(a) In Problem 49, find the beat frequency between the outgoing and reflected sound waves. 

(b) Show that the beat frequency is proportional to the speed of the blood cell if v ≪ u. [Hint: Use the binomial approximation from Appendix A.9.]

Data From Problem 49

Blood flow rates can be found by measuring the Doppler shift in frequency of ultrasound reflected by red blood cells (known as angiodynography). 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, show that the frequency fr of reflected waves detected by the apparatus is given by