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114 學年度臺灣大學轉學生招生考試｜第 66 題

普通物理學(A)

Consider a surface water wave with a wavelength λ (and thus a wavenumber k = 2π/λ) propagating in the ocean with a depth H. The gravitational acceleration is g. In the case of λ << H, the wave velocity is
  1. √(gH)
  2. √(H/k)
  3. √(gH2/k)
  4. √(gkH2)
  5. √(g/k)
提示
λ ≪ H is the deep-water limit, where the wave does not feel the bottom. In that limit the phase speed depends on g and k (or λ), not on the depth H.
正確答案

正解：E

詳解
A. (A) √(gH) — incorrect. This is the shallow-water (long-wave, λ ≫ H) limit, the opposite case to the one asked.
B. (B) √(H/k) — incorrect. Dimensionally inconsistent with a speed and wrongly retains the depth H, which drops out in deep water.
C. (C) √(gH²/k) — incorrect. Keeps a dependence on depth H, but deep-water waves do not feel the bottom.
D. (D) √(gkH²) — incorrect. Also retains H and has the wrong power of k for the deep-water dispersion relation.
E. (E) √(g/k) — correct. The full gravity-wave dispersion is v = √((g/k)·tanh(kH)). For λ ≪ H, kH ≫ 1 so tanh(kH) → 1, giving the deep-water phase speed v = √(g/k).

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