Pressure Variance, Displacement, and a Soft Drink Bottle

[Ethan Godden]

Homework Statement

If you blow across the top of an empty soft-drink bottle, a pulse of sound travels down through the air in the bottle. At the moment the pulse reaches the bottom of the bottle, what is the correct description of the displacement of the elements of air from their equilibrium positions and the pressure of the air at this point. (a) the displacement and pressure are both at a max (b)both at a minimum (c)the displacement is zero and the pressure is a max (d) The displacement is zero and the pressure is a minimum

Homework Equations

s(x,t)=s_maxcos(kx-ωt) where s is the displacement
ΔP=ΔP_maxsin(kx-ωt)=Bs_maxksin(kx-ωt) where B is the bulk modulus
→ΔP_max=Bs_maxk

The Attempt at a Solution

I am greatly confused about how to think for this question. I know both the displacement and ΔP functions are out of phase because one uses sine and the other uses cosine which mean (c) and (d) are the only viable answers. I do not know how to determine if the pressure is a minimum or a maximum. My guess is that since the sound wave is at the bottom of the bottle, the pressure would be at a maximum because of the soundwave pushing the bottle. Is this correct?

Thank you

 

[haruspex]

c) and (d) are the only viable answers.

I agree with that, but more because the bottom of the bottle is a fixed position, the air there necessarily stays put.
But to choose between max and min, you need to consider what happens at the top of the bottle when a sudden gust of air goes across it. Does more air enter the bottle or does some air leave it?

 

[Ethan Godden]

I agree with that, but more because the bottom of the bottle is a fixed position, the air there necessarily stays put.

What do you mean by fixed position? Are you saying since sound waves a longitudinal, the elements of the air at the botoom of the bottle cannot move any father? This would suggest a place of high pressure if what I am thinking this means is true.

But to choose between max and min, you need to consider what happens at the top of the bottle when a sudden gust of air goes across it. Does more air enter the bottle or does some air leave it?

If this were real life, I would think air is added to the bottle for a period of time. That being said, I would think since this is a question out of the text, blowing air across the top of the bottle means no elements of air are added to the bottle: just moving across the top.

 

[haruspex]

What do you mean by fixed position? Are you saying since sound waves a longitudinal, the elements of the air at the botoom of the bottle cannot move any father?

Yes, and conversely that the layer of air immediately adjacent to the bottom of the bottle can never move up, as that would creat a perfect vacuum. That a node arises at the closed end of a tube is a standard assumption in acoustics.

to the bottle for a period of time. That being said, I would think since this is a question out of the text, blowing air across the top of the bottle means no elements of air are added to the bottle: just moving across the top.

No, some air must enter or leave to create a pulse of sound in the bottle. Are you familiar with the Venturi effect?

 

[Ethan Godden]

No, some air must enter or leave to create a pulse of sound in the bottle. Are you familiar with the Venturi effect?

I am not.

 

[haruspex]

I am not.

Google.

 

[Ethan Godden]

Okay, my only problem with this is doesn't the Venturi effect cause less air to be in the bottle? Since air is flowing over the top, does that means air is pulled out of the bottle to fill the air above the bottle that has been pushed away.

I am assuming this interpretation is wrong as then wouldn't there be less air at the bottom of the bottle?

 

[haruspex]

Okay, my only problem with this is doesn't the Venturi effect cause less air to be in the bottle? Since air is flowing over the top, does that means air is pulled out of the bottle to fill the air above the bottle that has been pushed away.

I am assuming this interpretation is wrong as then wouldn't there be less air at the bottom of the bottle?

Why does that make it wrong? Isn't that one of the two offered options?

 

[Ethan Godden]

I was assuming before that air was added to the bottle pushes the air down in the bottle creating a region of high pressure at the bottom.

Now I am thinking that if air is pulled out, then air from the bottom has less pressure meaning there is a minimum instead of a maximum at the bottom.

 

[haruspex]

I was assuming before that air was added to the bottle pushes the air down in the bottle creating a region of high pressure at the bottom.

Now I am thinking that if air is pulled out, then air from the bottom has less pressure meaning there is a minimum instead of a maximum at the bottom.

That would be my answer.

 

[Ethan Godden]

That would be a problem as the back of my book says its a maximum (I probably should have disclosed the book answer before).

 

[haruspex]

That would be a problem as the back of my book says its a maximum (I probably should have disclosed the book answer before).

Then the author did not think through the mechanism by which blowing across the top of the bottle generates movement in the bottle.