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AS Daily A Level Physics question

2026-06-01 OCR A Waves II: stationary waves; harmonics; boundary conditions; string/air column examples OCR-A 4.4.2 Superposition: stationary waves and harmonics OCR-A 4.4.2 Boundary conditions: nodes/antinodes on strings and air columns

In a lab sonometer, a steel string of length L is fixed at both ends and under tension T. It is driven at a clear third-harmonic resonance with frequency f3. A light clamp is then added exactly at the midpoint to hold that point stationary, and at the same time the tension is increased to 2T. The driver is swept to find the lowest-frequency resonance of the modified setup. Compared with the original f3, what is the new resonance frequency, and why?

  1. A It decreases to about 0.94 f3, because the clamp forces a node at the centre so the lowest allowed mode is n = 2, and doubling T increases the wave speed by a factor √2. (correct)
  2. B It increases to about 1.33 f3, because the clamp makes the lowest allowed mode n = 2 and doubling T doubles the wave speed.
  3. C It decreases to about 0.67 f3, because moving from the third to the second harmonic lowers the frequency by 2/3 and the wave speed is unchanged.
  4. D It stays approximately at f3, because clamping the midpoint just splits the string into two halves each resonating at its own fundamental, matching the previous third harmonic.

Answer

The correct answer is A.

Correct: A — It decreases to about 0.94 f3, because the clamp forces a node at the centre so the lowest allowed mode is n = 2, and doubling T increases the wave speed by a factor √2. A is correct since only even modes survive when the midpoint is fixed (lowest n = 2) and f ∝ n × wave speed, with the speed scaling as √T, giving f_new/f3 = (2√2)/3 ≈ 0.94. B assumes the wave speed is proportional to T (so doubling T would double the speed), leading to a ratio of about 4/3 ≈ 1.33, which ignores the √T dependence. C ignores the change in tension, taking only the harmonic-number change (2/3) and missing the √2 increase from the higher tension. D incorrectly claims little or no change: the third harmonic has an antinode at the centre (so it is disallowed when clamped), and the new lowest mode is n = 2 with frequency v′/L, which is not equal to the old f3.

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