Deep water wave dispersion relation

Steps:

• The phase speed v of deep water waves follows the dispersion relation ω² = g k, where k = 2π/L is the wave number.
• Phase speed v = ω / k, so ω = v k and substituting gives v² k² = g k.
• Simplifying, v² = g / k = g L / (2π).
• Thus, v = √(g L / 2π), matching option A.

Why A is correct:

• It derives directly from the linear gravity wave dispersion relation for deep water, where wavelength L replaces λ.

Why the others are wrong:

• B: Gives v proportional to L (not √L) and lacks square root, yielding incorrect dimensions for speed.
• C: Inverts the wavelength dependence, predicting slower waves for longer L, opposite to reality.
• D: Simplifies to √(g/L), same error as C with inverse L dependence.

Final answer: A