How many open ends does the pipe have (it has at least one)? 1 2 3 (b) Is there a – [Free] B43
The sixth harmonic is set up in a pipe. (a) How many open ends does the pipe have (it has at least one)? 1 2 3 (b) Is there a node, antinode, or some intermediate state at the midpoint? node antinode some intermediate state Search
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Question:
The sixth harmonic is set up in a pipe.
- How many open ends does the pipe have (it has at least one)?
- Is there a node, antinode, or some intermediate state at the midpoint?
Answer with Full Explanation:
Step 1: Identify the Type of Pipe
In acoustics, the harmonics that a pipe supports depend on whether the pipe has:
- Both ends open or both ends closed — supports all harmonics: 1st, 2nd, 3rd, etc.
- One end open and one end closed — supports only odd-numbered harmonics: 1st, 3rd, 5th, etc.
Since the sixth harmonic is set up in the pipe, the pipe must support even harmonics. Therefore, it cannot have only one open end.
Answer to (a): The pipe has 2 open ends.
Step 2: Displacement at the Midpoint of the Pipe
For a pipe with two open ends, the displacement wave pattern follows the standing wave equation:
y(x) = A · sin(nπx / L)
Where:
- n = harmonic number
- L = length of the pipe
- x = position along the pipe
Now plug in values for the sixth harmonic at the midpoint (x = L/2):
y(L/2) = A · sin(6π(L/2) / L) = A · sin(3π)
We know that:
sin(3π) = 0
So the displacement at the midpoint is zero. A point where the displacement is zero is called a node.
Answer to (b): There is a node at the midpoint.
Final Answers:
- (a) The pipe has 2 open ends.
- (b) The midpoint of the pipe is a node.