Chapter 17, Problem 80EAP

Ultrasound has many medical applications, one of which is to monitor fetal heartbeats by reflecting ultrasound off a fetus in the womb.

1. Consider an object moving at speed towards and at-rest source that is emitting sound waves of frequency. Show that the reflected wave (ie., the echo) that returns to the source has a Doppler-shifted frequency

where v is the speed of sound in the medium.

Suppose the object’s speed is much less than the wave speed: v₀< 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

$f_{beat}\approx (2v_0/v)f_0$

The reflection of 2.4 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.

Suppose the surface of the heart moves in simple harmonic motion at 90 beats/min. What is the amplitude in mm of the heartbeat?