Relationship between Frequency and Volume in Blowing Air into a Bottle

[mtr]

Homework Statement

There is a plastic bottle into which we blow in such a way, that we receive a sound of freguency f. The bottle is filled with water that has volume V. Prove, that $$f=f_{0}(\frac{V}{V_{0}})^{\alpha}$$, where $$f_{0}, V_{0}, \alpha$$ are constants.

Homework Equations

The Attempt at a Solution

I always thought, that the wavelength is directly proportional to height of an empty part of the bottle, but then f is inversely proportional to V. Is there any mistake in this task?

 

[alphysicist]

Hi mtr,

Homework Statement

There is a plastic bottle into which we blow in such a way, that we receive a sound of freguency f. The bottle is filled with water that has volume V. Prove, that $$f=f_{0}(\frac{V}{V_{0}})^{\alpha}$$, where $$f_{0}, V_{0}, \alpha$$ are constants.

Homework Equations

The Attempt at a Solution

I always thought, that the wavelength is directly proportional to height of an empty part of the bottle,

As the height of the air space decreases, the wavelength will decrease also.

but then f is inversely proportional to V.

What is your reasoning for saying that?

 

[mtr]

As the height of the air space decreases, the wavelength will decrease also.

That's right, but considering the wavelength, we know, that $$\lambda = 2L$$ so when L decreases, lambda decreases proportionally to the 1st power of L and inversely proportionally to the 1st power of V.

What is your reasoning for saying that?

I can see my mistake. In fact it is lambda inversly proportional to V so f is directly proportional to V.

Still, I made an experiment and it looks like there is such a formula: $$f=f_{0} \alpha^{\frac{V}{V_{0}}}$$
Actually I'm even more confused with this. Still the formula given in the task cannot be correct, because when V=0, f does NOT equal 0.

 

[alphysicist]

That's right, but considering the wavelength, we know, that $$\lambda = 2L$$ so when L decreases, lambda decreases proportionally to the 1st power of L and inversely proportionally to the 1st power of V.

Think of the bottle as a cylinder. This would be like a pipe with one end closed, so the fundamental wavelength is 4L, if L is the height of the air column.

But I don't think the wavelength is inversely proportional to V, as you've seen when you looked at the frequency. As V increases, the wavelength decreases, but not proportionally.

I can see my mistake. In fact it is lambda inversly proportional to V so f is directly proportional to V.

Still, I made an experiment and it looks like there is such a formula: $$f=f_{0} \alpha^{\frac{V}{V_{0}}}$$
Actually I'm even more confused with this. Still the formula given in the task cannot be correct, because when V=0, f does NOT equal 0.

Was the problem that you had in the original post all that there was? There were no diagrams, or approximate ranges given? By treating the bottle as a cylinder you can get an exact result, which does not match with $f\propto V^{\alpha}$ from your original post.

If I'm reading the problem correctly, and you haven't overlooked any parts of the problem in your post, then I would agree that the formula in the original post does not seem to be an exact answer.

 

[mtr]

But I don't think the wavelength is inversely proportional to V, as you've seen when you looked at the frequency. As V increases, the wavelength decreases, but not proportionally.

Well, I won't argue with that ;)


There were no additional information. They also do not say anything about the bottle's shape, so I suppose it can be any.