Question

Ultrasound, which consists of sound waves with frequencies above the human audible range, can be used to produce an image of the interior of a human body. Moreover, ultrasound can be used to measure the speed of the blood in the body; it does so by comparing the frequency of the ultrasound sent into the body with the frequency of the ultrasound reflected back to the body’s surface by the blood. As the blood pulses, this detected frequency varies.

Suppose that an ultrasound image of the arm of a patient shows an artery that is angled at$\mathbf{\theta }\mathbf{}\mathbf{=}\mathbf{20}\mathbf{°}$to the ultrasound’s line of travel (Fig. 17-47). Suppose also that the frequency of the ultrasound reflected by the blood in the artery is increased by a maximum of$\mathbf{5495}\mathbf{\text{\hspace{0.17em}Hz}}\mathbf{}$from the original ultrasound frequency of$\mathbf{5.000}\mathbf{}\mathbf{000}\mathbf{\text{\hspace{0.17em}MHz}}$

(a) In Fig. 17-47, is the direction of the blood flow rightward or leftward?

(b) The speed of sound in the human arm is$\mathbf{1540}\mathbf{\text{\hspace{0.17em}m}}\mathbf{/}\mathbf{\text{s}}$. What is the maximum speed of the blood?

(c) If angleuwere greater, would the reflected frequency be greater or less?

Short answer

1. The blood flow is in rightward direction.
2. The maximum speed of the blood is$0.90\text{\hspace{0.17em}m}/\text{s}$
3. The reflected frequency is less for angle u being greater.

Step-by-step solution

The given data

1. Speed of sound in the human arm,$\mathrm{v}=1540\text{\hspace{0.17em}m}/\text{s}$.
2. Frequency of original ultrasound,$\mathrm{f}=5\text{\hspace{0.17em}MHzor}5\times {10}^6\text{\hspace{0.17em}Hz}$.
3. Increased frequency$\mathrm{\Delta f}=5495\text{\hspace{0.17em}Hz}$.
4. Angle between human arm and ultrasound’s travel line, $\mathrm{\theta }={20}^0$.

Understanding the concept of the Doppler Effect

Using the formula of the Doppler Effect, we can find the direction of the blood flow, and also, we can find the maximum speed of the blood.

Formula:

The frequency received by the observer according to Doppler’s Effect,

$\mathbf{f}\mathbf{+}\mathbf{\Delta f}\mathbf{}\mathbf{=}\mathbf{f}\left(\frac{v+v_x}{v-v_x}\right)$ …(i)

a) Calculation of the direction of the blood flow

Since we are given that the frequency increases in Doppler shift, the blood is moving towards the right.

Therefore, the blood flow is rightward.

b) Calculation of maximum speed 

For the reception of the ultrasound by the blood and the subsequent remitting of the signal by the blood, we write the frequency relation as equation (ii):

$\left(\frac{\mathrm{f}+\mathrm{\Delta f}}{\mathrm{f}}\right)=\frac{\mathrm{f}}{\mathrm{f}}\left(\frac{\mathrm{v}+{\mathrm{v}}_{\mathrm{x}}}{\mathrm{v}-{\mathrm{v}}_{\mathrm{x}}}\right)$

Let,$\mathrm{K}=\frac{\mathrm{f}+\mathrm{\Delta f}}{\mathrm{f}}$

$\begin{array}{c}\mathrm{K}=\left(\frac{\mathrm{v}+{\mathrm{v}}_{\mathrm{x}}}{\mathrm{v}-{\mathrm{v}}_{\mathrm{x}}}\right)\\ (\mathrm{v}-{\mathrm{v}}_{\mathrm{x}})\mathrm{K}=\mathrm{v}+{\mathrm{v}}_{\mathrm{x}}\\ \mathrm{vK}-{\mathrm{v}}_{\mathrm{x}}\mathrm{K}=\mathrm{v}+{\mathrm{v}}_{\mathrm{x}}\\ {\mathrm{v}}_{\mathrm{x}}\mathrm{K}+{\mathrm{v}}_{\mathrm{x}}=\mathrm{vK}-\mathrm{v}\\ {\mathrm{v}}_{\mathrm{x}}(\mathrm{K}+1)=\mathrm{v}(\mathrm{K}-1)\end{array}$

The velocity of the blood is expressed as,

$\begin{array}{c}{\mathrm{v}}_{\mathrm{blood}}{\mathrm{cos\theta }}^0(\mathrm{K}+1)=\mathrm{v}(\mathrm{K}-1)\\ {\mathrm{v}}_{\mathrm{blood}}=\frac{\mathrm{v}(\mathrm{K}-1)}{{\mathrm{cos\theta }}^0(\mathrm{K}+1)}\end{array}$

Where, the value of K is found to be

$\begin{array}{c}\mathrm{K}=\frac{\mathrm{f}+\mathrm{\Delta f}}{\mathrm{f}}\\ =\frac{(5\times {10}^6\text{\hspace{0.17em}Hz})+\left(5495\text{\hspace{0.17em}Hz}\right)}{(5\times {10}^6\text{\hspace{0.17em}Hz})}\\ =1.001099\end{array}$

Hence, substituting the given values in the above equation , we get

$\begin{array}{c}{\mathrm{v}}_{\mathrm{blood}}=\frac{(1540\text{\hspace{0.17em}m}/\text{s})(1.001099-1)}{\cos {20}^0(1.001099+1)}\\ {\mathrm{v}}_{\mathrm{blood}}=0.90\text{\hspace{0.17em}m}/\text{s}\end{array}$

Therefore, the maximum speed of the blood is $0.90\text{\hspace{0.17em}m}/\text{s}$.

c) Checking the behaviour of reflected frequency with respect to the angle

As the angle increases, the horizontal component of velocity decreases. Therefore, it is expected that the frequency$∆\mathrm{f}$decreases whentheangle increases from$0°$ to $90°$.

Hence, the reflected frequency is less for larger angles.