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Subtract the constant equilibrium magnetic energy density. Since ${\mathbf{B}}_0\cdot \mathbf{b}=0$, the perturbation densities areThe traveling-wave amplitude relation from part a givesThis 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 $\mathbf{v}=\mathrm{Re}(\hat{\mathbf{v}}{e}^{i(kz-\omega t)})$ and similarly for $\mathbf{b}$, the period averages areThe 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.

In ideal magnetohydrodynamics, the first-order electric field is ${\mathbf{E}}_1=-\mathbf{v}\times B_0{\mathbf{e}}_z$. Retaining its product with both the background and first-order magnetic field gives the Poynting vectorThe first term is transverse and oscillatory, with zero period average. The second is quadratic and transports wave energy. A potential quadratic ideal electric-field term $-\mathbf{v}\times \mathbf{b}$ vanishes here because the real traveling-wave fields are parallel or antiparallel.

For the $+z$ branch, $\mathbf{b}=-\sqrt{{\mu }_0{\rho }_0}\mathbf{v}$, soFor the opposite branch, its sign reverses. The averaged Poynting flux is the total wave energy density, including equal kinetic and magnetic shares, multiplied by the group velocity. This agrees with Alfvén wave energy transport. Although the flux in this expression is electromagnetic, it supplies the transport of both components of the coupled fluid-field wave.