Circularly polarized nonlinear Alfvén wave (${\mathbf{u}}_{\perp }=-{\mathbf{B}}_{\perp }/\sqrt{{\mu }_0{\rho }_0}$)

In homogeneous ideal magnetohydrodynamics, a constant longitudinal field $B_0>0$ and transverse field $a{B}_0(\cos \zeta ,\sin \zeta )$, $\zeta =k(z-v_{a}t)$, 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 $|{\mathbf{B}}_{\perp }{|}^2=a^2{B}_0^2$ is constant, there is no longitudinal magnetic pressure gradient. All nonlinear terms are compatible with the constant-density solution, even in a compressible fluid.