Question

The ultrasound is a medical application of the Doppler-effect. It is utilized to measure fetal heartbeats by reflecting ultrasound waves off a fetus in the womb. Let f? be the frequency transmitted by the at-rest source and let f???? be the frequency of the echo (from the beating heart) received back by the source. Consider an instant when the beating heart is effectively moving towards the source with speed vH. Let v be the speed of sound in the human body. a. Draw a clear and neat figure that depicts the physical situation. Your figure must include labels for various quantities, etc. b. Write an expression, in terms of variables, for the frequency of the ultrasound waves received by the heart. c. Show that f???? = ((v+vH)/(v-vH))f?. d. Utilize the expression in part (c) to show that, for vH << v, the difference, ?f, between the transmitted and received frequencies at the source can be approximated as, ?f ? (2vH/v)f?. e. Suppose for a heart, the maximum value of ?f turns out to be 65 Hz. What is then the maximum speed of the beating heart, if f? = 2.40 MHz, and v = 1540 m/s? f. If the beating heart in part (e) is modeled to be a simple harmonic oscillator, with displacement x(t) = A cos(?t) and a heartbeat of 90 beats/min, what is the amplitude in mm of the heartbeat?

          The ultrasound is a medical application of the Doppler-effect. It is utilized to measure fetal heartbeats by reflecting ultrasound waves off a fetus in the womb. Let f? be the frequency transmitted by the at-rest source and let f???? be the frequency of the echo (from the beating heart) received back by the source. Consider an instant when the beating heart is effectively moving towards the source with speed vH. Let v be the speed of sound in the human body.
a. Draw a clear and neat figure that depicts the physical situation. Your figure must include labels for various quantities, etc.
b. Write an expression, in terms of variables, for the frequency of the ultrasound waves received by the heart.
c. Show that f???? = ((v+vH)/(v-vH))f?.
d. Utilize the expression in part (c) to show that, for vH << v, the difference, ?f, between the transmitted and received frequencies at the source can be approximated as, ?f ? (2vH/v)f?.
e. Suppose for a heart, the maximum value of ?f turns out to be 65 Hz. What is then the maximum speed of the beating heart, if f? = 2.40 MHz, and v = 1540 m/s?
f. If the beating heart in part (e) is modeled to be a simple harmonic oscillator, with displacement x(t) = A cos(?t) and a heartbeat of 90 beats/min, what is the amplitude in mm of the heartbeat?

The ultrasound is a medical application of the Doppler-effect. It is utilized to measure fetal heartbeats by reflecting ultrasound waves off a fetus in the womb. Let f? be the frequency transmitted by the at-rest source and let f???? be the frequency of the echo (from the beating heart) received back by the source. Consider an instant when the beating heart is effectively moving towards the source with speed vH. Let v be the speed of sound in the human body.
a. Draw a clear and neat figure that depicts the physical situation. Your figure must include labels for various quantities, etc.
b. Write an expression, in terms of variables, for the frequency of the ultrasound waves received by the heart.
c. Show that f???? = ((v+vH)/(v-vH))f?.
d. Utilize the expression in part (c) to show that, for vH << v, the difference, ?f, between the transmitted and received frequencies at the source can be approximated as, ?f ? (2vH/v)f?.
e. Suppose for a heart, the maximum value of ?f turns out to be 65 Hz. What is then the maximum speed of the beating heart, if f? = 2.40 MHz, and v = 1540 m/s?
f. If the beating heart in part (e) is modeled to be a simple harmonic oscillator, with displacement x(t) = A cos(?t) and a heartbeat of 90 beats/min, what is the amplitude in mm of the heartbeat?

Added by Tom-S C.

Question

Please give Ace some feedback