# Longitudinal Waves (Heart Wall of unborn)

## Homework Statement

A vet is using an ultrasonic motion detector to detect the heartbeat of an unborn chimp baby and claims that the speed of sound in the soft tissue of a chimp should be 1489 m/s. According to the manual which comes with the motion detector, the emitted frequency of sound waves from the motion detector is 1.97 MHz (M is mega or 106). You ask the vet to estimate the number of heartbeats per minute for the tiny heart of the chimp fetus, and you also ask her to estimate the range of motion of the heart wall as it goes back and forth. She says that the expected values would be about 121 heartbeats per minute and, for the range of motion, about 2.05 mm. Assuming that the heart wall moves in Simple Harmonic Motion, what is the maximum speed of the heart wall during any cycle, in mm/s?

## The Attempt at a Solution

I calculated the amplitude of this longitudinal wave by dividing the range of the motion by 2. When i first read this problem, I just thought that the maximum speed is the speed of sound, but that's not right. It doesn't have to do with intensity does it? Jeez i feel dumb haha.

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Okay, I'm guessing that there's more to this question than just this first part, right?

How much do you know about SHM? The question gives you enough information to find the frequency of the motion and the amplitude. You're going to need a formula for the velocity of a harmonic oscillator in terms of the oscilator's position (i.e. displacement from the equilibrium position). Then you'll have to think about when this velocity is maximum.

This was the only information given in the problem.

As for SHM, I know that the maximum velocity of a point on the wave occurs at the moment of equilibrium because of transfers of energy. I do have an equation maximum kinetic energy. E(mech) = (1/2) k A^2 = U(s, max), but I don't know how to calculate the mechanical energy that is preserved during the motion of the heart wall? I am given properties of the wave of the sound of the heartbeat, but do those apply directly in accordance with the motion of the heart wall since this is a longitudinal wave?

I am having a hard time organizing all my thoughts.

Does this question involve the velocity of the oscillator at all? I think it is simply asking for the wave being emitted by the heart wall of the unborn chimp. There is more to this question later on about applying the Doppler effect to find the change in frequency measured by the ultrasonic motion detector due to the interaction between the two waves , but this is preliminary to all those...

Well the question asks for the maximum speed of the heart wall and we are told that the heart wall acts as a harmonic oscillator with a certain amplitude and frequency.

You have an equation for the potential energy of the system when the oscillator is at it's maximum displacment. At this point the speed is zero so all of the energy is potential energy and there is no kinetic energy. This means that you're quite correct to say that the total mechanical energy of the system is

<< almost-complete solution deleted by berkeman >>

You've correctly stated that the velocity is maximised at the equlibrium position, i.e. when x = 0. So using this last equation you can just plug in values for A and f and calculate the maximum speed of the oscilator.

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berkeman
Mentor
Well the question asks for the maximum speed of the heart wall and we are told that the heart wall acts as a harmonic oscillator with a certain amplitude and frequency.

You have an equation for the potential energy of the system when the oscillator is at it's maximum displacment. At this point the speed is zero so all of the energy is potential energy and there is no kinetic energy. This means that you're quite correct to say that the total mechanical energy of the system is

<< almost-complete solution deleted by berkeman >>

You've correctly stated that the velocity is maximised at the equlibrium position, i.e. when x = 0. So using this last equation you can just plug in values for A and f and calculate the maximum speed of the oscilator.
Please don't solve the OP's homework problem for them. We need to stay within the PF Rules and offer only tutorial help, to help the OP figure out the problem on their own. The OP must do the bulk of the work.

Sorry. Going by what I've seen on other threads I was going to just suggest the formula for velocity in terms of displacement but since I had some time on my hands and wanted to practive posting some LaTex code I thought I'd derive it as well. It didn't seem that the derivation was part of the problem being solved. In the few problems I've seen solved here it seemed that providing a correct, standard result was permitted. I didn't see the harm in expanding on a standard result with some reasoning.

On reflection though, I agree that it would have been better to restrict myself to providing general hints rather than setting everything out in detail.

berkeman
Mentor
Sorry. Going by what I've seen on other threads I was going to just suggest the formula for velocity in terms of displacement but since I had some time on my hands and wanted to practive posting some LaTex code I thought I'd derive it as well. It didn't seem that the derivation was part of the problem being solved. In the few problems I've seen solved here it seemed that providing a correct, standard result was permitted. I didn't see the harm in expanding on a standard result with some reasoning.

On reflection though, I agree that it would have been better to restrict myself to providing general hints rather than setting everything out in detail.
No worries. So what would be a couple good hints for the OP to get them jump-started... ?

I think it would have been useful to focus the OP's mind on the facts at hand. They seem to have been a little confused about exactly what this first part of the question required and had difficulty in realising that this is a problem in SHM.

Does this question involve the velocity of the oscillator at all?
Having done this, it would probably have been better to suggest that thergy conservation could be applied to the given equation for the maximum potential energy and that this could lead to an expression for the maximum velocity.

Would this have been okay?