Randall D. Knight — Superposition: Problem Challenge_Problem_81

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 \(v_0\) toward an at-rest source that is emitting sound waves of frequency \(f_0\). Show that the reflected wave (i.e., the echo) that returns to the source has a Doppler-shifted frequency \[f_{\text{echo}} = \left( \frac{v + v_0}{v - v_0} \right) f_0\] where \(v\) is the speed of sound in the medium. b. Suppose the object’s speed is much less than the wave speed: \(v_0 \ll v\). Then \(f_{\text{echo}} \approx f_0\), and a microphone that is sensitive to these frequencies will detect a beat frequency if it listens to \(f_0\) and \(f_{\text{echo}}\) simultaneously. Use the binomial approximation and other appropriate approximations to show that the beat frequency is \(f_{\text{beat}} \approx (2v_0/v) f_0\). 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?