How can sound move faster in one pipe than another? If both in air?

[gauss44]

(A question from The Berkeley Review MCAT prep book. The book gives the answer in the form of the equation below. I'm still having trouble figuring out how an example of this would look in real life. If someone would provide a hypothetical narrative where the following could take place, that would be extremely helpful. Thanks!)

1. The problem statement, all variables and given

In studying the relationship between the variable lengths of two open resonating pipes and the number of harmonics each pipe can produce at different lengths, a musicologist notices the following trends in her data:

A graph compares pipe A to pipe B. Pipe A is 2 meters long and has 4 nodes, while pipe B is 3 meters long and has 4 nodes. On a graph of nodes versus length, pipe A has a steeper slope.

Question: Pipes A and B are both telescoping pipes. What could explain the difference in slope for the lines representing each pipe?

Homework Equations

n=2Lfn/v

The Attempt at a Solution

I understand the solution based on the equation, but cannot visualize what's going on. (Again the answer was given in the book as n=2Lfn/v. I get that n and v are inversely proportional.)

 

[mfb]

If sound is faster in one pipe, you need a longer pipe to have the same amount of nodes at the same frequency. What is unclear about that?

 

[gauss44]

If sound is faster in one pipe, you need a longer pipe to have the same amount of nodes at the same frequency. What is unclear about that?

I am missing an understanding of how or why sound can be faster in one pipe than the other? Maybe it's the width of the pipe? Or how fast the air is moving, like if you blow on a flute hard or just barely blow into it?

 

[mfb]

Do we have air in both pipes? If yes, do they have the same temperature and pressure?

Can you quote the full textbook solution?
Different pipe ends (open/closed) could be related to the solution, but then the problem statement is poorly written.

 

[gauss44]

Do we have air in both pipes? If yes, do they have the same temperature and pressure?

Can you quote the full textbook solution?
Different pipe ends (open/closed) could be related to the solution, but then the problem statement is poorly written.

The answer is yes to your first two questions.

The book's full answer explanation is here (however, I believe it adds nothing helpful, and could distract from my question): "Using the equation for the resonant frequencies of an open pipe, you can relate n and L as n=2Lf/v, which means that n and L are linearly related, and more importantly that a bigger sound speed v results in a smaller slope. This makes choice B (B. The speed of sound is slower for Pipe A than for Pipe B) correct. The pipe width doesn't not affect frequencies and resonances other than to decrease the overall intensity of resulting sounds. If you didn't remember the precise equation, at least know the basic variables used in it (e.g. realize that width is not an important factor). This way, you've increased your chances of guessing, should you need to."

 

[mfb]

Well, there are just 3 ways to get a different speed of sound in a gas:
- a different gas composition
- a different pressure
- a different temperature

 

[Borek]

I agree with OP that different speed - while technically a correct answer - jumps out of nowhere. If I were to solve the problem I would assume both pipes are tested in the same conditions, and I would conclude from that speed of gas is identical in both - which would prevent me from getting the right answer.

Lousy question IMHO.

 

[gauss44]

Well, there are just 3 ways to get a different speed of sound in a gas:
- a different gas composition
- a different pressure
- a different temperature

It must be pressure... somehow?

 

[TSny]

Interestingly, the speed of sound in an ideal gas is independent of pressure.

 

[mfb]

Air is not an ideal gas, but I agree that the changes are small for reasonable pressures.