Calculating Pipe Ratios: Solving an Open/Closed and Open/Open Resonance Problem

[erok81]

Homework Statement

An open organ pipe (an open/closed configuration) is tuned to a given frequency. A second pipe with both ends open resonates with twice this frequency. What is the ratio of the length of the first pipe to the second pipe?

Homework Equations

I used $$f_{1}=\frac{v_{1}}{4L_{1}}, f_{2}=\frac{v_{2}}{2L_{2}}$$ to relate the lengths and frequencies. In this case f_1 is the open/closed system and f_2 is the open/open system.

The Attempt at a Solution

Since the second frequency is twice the first I tried f_2=2f_1 and inserted those into their respective equations - setting f_2 as 2f_1. So I had:

$$f_{1}=\frac{v_{1}}{4L_{1}}, 2f_{1}=\frac{v_{2}}{2L_{2}}$$

Multiplied "f_2" by 2 to get f_1 alone in that equation in order to set them both equal to f_1.

f_2 now becomes $$f_{1}=\frac{v_{2}}{4L_{2}}$$ after the above operation.

Both equations are now equal to f_1 so the following is true:

$$f_{1}=\frac{v_{1}}{4L_{1}}, f_{1}=\frac{v_{2}}{4L_{2}}$$

Then if do a bit of manipulation I get 4L_2/4L_2.

That is how I arrive at my final answer of 1:1 ratio for the two lengths.

Does this look completely wrong? I've always been poor at calculating ratios for some reason.

 

[ehild]

Homework Equations

I used $$f_{1}=\frac{v_{1}}{4L_{1}}, f_{2}=\frac{v_{2}}{2L_{2}}$$ to relate the lengths and frequencies. In this case f_1 is the open/closed system and f_2 is the open/open system.

And what are v₁ and v₂?

ehild

 

[erok81]

That's where I don't think my method works. I'll end up with one on one side since they are the same value.

Since they are sound waves in the same medium and temp, let's say they are traveling at 343 m/s.

 

[ehild]

Yes, v1 and v2 are the speed of sound. The two pipes sound at the same place and time, the speed of sound must be the same for both. As v1=v2, your derivation leads to

You derived that $$f_{1}=\frac{v}{4L_{1}}, f_{1}=\frac{v}{4L_{2}}$$

so what is the relation between the lengths of the pipes?


ehild

 

[erok81]

I'd say 1:1 based on those two equations being equal now...ratio-wise anyway.

 

[ehild]

Yes. If two pipes are of equal lengths, and one is open at both ends, its frequency is twice of the one which is open at one end only.

ehild

 

[erok81]

Perfect. Thank you for the help.