A guitar string playing a note shows a stationary wave. At the centre is:

Question

BlackTom AI · Accepted Answer

A guitar string fixed at both ends forms standing waves; for the lowest allowed mode (and any odd harmonic) the centre of the string is a point of maximum transverse displacement — an antinode.

1) Relevant knowledge points and concepts
- A string fixed at both ends must have nodes (zero displacement) at those ends.
- Allowed standing-wave modes have displacement y(x,t) = A sin(kx) cos(ωt) with k = nπ/L, where n is a positive integer (mode number) and L is string length.
- An antinode is a point where the amplitude is maximum; a node is where amplitude is zero.

2) Complete reasoning process step by step
- Use the standing-wave spatial factor: sin(kx) with k = nπ/L.
- Evaluate at the centre x = L/2: sin(kL/2) = sin(nπ/2).
- For the fundamental (n = 1): sin(1·π/2) = sin(π/2) = 1 → maximum amplitude at the centre.
- For n = 3, 5, ... (odd harmonics) sin(nπ/2) alternates between ±1 → antinodes at the centre.
- For even n (2, 4, ...) sin(nπ/2) = 0 → centre would be a node for those specific harmonics.

3) Intermediate calculation steps shown
- k = nπ/L
- At x = L/2: sin(kx) = sin(nπ/2)
- n = 1: sin(π/2) = 1 (antinode)
- n = 2: sin(π) = 0 (node)

4) Why the correct answer is correct
- A guitar string sounding its fundamental note mainly displays the n = 1 mode; the mathematical evaluation above shows the centre has maximum amplitude in that mode, which by definition is an antinode.

5) Why the other options are incorrect
- Option a (“no string movement”) describes a node; option b explicitly says “a node.” Both are incorrect for the fundamental mode because the centre moves maximally (not zero).
- Only for even-numbered harmonics would the centre be a node; the usual musical note implies the fundamental or odd harmonics where the centre is an antinode.

6) Conclusion
- Therefore the centre is an antinode. This confirms the correct choice: c. an antinode.

A guitar string playing a note shows a stationary wave. At the centre is:单项选择题

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