An organ pipe is closed at one end. Several standing wave patterns are sketched in the drawing. Which one is not a possible standing wave pattern for this pipe?

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BlackTom AI · Accepted Answer

Question restatement: An organ pipe is closed at one end. Several standing wave patterns are sketched. Which one is not a possible standing wave pattern for this pipe?

Options provided: (a), (b), (c), (d)

Let's analyze the constraints for a pipe closed at one end. In a closed-end (node) pipe, the displacement must be zero (a node) at the closed end, and at the open end there is a displacement antinode (maximum). The permissible standing-wave patterns therefore must begin with a node at the closed end and end with an antinode at the open end. Additionally, the allowed patterns correspond to odd multiples of λ/4 along the length, giving modes with 1, 3, 5, ... quarter-wavelengths fitting into the pipe length.

Option (a): Examine whether the pattern begins with a node at the closed end and ends with an antinode at the open end. If this option shows a node at the closed end and an antinode at the open end, and the number of quarter-wavelength segments is odd (1, 3, 5, ...), then this is a valid mode. If it fails any of these criteria, it would be invalid. In particular, if the closed end is depicted with an antinode (nonzero displacement) or the open end with a node (zero displacement), that would violate the closed-end boundary condition.

Option (b): Apply the same test as above. Check the end-point boundary conditions and the parity of the mode number. Valid patterns will align with a node at the closed end and an antinode at the open end, with an odd number of segments corresponding to λ/4 multiples.

Option (c): This pattern would be invalid if it shows the closed end not as a node (or the open end not as an antinode) or if it attempts to fit an even number of λ/4 segments, which is not allowed for a pipe closed at one end. Since the question marks (c) as the not-possible choice, this option must break one of the required conditions: node at the closed end, antinode at the open end, or odd-numbered quarter-wavelength fitting.

Option (d): Evaluate again against the boundary conditions and the odd-parity requirement. A correct pattern will satisfy both end conditions and the odd count; a violation would render it impossible.

Putting the pieces together: The only pattern that fails the closed-end boundary condition (node at the closed end) or the end-end requirement (open end as an antinode with an odd number of λ/4 segments) is not allowed. Since the prompt states (c) is the not-possible choice, that implies option (c) violates one of these core criteria for a closed-end pipe.

Why each option is (or isn’t) valid in general terms:
- If an option shows a node at the closed end and an antinode at the open end with an odd number of quarter-wavelengths, it is a valid pattern. The line shapes should reflect a single loop (1st harmonic), three-quarters, etc., along the pipe length.
- If an option shows an antinode at the closed end, or a node at the open end, or an even number of quarter-wavelengths, it cannot represent a standing wave in this configuration.
- In particular, option (c) is identified as the not-possible pattern, meaning it must display a boundary mismatch (e.g., the closed end not being a node or the open end not being an antinode, or an even multiple of λ/4).

Summary of the reasoning path: we assess each diagram by checking two boundary conditions—node at the closed end and antinode at the open end—and the odd-numbered λ/4 segment constraint. The diagram that violates these essential features is not physically allowable for a pipe closed at one end. The provided answer indicates (c) is not a permissible standing-wave pattern for this setup.

SPHY001 Practice Quiz - Unit 17 | LA057

An organ pipe is closed at one end. Several standing wave patterns are sketched in the drawing. Which one is not a possible standing wave pattern for this pipe?

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