Question

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. A. (5 points) Consider an object moving at speed $v_0$ toward an at rest source that is emitting sound waves of frequency $f_0$. Show that a reflected wave that returns to the source has a doppler shifted frequency of $f_{echo} = \left(\frac{v + v_0}{v - v_0}\right) f_0$ where $v$ is the speed of sound in the medium. B. (5 points) 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, determine an expression for $f_{echo}$ up to first order. The binomial approximation to first order is given by if $x << 1$. $(1 + x)^n \approx 1 + nx$ C. (5 points) Suppose the objects speed is much less than the wave speed $v_0 << v$. When this happens, the frequency $f_{echo} \approx f_0$. The microphone that is sensitive to these frequencies will detect a beat frequency if it listens to $f_{echo}$ and $f_0$ simultaneously. Use the binomial approximation to show that (up to first order) the beat frequency is $f_{beat} = (2v_0/v) f_0$ D. (5 points) The reflection from the surface of a fetus's beating heart combines with the 2.40 MHz ultrasound waves to produce a beat frequency of 65 Hz. The speed of ultrasound waves within the human body is 1540 m/s. What is the maximum speed of the surface of the fetus heart? E. (5 points) Suppose the surface of the heart moves with simple harmonic motion at 90 beats/min. What is the amplitude in mm of the heartbeat? Does your answer seem reasonable? What are the typical dimensions of a fetus heart? 2

          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.
A. (5 points) Consider an object moving at speed $v_0$ toward an at rest source that is emitting sound
waves of frequency $f_0$. Show that a reflected wave that returns to the source has a doppler shifted
frequency of
$f_{echo} = \left(\frac{v + v_0}{v - v_0}\right) f_0$
where $v$ is the speed of sound in the medium.
B. (5 points) 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, determine an expression for $f_{echo}$ up to first order. The binomial approximation to
first order is given by
if $x << 1$.
$(1 + x)^n \approx 1 + nx$
C. (5 points) Suppose the objects speed is much less than the wave speed $v_0 << v$. When this
happens, the frequency $f_{echo} \approx f_0$. The microphone that is sensitive to these frequencies will
detect a beat frequency if it listens to $f_{echo}$ and $f_0$ simultaneously. Use the binomial approximation
to show that (up to first order) the beat frequency is
$f_{beat} = (2v_0/v) f_0$
D. (5 points) The reflection from the surface of a fetus's beating heart combines with the 2.40 MHz
ultrasound waves to produce a beat frequency of 65 Hz. The speed of ultrasound waves within the
human body is 1540 m/s. What is the maximum speed of the surface of the fetus heart?
E. (5 points) Suppose the surface of the heart moves with simple harmonic motion at 90 beats/min.
What is the amplitude in mm of the heartbeat? Does your answer seem reasonable? What are the
typical dimensions of a fetus heart?
2

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.
A. (5 points) Consider an object moving at speed v0 toward an at rest source that is emitting sound
waves of frequency f0. Show that a reflected wave that returns to the source has a doppler shifted
frequency of
fecho = ((v + v0)/(v - v0)) f0
where v is the speed of sound in the medium.
B. (5 points) 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, determine an expression for fecho up to first order. The binomial approximation to
first order is given by
if x << 1.
(1 + x)^n ≈ 1 + nx
C. (5 points) Suppose the objects speed is much less than the wave speed v0 << v. When this
happens, the frequency fecho≈ f0. The microphone that is sensitive to these frequencies will
detect a beat frequency if it listens to fecho and f0 simultaneously. Use the binomial approximation
to show that (up to first order) the beat frequency is
fbeat = (2v0/v) f0
D. (5 points) The reflection from the surface of a fetus's beating heart combines with the 2.40 MHz
ultrasound waves to produce a beat frequency of 65 Hz. The speed of ultrasound waves within the
human body is 1540 m/s. What is the maximum speed of the surface of the fetus heart?
E. (5 points) Suppose the surface of the heart moves with simple harmonic motion at 90 beats/min.
What is the amplitude in mm of the heartbeat? Does your answer seem reasonable? What are the
typical dimensions of a fetus heart?
2

Added by Elizabeth I.

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