What is the speed of sound in a rigid body?

[abhishek4]

Homework Statement

What will be the speed of sound in a perfectly rigid body?
Given : In a hypothetical situation we have a rigid body find the speed of sound when it passes through the material.
Elasticity of medium = ∞

Homework Equations

velocity = √E/ρ
ρ = density
E = elasticity of medium

The Attempt at a Solution

My attempt:
An object which does not allow the movement of particles means that the particles in the body cannot vibrate which does not allow transfer of energy through the body. Hence sound will not travel through the body and hence the velocity of sound becomes 0.
My teachers attempt:
v = √∞/ρ
v = ∞
My doubt :
A wave cannot travel faster than the speed of light and when i asked my teacher as is this infinity referring to such a body not existing she said that it has a very high speed.

 

[Wee-Lamm]

Elasticity is the measure of an objects ability to resist a distorting influence and return to it's original shape. Elasticity does not directly imply that the particles within the body cannot vibrate.

Perhaps, the decay rate of the excited particles vibration is also infinite which would suggest they would stop vibrating at the removal of the stress, just as quickly as they began to vibrate when the stress was introduced?

 

[Isaac0427]

I find this very confusing. Physics does not always work that well with solutions of infinity, because infinity is undefined. That is why we don't even make up hypothetical situations about the frame of reference of light, because time moves infinitely slow, and therefore both time and the frame of reference is undefined, and therefore the reference frame does not exist. In this case, if E is undefined, the velocity will be undefined, and doing this kind of problem is completely a waist of time in my opinion, unless the lesson was about undefined solutions which I'm inferring it wasn't. In science (especially in classical mechanics), if a property of something is undefined, something is probably going wrong. I would think that the infinity does mean the body does not exist.

 

[abhishek4]

I find this very confusing. Physics does not always work that well with solutions of infinity, because infinity is undefined. That is why we don't even make up hypothetical situations about the frame of reference of light, because time moves infinitely slow, and therefore both time and the frame of reference is undefined, and therefore the reference frame does not exist. In this case, if E is undefined, the velocity will be undefined, and doing this kind of problem is completely a waist of time in my opinion, unless the lesson was about undefined solutions which I'm inferring it wasn't. In science (especially in classical mechanics), if a property of something is undefined, something is probably going wrong. I would think that the infinity does mean the body does not exist.

i would like to make it clear to you that physics from the start has been the study of unknown and as per me no question is a waste of time. I would like to thank you for your reply but it wasn't what i was looking for as i did not ask for a lecture on what science is. Also as you are saying that the velocity is undefined that clearly means that such an object does not exist which could have been explained clearly without further explaining to me the meaning of science which i already know.

 

[abhishek4]

Elasticity is the measure of an objects ability to resist a distorting influence and return to it's original shape. Elasticity does not directly imply that the particles within the body cannot vibrate.

Perhaps, the decay rate of the excited particles vibration is also infinite which would suggest they would stop vibrating at the removal of the stress, just as quickly as they began to vibrate when the stress was introduced?

Thanks a lot for your reply but i was wondering in case of a rigid body shouldn't the particles resist vibrations. It was just a hunch but in any case shouldn't the sound be 0 or undefined.

 

[Samy_A]

Homework Statement

What will be the speed of sound in a perfectly rigid body?
Given : In a hypothetical situation we have a rigid body find the speed of sound when it passes through the material.
Elasticity of medium = ∞

Homework Equations

velocity = √E/ρ
ρ = density
E = elasticity of medium

The Attempt at a Solution

My attempt:
An object which does not allow the movement of particles means that the particles in the body cannot vibrate which does not allow transfer of energy through the body. Hence sound will not travel through the body and hence the velocity of sound becomes 0.
My teachers attempt:
v = √∞/ρ
v = ∞[/B]
My doubt :
A wave cannot travel faster than the speed of light and when i asked my teacher as is this infinity referring to such a body not existing she said that it has a very high speed.

You are mixing classical Physics with Special Relativity. In Special Relativity, the concept of rigid body is problematic. And in classical Physics it's an idealization too.
Read at the bottom of page 2 of http://arxiv.org/pdf/1105.3899.pdf

 

[Isaac0427]

no question is a waste of time.

I understand where you are coming from, but I would say that questions about a situation that could never happen where the point of the question was not to show that the situation would never happen are a waist of time.

Also as you are saying that the velocity is undefined that clearly means that such an object does not exist which could have been explained clearly without further explaining to me the meaning of science which i already know.

I didn't know how much you knew of what I said. I answered the question in detail in case anyone without a lot of background (which I wasn't sure if you were in that category) wanted to know the answer.

