How Does the Doppler Effect Apply to Ultrasound Fetal Heart Monitoring?

[jpnnngtn]

Here it goes:

Expectant parents are thrilled to hear their unborn baby's heartbeat, revealed by an ultrasonic motion detector. Suppose the fetus's ventricular wall moves in simple harmonic motion with an amplitude 1.80 mm and a frequency of 115 per minute.
(a) find the maximum linear speed of the heart wall. Suppose the motion detector in contact with the mother's abdomen produces sound at precisely 2MHz, which travels through tissue at 1.5 km/s
(b) Find the maximum frequency at which the sound arrives at the wall of the baby's heart.
(c) Find the maximum frequency at which reflected sound is received by the motion detector.


So far i have realized that this would be a problem relating to the Doppler Effect. This is where i have started. However I don't no what is what. A = 1.8 mmm (What would I use that for?)
F(baby) = 115 (Again, What or Where does this go?) F

F(motion detector) = 2MHz (I am sure this goes in the Doppler Effect Formula)
V (tissue) = 1.5 km/s

What direction do I go to answer these questions? I am truly lost.

 

[quantumdude]

Originally posted by jpnnngtn
So far i have realized that this would be a problem relating to the Doppler Effect.

That's right.

This is where i have started. However I don't no what is what. A = 1.8 mmm (What would I use that for?)

That is the amplitude of the harmonic motion of the ventricular wall. You need that to find the maximum linear speed of the wall. This would have been covered in a previous unit on wave motion or simple harmonic motion.

Once you figure out the maximum linear speed of the ventricular wall, you have to deal with a problem in which sound is reflected off a moving surface (the wall itself). Some sound waves will be reflected when the wall is moving towards the source, and some will be reflected when the wall is moving away from the source. That is how the doppler effect comes in, and you will need to work along those lines for part b.

F(baby) = 115 (Again, What or Where does this go?) F

Try to think about the problem and visualize what is happening, not just look to plug numbers into formulas. The frequency of the baby's heart is part of what determines its maximum linear speed. Again, this would come from a previous unit on wave motion or harmonic motion.

F(motion detector) = 2MHz (I am sure this goes in the Doppler Effect Formula)

Yes, you're right about that.

 

[jpnnngtn]

Got this far, still need help

O.K., I know that the Amplitude of the baby's ventrical wall is 1.8 mm. Does this have anything to do with the wavelength? Or am i going in the wrong direction yet again? I (we) have established that the frequency is 115 beats/min? (looking for the linear speed now)
Am I wrong by saying:

Wavelength = Amplitude / Frequency

V (linear Speed) = Wavelength * Frequency

?

As for the Doppler Effect portion of the problem, I think I can figure that out once I get over this hump. As for now, this is where I am at. Thanks!


This is from the (This one is tough(Need some advice ) post)

 

[quantumdude]

Originally posted by jpnnngtn
O.K., I know that the Amplitude of the baby's ventrical wall is 1.8 mm. Does this have anything to do with the wavelength? Or am i going in the wrong direction yet again?

The amplitude has nothing to do with the wavelength of anything. It does, however, have to do with the velocity. You should have learned earlier about harmonic motion. Let y(t) be the displacement of the ventricular wall from its eqilibrium position. If it undergoes harmonic motion, then y(t) will be a function of the form:

y(t)=Acos(ωt+φ)

A=amplitude
ω=angular frequency
t=time
φ=phase angle

You can find the maximum linear speed from that.

Any of this looking familiar?

I (we) have established that the frequency is 115 beats/min? (looking for the linear speed now)
Am I wrong by saying:

Wavelength = Amplitude / Frequency

That is wrong on two counts.

First, you are mixing up the frequency of the sound wave with the frequency of the harmonic motion of the ventricular wall.

Second, for the wave the correct relation is v=λf, not λ=A/f.

V (linear Speed) = Wavelength * Frequency

?

That is true for the sound wave. It is not true for the harmonic motion of the ventricular wall.

As for the Doppler Effect portion of the problem, I think I can figure that out once I get over this hump. As for now, this is where I am at. Thanks!

Yes, you do need to get over this hump before doing the Doppler part of the problem. Just remember that the sound wave and the harmonic motion of the ventricular wall are two different things.

 

[jpnnngtn]

That helped but I still am alost

When you speak of time, are you talking about the period ?

And as far as the phase angle, I have never heard of that, my book doesn't even mention that until Chapter 21 (Alternating Current Circuits and Electromagnetic Waves). I am only on Chapter 14 (Sound). How do I use this?

 

[quantumdude]

Originally posted by jpnnngtn
When you speak of time, are you talking about the period ?

No.

And as far as the phase angle, I have never heard of that, my book doesn't even mention that until Chapter 21 (Alternating Current Circuits and Electromagnetic Waves). I am only on Chapter 14 (Sound).

Even so, you should have learned about about translation of functions to the left and right in your math courses. If you have a function f(x), then f(x-a) is the same function moved 'a' units to the right and f(x+a) is the same function moved 'a' units to the left.

How do I use this?

The phase angle is determined by the initial conditions of the motion. But you don't need to use it for this problem, because you just need the maximum linear speed. You find that by taking dy/dt and noting that sin(x) oscillates between ±1.

 

[quantumdude]

By the way, what book are you using?

 

[jpnnngtn]

I am using "College Physics" by Serway and Faughn

 

[jpnnngtn]

So you mean i just need to take the derivative of that function to find the velocity......? After all of that? WOW! So that's it right?

 

[jpnnngtn]

but remember, All that I have to get the maximum linear speed is the Amplitude (1.8mm) and the frequency (115 b/m) is that enough?

 

[quantumdude]

Originally posted by jpnnngtn
So you mean i just need to take the derivative of that function to find the velocity......?

Yes, that's all you have to do to find the maximum linear speed.

but remember, All that I have to get the maximum linear speed is the Amplitude (1.8mm) and the frequency (115 b/m) is that enough?

Yes, that is enough.

When you take the derivative of the general solution for harmonic motion, you get:

dy/dt=-ωAsin(ωt+φ)

It doesn't matter what φ is because you want the maximum. Now note that the maximum value of sin(ωt+φ) is 1 and you've got it.

 

[jpnnngtn]

But what about the time, where would I go to get that?

 

[quantumdude]

Originally posted by jpnnngtn
But what about the time, where would I go to get that?

You don't need it. All you need to know is that sin(x) oscillates between ±1. Remember, you are only interested in the maximum speed. The maximum speed is reached when sin(ωt+φ) hits its extreme values.

 

[jpnnngtn]

Got it. Thanks!

 

[jpnnngtn]

Ok, is this right?

to find the angular frequency:

ù = 2 * (Pi) * 115 b/m

I am going the right way here ...?

 

[quantumdude]

Originally posted by jpnnngtn
to find the angular frequency:

ù = 2 * (Pi) * 115 b/m

I am going the right way here ...?

Yes, that's right.