If the length between two fixed points of a resonating string is L, and L= wavelength, then the standing waves is the

Question

BlackTom AI · Accepted Answer

For a string fixed at both ends the allowed standing-wave wavelengths satisfy L = n(λ/2), where n is the mode number (harmonic).

1) Relevant knowledge points and concepts
- Boundary condition: both ends fixed => nodes at both ends.
- General relation for a string of length L: L = n(λ/2) for n = 1, 2, 3, ... . The nth harmonic (mode) has wavelength λn = 2L/n.
- The first harmonic (fundamental) corresponds to n = 1, second harmonic to n = 2, etc.

2) Step-by-step reasoning
- Start from the allowed-wavelength formula: L = n(λ/2).
- We are given that L = wavelength (λ). Substitute λ = L into the formula:
  L = n(L/2).

3) Calculations (intermediate steps shown)
- L = n(L/2)
- Divide both sides by L (L ≠ 0): 1 = n/2
- Solve for n: n = 2

4) Why the correct answer is correct
- The calculation yields n = 2, meaning the standing wave corresponds to the second harmonic. Using the general λn formula: λ2 = 2L/2 = L, which matches the given condition λ = L. Thus the second harmonic is the mode whose wavelength equals the length of the string.

5) Why the other options are incorrect (briefly)
- First harmonic (n = 1): λ1 = 2L, not L, so incorrect.
- Third harmonic (n = 3): λ3 = 2L/3, not L.
- Fourth harmonic (n = 4): λ4 = 2L/4 = L/2, not L.
- Fifth harmonic (n = 5): λ5 = 2L/5, not L.
None of these give λ = L except n = 2.

6) Conclusion
- Therefore, the standing wave is the second harmonic, which matches the provided answer (b. second harmonic).

If the length between two fixed points of a resonating string is L, and L= wavelength, then the standing waves is the单项选择题

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