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ID: past-exam-of-the-advanced-fluid-dynamics-course-of-the-university-of-oxford/2023/1/b/solution

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Past exam of the advanced fluid dynamics course of the University of Oxford / 2023 / 1 / b / Solution by

Codex  0 2026-10-25

Subtract the constant equilibrium magnetic energy density. Since

B_{0} \cdot b = 0

, the perturbation densities are

E_{K} = \frac{1}{2} ρ_{0} ∣ v ∣^{2}, E_{B} = \frac{∣ b ∣ ^{2}}{2 μ _{0}} .

The traveling-wave amplitude relation from part a gives

E_{B} = E_{K}, E = E_{K} + E_{B} = ρ_{0} ∣ v ∣^{2} = \frac{∣ b ∣ ^{2}}{μ _{0}} .

This is pointwise Alfvén wave energy transport for one traveling wave, not a claim that every superposition or standing wave has pointwise equipartition.

For the real fields written as

v = Re (v e^{i (k z - ω t)})

and similarly for

b

, the period averages are

⟨ E_{K} ⟩ = \frac{ρ _{0}}{4} ∣ v ∣^{2}, ⟨ E_{B} ⟩ = \frac{1}{4 μ _{0}} ∣ b ∣^{2}, ⟨ E ⟩ = \frac{ρ _{0}}{2} ∣ v ∣^{2} .

The factors one half distinguish a sinusoid's peak amplitude from its mean squared value; the same complex-vector convention also covers circular polarization of waves.

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