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 \(f_{r}\) of reflected waves detected by the apparatus is given by $$f_{\mathrm{r}}=f \frac{1+v / u}{1-v / u}$$ [Hint: There are two Doppler shifts. A red blood cell first acts as a moving observer; then it acts as a moving source when it reradiates the reflected sound at the same frequency that it received.]

Step by step solution

01

Doppler shift for moving observer

First, let's consider the red blood cells as moving observers. When an observer is moving toward a source, the observed frequency increases. The formula for the observed frequency \(f_{o}\) in this case is: $$f_{\mathrm{o}} = f \left(1 + \frac{v}{u}\right)$$ where \(f\) is the frequency of the source, \(v\) is the speed of the observer (in this case, the red blood cells), and \(u\) is the speed of sound in blood.

02

Doppler shift for moving source

Next, let's consider the red blood cells as moving sources. We have found the observed frequency \(f_o\) in step 1, and now we treat the red blood cells as a moving source emitting waves at the frequency \(f_o\). When a source is moving away from an observer (in our case, the apparatus), the observed frequency decreases. The formula for the reflected wave frequency \(f_r\) is: $$f_{\mathrm{r}} = f_{\mathrm{o}} \left(1 - \frac{v}{u}\right)$$

03

Combine the Doppler shifts

Now, we need to substitute the expression of \(f_o\) from step 1 into the formula for \(f_r\) in step 2: $$f_{\mathrm{r}} = f \left(1 + \frac{v}{u}\right) \left(1 - \frac{v}{u}\right)$$

04

Simplify the expression

To simplify the expression, we will perform the algebraic multiplication: $$f_{\mathrm{r}} = f\left(1 - \frac{v^2}{u^2}\right)$$ Now, we will modify the denominator to be in the form of \(1 - \frac{v^2}{u^2}\): $$f_{\mathrm{r}}=f \frac{u^2 - v^2}{u^2}$$ Finally, we will factor out the denominator and use the difference of squares identity: $$f_{\mathrm{r}}= f \frac{(u + v)(u - v)}{u^2}$$ After dividing both the numerator and denominator by \(u\), we obtain the final expression for the reflected wave frequency: $$f_{\mathrm{r}}=f \frac{1+v / u}{1-v / u}$$

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