 

[abhishek4]

I understand where you are coming from, but I would say that questions about a situation that could never happen where the point of the question was not to show that the situation would never happen are a waist of time.

I didn't know how much you knew of what I said. I answered the question in detail in case anyone without a lot of background (which I wasn't sure if you were in that category) wanted to know the answer.

i have researched in this topic and was looking for a to the point answer plus i have been looking for about 12 hours for this solution which made me think that you were not trying to give an answer. I am sorry.

 

[abhishek4]

You are mixing classical Physics with Special Relativity. In Special Relativity, the concept of rigid body is problematic. And in classical Physics it's an idealization too.
Read at the bottom of page 2 of http://arxiv.org/pdf/1105.3899.pdf

Thank you so much. This helped a lot, i didnt think of relating it to special relativity, thanks a lot.

 

[Isaac0427]

i have researched in this topic and was looking for a to the point answer plus i have been looking for about 12 hours for this solution which made me think that you were not trying to give an answer. I am sorry.

That's all good. Here's a to-the-point answer:
The situation can not exist because the answer is undefined.

 

[abhishek4]

That's all good. Here's a to-the-point answer:
The situation can not exist because the answer is undefined.

Thanks.

 

[Wee-Lamm]

That's all good. Here's a to-the-point answer:
The situation can not exist because the answer is undefined.

Except, the sound did not originate in the rigid body which also happens to have an infinitely high elasticity modulus, the sound would not simply cease to exist. The question posed did specify t as being, "When the sound passes through". I could agree that a "Perfectly Rigid Body" cannot exist, but disagree with the notion that the defined situation could not.

It is worth noting that Elasticity is a property of the material that a body consists of, while rigidity or stiffness is a property of the structure of the body, or how it is put together. These properties can serve to negate or enhance the other, depending on the other qualities present in both as well as the body, but knowing only these 2 qualities cannot yield an immutable result that is applicable to all similar situations.

 

[Wee-Lamm]

Except, the sound did not originate in the rigid body which also happens to have an infinitely high elasticity modulus, the sound would not simply cease to exist. The question posed did specify t as being, "When the sound passes through". I could agree that a "Perfectly Rigid Body" cannot exist, but disagree with the notion that the defined situation could not.

It is worth noting that Elasticity is a property of the material that a body consists of, while rigidity or stiffness is a property of the structure of the body, or how it is put together. These properties can serve to negate or enhance the other, depending on the other qualities present in both as well as the body, but knowing only these 2 qualities cannot yield an immutable result that is applicable to all similar situations.

Sorry to reply to my own post, an edit seemed awkward.

I would expect the sound at the time "when it passed through" would be measurable by comparing the sound before it went through against after it passed through. For sound to not be present it must be either completely absorbed or reflected, or converted into some other form such as vibration. Perhaps the structure of the body would disallow sympathetic vibration of it's particles, meaning they did not continue to vibrate and transfer that energy to other neighboring particles, but I suspect they would simply reflect much of the energy back to the sound wave, and perhaps lose some to friction as it passes through?

 

[Chestermiller]

The answer to your original question is that the speed of sound would be infinite if the body were perfectly rigid. But there is no such thing as a perfectly rigid body. For any realistic value of the modulus of elasticity of, say, a metal, the predicted speed of sound would be orders of magnitude lower than the speed of light. Why don't you just calculate how high the modulus of elasticity would have to be in order for the speed of sound to equal the speed of light? How does this compare with the highest values you can find for solids? Or why don't you calculate the speed of sound for a typical value of the density and modulus of elasticity of a solid? What fraction is this of the speed of light?

 

[Isaac0427]

Except, the sound did not originate in the rigid body which also happens to have an infinitely high elasticity modulus, the sound would not simply cease to exist. The question posed did specify t as being, "When the sound passes through". I could agree that a "Perfectly Rigid Body" cannot exist, but disagree with the notion that the defined situation could not.

It is worth noting that Elasticity is a property of the material that a body consists of, while rigidity or stiffness is a property of the structure of the body, or how it is put together. These properties can serve to negate or enhance the other, depending on the other qualities present in both as well as the body, but knowing only these 2 qualities cannot yield an immutable result that is applicable to all similar situations.

Yes, but how can sound that can't exist travel at an infinite speed? That, I believe, makes the situation undefined, and non-existent.

 

[Chestermiller]

Yes, but how can sound that can't exist travel at an infinite speed? That, I believe, makes the situation undefined, and non-existent.

Do you understand what the term "speed of sound" means in terms of a compression wave traveling through a medium?

