Chapter 12, Problem 57P

To determine

To Calculate: The maximum velocity of the surface of the beating heart.

Answer to Problem 57P

  $0.17\text{m/s}$

Explanation of Solution

Given:

Ultrasonic wave frequency, $\left(f_1\right)$

  $=2.25\times {10}^6\text{Hz}$

Beat frequency, $\left(f_b\right)$

  $=500\text{ Hz}$

Speed of sound in tissue, $\left(v\right)$

  $=1.54\times {10}^3\text{m/s}$

Formula used:

Beat frequency:

  $f_b=f_1-f_2$

From Doppler effect:

  $f_1=f_2\left(\frac{v_s+v_h}{v_s-v_h}\right)$

Calculation:

To find the maximum velocity $\left(v_o\right)$ , use the Doppler formula.

  $\begin{array}{l}{f}_b=500\text{Hz}\\ f_2=f_1-500\text{Hz}\\ f_2=2.25\times {10}^6-500\\ f_1=f_2\left(\frac{v_s+v_h}{v_s-v_h}\right)\\ 2.25\times {10}^6=\left(2.25\times {10}^6-500\right)\left(\frac{1.54\times {10}^3+v_h}{1.54\times {10}^3-v_h}\right)\\ v_h=0.17\text{m/s}\end{array}$

Conclusion:

The maximum velocity is for the surface of the beating heart is $0.17\text{m/s}$ .