= Circularly polarized nonlinear Alfvén wave
{title2=$\mathbf u_\perp=-\mathbf B_\perp/\sqrt{\mu_0\rho_0}$}
In homogeneous <ideal magnetohydrodynamics>, a constant longitudinal field $B_0>0$ and transverse field $aB_0(\cos\zeta,\sin\zeta)$, $\zeta=k(z-v_at)$, give an exact traveling wave with constant <mass density> and <pressure>, $u_z=0$, $v_a=B_0/\sqrt{\mu_0\rho_0}$ and the displayed transverse <velocity>. Induction and transverse momentum reduce to the same wave relation. Since $|\mathbf B_\perp|^2=a^2B_0^2$ is constant, there is no longitudinal <magnetic pressure> <gradient>. All nonlinear terms are compatible with the constant-density solution, even in a compressible fluid.
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