4. Two closed organ pipes of length 0.5 m and 0.525 m are sounded together and produce 6 beats per second.

Question

4. Two closed organ pipes of length 0.5 m and 0.525 m are sounded together and produce 6 beats per
second. What is the velocity of sound in the medium with which the organ pipes are filled?

Answer 1

Hi Kait!

 

The answer to this depends on a couple of pieces of information:

 

1.  Are the pipes closed at one end, and open at the other? or open at both ends?

 

Usually, when organ pipes are mentioned in these kind of problems, they are meant to exemplify pipes open at one end and closed at the other.  But we can work it both ways.

 

2.  What harmonic is being sounded?

 

In the absence of other instructions, we can usually assume the first (or fundamental) harmonic (frequency) is being sounded, since that tends to be the dominant one in these situations.

 

Let's assume both pipes are sounding their fundamental harmonic.  For a pipe closed at one end, the harmonic frequencies have the expression:

 

f_n = nv/4L  (with n = 1,3,5,...) with the fundamental frequency given when n = 1

where v = the speed of sound in air and L is the length of the pipe

 

The beat (or difference) frequency between the two frequencies f₁ and f₂ when the are sounded together (or heterodyned) is:

 

f_beat = |f₁ - f₂|

 

So, the beat frequency between the two pipes is given by the expression:

 

f_beat = |v/4L₁ - v/4L₂|

 

assuming the conditions are the same for both pipes such that speed of sound is the same for each.  With the information given, this is one equation with one unknown, v, which is what the problem is asking for.

 

If the pipes are meant to be open at both ends, the only change is that the harmonic frequency expression for the pipe becomes:

 

f = nv/2L (with n = 1 for the fundamental).  Everything else is exactly the same.

 

Just let me know if you have more questions about this, or would like to check an answer.  I hope this helps you put the question in order!