= Magnetic-pressure obstruction to a linearly polarized Alfvén wave
A finite sinusoidal transverse field $B_x=aB_0\cos[k(z-v_at)]$, $B_y=0$, has varying <magnetic pressure>. In a purely transverse traveling <Alfvén wave>, longitudinal momentum requires $\partial_z[p+B_x^2/(2\mu_0)]=0$, but the adiabatic <pressure> equation with $u_z=0$ requires $\partial_t p=0$. A traveling compensating gas <pressure> cannot satisfy both conditions. The uncompensated force is quadratic in amplitude and drives a compressive response. A regular periodic one-dimensional traveling solution cannot evade this by varying <mass density>: its mass, transverse momentum and induction integrals force constant <mass density> at the zeros of the transverse field and hence throughout the nontrivial wave.
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