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For the circular MHD wave polarization, $B_x^2+B_y^2=a^2{B}_0^2$ is constant. Choose any constant ${\rho }_0>0,p_0>0$, put $v_a=B_0/\sqrt{{\mu }_0{\rho }_0}$ and takeThis gives a circularly polarized nonlinear Alfvén wave of arbitrary amplitude. All fields depend only on $z,t$, and $u_z=0$, so $\mathbf{u}\cdot \nabla \mathbf{u}=0$ and $\nabla \cdot \mathbf{u}=0$. Constant mass density and pressure then satisfy continuity and the adiabatic pressure equation, and $\nabla \cdot \mathbf{B}=0$ because $B_z$ is constant.

The remaining ideal magnetohydrodynamic induction equation and transverse magnetohydrodynamic momentum equation areSince ${\partial }_t=-v_a{\partial }_z$ on the traveling profiles and ${\mathbf{u}}_{\perp }=-(v_a/B_0){\mathbf{B}}_{\perp }$, both identities hold exactly when $v_a^2=B_0^2/({\mu }_0{\rho }_0)$. The longitudinal magnetohydrodynamic momentum equation holds because $p+|{\mathbf{B}}_{\perp }{|}^2/(2{\mu }_0)$ is constant. The circularly polarized wave is therefore an exact nonlinear solution even though the fluid is compressible. Compressibility permits mass density changes; it does not require every possible motion to compress the gas.