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

v = 8.89 cm/s

Step-by-step 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 ((c+ v)/(c))`..........(1)

`f_2 = f_1 ((c)/(c-v))`..........(2)

now, from equation (1) and (2)

`f_2 = f_0 ((c + v)/(c-v))`

f₂ = 2.23 x 10⁶ + 258 = 2230258 Hz

f₀ = 2.23 x 10⁶ Hz

c = 1540 m/s

`2230258= 2230000((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