= Solution
Subtract the constant equilibrium <magnetic energy> density. Since $\mathbf B_0\cdot\mathbf b=0$, the perturbation densities are
$$
\mathcal E_K=\frac12\rho_0|\mathbf v|^2,\qquad
\mathcal E_B=\frac{|\mathbf b|^2}{2\mu_0}.
$$
The traveling-wave <amplitude> relation from part <1/a> gives
$$
\boxed{\mathcal E_B=\mathcal E_K,\qquad\mathcal E=\mathcal E_K+\mathcal E_B=\rho_0|\mathbf v|^2=\frac{|\mathbf b|^2}{\mu_0}.}
$$
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 $\mathbf v=\operatorname{Re}(\widehat{\mathbf v}e^{i(kz-\omega t)})$ and similarly for $\mathbf b$, the period averages are
$$
\langle\mathcal E_K\rangle=\frac{\rho_0}{4}|\widehat{\mathbf v}|^2,\qquad
\langle\mathcal E_B\rangle=\frac1{4\mu_0}|\widehat{\mathbf b}|^2,\qquad
\boxed{\langle\mathcal E\rangle=\frac{\rho_0}{2}|\widehat{\mathbf v}|^2.}
$$
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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