 

[Wee-Lamm]

Yes, but how can sound that can't exist travel at an infinite speed? That, I believe, makes the situation undefined, and non-existent.

The sound was not produced by the rigid body, it already existed as it was propagated from an external source.

At the collision point where the already existing sound wave contacts the surface of the rigid body, 1 of 3 things can happen (or perhaps a combination of any 2 or all 3 of them. The wave can be reflected, absorbed, or allowed to pass through. OP specified the sound does pass through, so we can attempt to measure how much of the sound energy does pass through, how much is absorbed or amplified on the way through, and how much was reflected.

ChesterMiller makes a good observation with considering "orders of magnitude", I suspect that using the speed of light would give us a constant reference to relate to.

Can we be sure that the structure of the object is rigid enough to not vibrate in sympathy with the sound wave for at least as long as the force of the wave is still generating points of collision?

Does the Infinite Elasticity of the medium discourage the excitation of neighboring particles, will it encourage the energy to be shared more equally with neighboring particles, and what is the decay rate of their active state once the force of the wave becomes 0?

 

[Chestermiller]

Can we be sure that the structure of the object is rigid enough to not vibrate in sympathy with the sound wave for at least as long as the force of the wave is still generating points of collision?

It doesn't have to be a wave that hits the body. You can just bang the end of the body with a hammer. This will create the required traveling compression pulse. Even if the rod is a mile long, the sound pulse will propagate to the other end, and you'll will hear a clang. Hey, you hear thunder from miles away. The dissipation in an elastic metal rod will be much less than that.

 

[Isaac0427]

The sound was not produced by the rigid body, it already existed as it was propagated from an external source.

Oh ok, so it's just that the sound can't exist. Still, the teacher in the situation would be incorrect, and the speed of sound is undefined, not "really fast" as the teacher had said.

 

[Wee-Lamm]

It doesn't have to be a wave that hits the body. You can just bang the end of the body with a hammer. This will create the required traveling compression pulse. Even if the rod is a mile long, the sound pulse will propagate to the other end, and you'll will hear a clang. Hey, you hear thunder from miles away. The dissipation in an elastic metal rod will be much less than that.

I agree entirely with this, but the OP asked, "what will the speed of a sound be, as it passes through the rigid object".

 

[Wee-Lamm]

Oh ok, so it's just that the sound can't exist. Still, the teacher in the situation would be incorrect, and the speed of sound is undefined, not "really fast" as the teacher had said.

The sound existed before it collided with the rigid body and it passed through. Passing through implies that at least some of the sound did 'still' exist in some form, while passing through, and that at least some of the sound did still exist in some form, after it had passed through.

I presume that we still cannot create nor destroy energy, but we can manipulate it into another form of energy?

 

[Chestermiller]

Oh ok, so it's just that the sound can't exist. Still, the teacher in the situation would be incorrect, and the speed of sound is undefined, not "really fast" as the teacher had said.

The sound very much can exist. The teacher in this situation is absolutely correct, in the ideal mathematical limit of infinitely high elastic modulus. Real materials don't have infinitely high elastic moduli. Therefore, sound can't travel infinitely fast in them. So what! There is still sound, and it still travels.

It sounds like you have no background in this area, and you are just speculating, based on very limited knowledge. Please do not submit your personal theories to Physics Forums. This can be a reason for warnings and infraction points, which can lead to banning.

Chet

 

[abhishek4]

The sound very much can exist. The teacher in this situation is absolutely correct, in the ideal mathematical limit of infinitely high elastic modulus. Real materials don't have infinitely high elastic moduli. Therefore, sound can't travel infinitely fast in them. So what! There is still sound, and it still travels.

It sounds like you have no background in this area, and you are just speculating, based on very limited knowledge. Please do not submit your personal theories to Physics Forums. This can be a reason for warnings and infraction points, which can lead to banning.

Chet

Thank you sir, your answer was really helpful and I completely understand the fact of sound being infinite in case of this hypothetical situation but can we actually conclude from this that such an object does not exist just the way how we disprove theories in maths by making assumptions to their contradictory?

 

[Chestermiller]

Thank you sir, your answer was really helpful and I completely understand the fact of sound being infinite in case of this hypothetical situation but can we actually conclude from this that such an object does not exist just the way how we disprove theories in maths by making assumptions to their contradictory?

I don't think so. I think that if the force balance leading to the wave equation in a deformable solid were modified to permit relativistic velocities (as reckoned from a fixed frame of reference), the resulting equation would prevent the wave velocity from exceeding the speed of light. It would be similar to the relativistic modification to Newton's 2nd law such that, if you apply a constant force to a body, the velocity of the body could not exceed the speed of light.