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Please show all steps. Label all variable symbols such as wavelength, wave speed, period etc before using them in a formula. Please write out answer if possible. DO NOT TYPE ANSWERS WITHOUT MULTIPLICATION OR DIVISION SIGNS.
![1. Ultrasound is sound waves with frequencies higher than the upper audible limit for human
hearing. Bats use ultrasound to for locating prey and obstacles. Ultrasonic devices are also used to
detect objects, measure distances and have many important medical applications. One such
application is to monitor fetal heartbeats by reflecting ultrasound waves off the fetus in the womb.
In this problem we will explore the ideas of doppler shift and beat frequency to understand how
ultrasound is used to monitor fetal heart health.
Ultrasound
transducer
Uterus-
Fetus
Fetal Ultrasound
a.
Consider an object moving at speed vo toward an at rest source that is emitting sound
waves of frequency fo. Show that a reflected wave (i.e. the echo) that returns to the source has a
doppler shifted frequency of
feceno = ) fo
where v is the speed of sound in the medium.
b.
Rearrange your equation by factoring out the speed of sound in the medium from both
the numerator and the denominator, thus creating two binomials. Using the binomial approximation
(or a Taylor expansion), determine an expression for fecho up to first order. The binomial
approximation to first order is given by
(1+ x)" z 1+ nx
if x « 1.
Suppose the objects speed is much less than the wave speed vo « v. When this
happens, the frequency fecho fo. The microphone that is sensitive to these frequencies will
detect a beat frequency if it listens to fecho and fo simultaneously. Use the binomial approximation
to show that (up to first order) the beat frequency is
foeat = (2"0/») fo
The reflection fecho of the 2.40 MHz ultrasound waves from the surface of a fetus's
beatıng heart combines with the 2.40 MHz ultrasound waves to produce a beat frequency that
reaches a maximum of 65 Hz. The speed of ultrasound waves within the body is 1540 m/s. What is
the maximum speed of the surface of the fetus's heart? Does your answer seem reasonable?
d.
Suppose the surface of the heart moves with simple harmonic motion at 90 beats/min.
е.
What is the amplitude in mm of the heartbeat? Does your answer seem reasonable?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a08c9e3-0e65-4415-a871-b9fbe8e64d8e%2F4e3d7737-81ce-4aad-8873-ccaa60139e11%2F8o1irdj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Ultrasound is sound waves with frequencies higher than the upper audible limit for human
hearing. Bats use ultrasound to for locating prey and obstacles. Ultrasonic devices are also used to
detect objects, measure distances and have many important medical applications. One such
application is to monitor fetal heartbeats by reflecting ultrasound waves off the fetus in the womb.
In this problem we will explore the ideas of doppler shift and beat frequency to understand how
ultrasound is used to monitor fetal heart health.
Ultrasound
transducer
Uterus-
Fetus
Fetal Ultrasound
a.
Consider an object moving at speed vo toward an at rest source that is emitting sound
waves of frequency fo. Show that a reflected wave (i.e. the echo) that returns to the source has a
doppler shifted frequency of
feceno = ) fo
where v is the speed of sound in the medium.
b.
Rearrange your equation by factoring out the speed of sound in the medium from both
the numerator and the denominator, thus creating two binomials. Using the binomial approximation
(or a Taylor expansion), determine an expression for fecho up to first order. The binomial
approximation to first order is given by
(1+ x)" z 1+ nx
if x « 1.
Suppose the objects speed is much less than the wave speed vo « v. When this
happens, the frequency fecho fo. The microphone that is sensitive to these frequencies will
detect a beat frequency if it listens to fecho and fo simultaneously. Use the binomial approximation
to show that (up to first order) the beat frequency is
foeat = (2"0/») fo
The reflection fecho of the 2.40 MHz ultrasound waves from the surface of a fetus's
beatıng heart combines with the 2.40 MHz ultrasound waves to produce a beat frequency that
reaches a maximum of 65 Hz. The speed of ultrasound waves within the body is 1540 m/s. What is
the maximum speed of the surface of the fetus's heart? Does your answer seem reasonable?
d.
Suppose the surface of the heart moves with simple harmonic motion at 90 beats/min.
е.
What is the amplitude in mm of the heartbeat? Does your answer seem reasonable?
Expert Solution
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Step 1
a)
The frequency of 'f' is given by
Where v is the velocity of sound in the medium.
vs is the velocity of source.
vo is the velocity of the observer.
The relative frequency heard by the observer initially is,
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