Ultrasound has many medical applications, one of which is to monitor fetal heartbeats by reflecting ultrasound off a fetus in the womb.
a. Consider an object moving at speed v_otoward an at-rest source that is emitting sound waves of frequency f₀. Show that the reflected wave (i.e., the echo) that returns to the source has a Doppler-shifted frequency

where v is the speed of sound in the medium.

b. Suppose the object€™s speed is much less than the wave speed: v_o << v. Then f_echo ‰ˆ f₀, and a microphone that is sensitive to these frequencies will detect a beat frequency if it listens to f₀ and f_echo simultaneously. Use the binomial approximation and other appropriate approximations to show that the beat frequency is f_beat ‰ˆ (2v_o/v) f₀.
c. The reflection of 2.40 MHz ultrasound waves from the surface of a fetus€™s beating heart is combined with the 2.40 MHz wave to produce a beat frequency that reaches a maximum of 65 Hz.
What is the maximum speed of the surface of the heart? The speed of ultrasound waves within the body is 1540 m/s.
d. Suppose the surface of the heart moves in simple harmonic motion at 90 beats/min. What is the amplitude in mm of the heartbeat?

Transcribed Image Text:

v + v. | fo fecho v – v. Vo