If the length between two fixed points of a resonating string is L, and L= wavelength/2, then the standing waves is the

Question

BlackTom AI · Accepted Answer

For a string fixed at both ends the allowed standing-wave wavelengths satisfy λ_n = 2L / n (with n = 1, 2, 3, ...). This is the core relation for harmonics on a string: n counts the harmonic number (1 = fundamental, 2 = second harmonic, etc.), L is the distance between the fixed ends, and λ_n is the wavelength of the nth mode.

1) Relevant knowledge points and concepts
- Boundary conditions for a string fixed at both ends require nodes at both ends.
- Allowed modes form standing waves with wavelengths λ_n = 2L / n.
- The fundamental (first harmonic) corresponds to n = 1 and has the longest wavelength λ1 = 2L.
- Frequency relation: f_n = n v / (2L), where v is wave speed.

2) Step-by-step reasoning
Given L = wavelength / 2. Write this as L = λ / 2.
Rearrange to find λ in terms of L:
λ = 2L.

3) Intermediate algebraic steps
Start: L = λ / 2
Multiply both sides by 2: 2L = λ
So λ = 2L.

Compare this λ to the general allowed wavelengths λ_n = 2L / n. Set λ = λ_n:
2L = 2L / n
Divide both sides by 2L (nonzero): 1 = 1 / n
Thus n = 1.

4) Why the correct answer is correct
n = 1 identifies the mode as the first harmonic (the fundamental). The condition λ = 2L is exactly the defining wavelength of the fundamental standing wave on a string fixed at both ends.

5) Why other options are incorrect
- Second harmonic (n=2) would have λ = 2L/2 = L, not 2L.
- Third harmonic (n=3) would have λ = 2L/3.
- Fourth (n=4) gives λ = 2L/4 = L/2.
- Fifth (n=5) gives λ = 2L/5.
None of these equal λ = 2L, so they do not match the given condition.

6) Conclusion
Therefore the standing wave described is the first harmonic (fundamental). This confirms answer a.

If the length between two fixed points of a resonating string is L, and L= wavelength/2, then the standing waves is theSingle choice

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