Question

The Doppler effect using ultrasonic waves of frequency 2.23×106 Hz is used to monitor the heartbeat of a fetus. A (maximum) beat frequency of 258 Hz is observed. Part A Assuming that the speed of sound in tissue is 1.54×103m/s, calculate the maximum velocity of the surface of the beating heart.

Answer 1

Answer:

v = 8.89 cm/s

Explanation:

given,

frequency of ultrasonic wave = 2.23 x 10⁶ Hz

beat frequency = 258 Hz

speed of sound in the tissue =  1.54×10³ m/s

maximum velocity  = ?

using Doppler equation

$f_1 = f_0 (\dfrac{c+ v}{c})$..........(1)

$f_2 = f_1 (\dfrac{c}{c-v})$..........(2)

now, from equation (1) and (2)

$f_2 = f_0 (\dfrac{c + v}{c-v})$

f₂ = 2.23 x 10⁶ + 258 = 2230258 Hz

f₀ = 2.23 x 10⁶ Hz

c = 1540 m/s

$2230258= 2230000(\dfrac{1540+ v}{1540-v})$

1.00012(1540 - v ) = 1540+ v

1540.178 - 1.00012 v = 1540 + v

2.00012 v = 0.178

v = 0.08899 m/s

v = 8.89 cm/s