Blood flow rates can be found by measuring the Doppler shift in frequency of ult

Question

Blood flow rates can be found by measuring the Doppler shift in frequency of ultrasound reflected by red blood cells (known as angiodynography). Assume that the speed of the red blood cells is v, the speed of sound in blood is u, that the ultrasound emits waves of frequency f, and that the blood cells are moving directly toward the ultrasound source.The reflected sound interferes with the emitted sound, producing beats. Find the beat frequency between the outgoing and reflected sound waves and show that it is proportional to the speed of the blood cell if v/u << 1.

Explanation / Answer

First shift (moving observer) f2 = f*(u + v)/u Now as the moving source we have fr = f2*(u/(u - v)) so f2 = fr*(u - v)/u Subbing into the first eqn we have fr *(u - v)/u = f*(u + v)/u or fr*(1-v/u) = f*(1+ v/u) So fr = f*(1 + v/u)/(1 - v/u) given v/u

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