= Solution
In the local short-wavelength approximation, derivatives of the slowly varying coefficients are smaller than derivatives of the phase. For the perturbation $v e^{i(\mathbf k\cdot\mathbf r-\omega t)}$, replace $\partial_t$ by $-i\omega$ and $\mathbf B_p\cdot\nabla$ by $i\mathbf k\cdot\mathbf B_p$ in the <torsional Alfvén wave> equation. Cancel the common nonzero amplitude and $R^2$ to obtain
$$
\boxed{\omega^2=\frac{(\mathbf k\cdot\mathbf B_p)^2}{\mu_0\rho}.}
$$
Thus the local <dispersion relation> is an <Alfvén wave> relation with <Alfvén velocity> $\mathbf v_{A,p}=\mathbf B_p/\sqrt{\mu_0\rho}$. The approximation requires wavelength small compared with the background variation scales. A wave vector exactly perpendicular to $\mathbf B_p$ gives zero leading <frequency>, so cannot simultaneously obey the assumed large-<frequency> limit.
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