slightly more complicated than the situation we talked about in class. In the reflected case, the waves are Doppler-shifted twice: once when they strike the moving object and once when they leave the object. Assume that the waves travel through the body with a velocity of 1540 m/s.

Consider a 2.25 MHz wave sent by a stationary source being reflected off a blood cell moving at 30 cm/sec toward the source.In doing this problem you’ll need to keep roughly seven significant digits.

a)What frequency does the blood cell “see”?

b)This shifted frequency is what the blood cell now “emits”. Thus the stationary observer sees a moving source coming toward them. What frequency does the observer see from the blood cell?

c)This is a very small frequency difference, which can be difficult to detect. Instead what is usually done is to combine the original and the returning signals and observe the beat frequency between them. What is the beat frequency?

d)This technique is called Doppler echocardiography and is used to measure blood flow. Can we use this technique to measure changes in blood flow? If the blood starts flowing more rapidly, what happens to the frequency that we receive after it has reflected off the blood? What happens to the beat frequency? Explain, in words, the relationship between blood speed and beat frequency.

Solution

speed of the wave = 1540 m/s

frequency   f   = 2.25 MHz

observer speed = 30 cm/s = 0.3 m/s

source stationary observer approaching the source

frequency received by observer   f’ = (1540+0.3/1540) *2.25 MHz = 2250438 Hz

the blood cells act as source and emit frequency f’

Now source approaching the observer with speed of 0.3 m/s

final frequency =   1540f’/(1540.0.3)

                                 = 2.25*1540.3/1539.7 MHz

                                 = 2250876.8 Hz

frequency of the beats =2250877 – 2250000 = 877 Hz

frequency of the beats f_b = (v+u) f_s /(v-u)

velocity of the blood cells

u (f_b +f_s) = v(f_s-f_b)

u = v(f_s-f_b)/ (f_b +f_s)