= Harmonic quadratic plane-wave profile
{title2=$\psi=a(u)(x^2-y^2)+b(u)xy$}
A homogeneous quadratic <polynomial> in the transverse coordinates $x,y$ is a <harmonic function> exactly when the sum of its $x^2$ and $y^2$ coefficients is zero. Thus it has the displayed form. For the <Minkowski metric> $du\,dv+dx^2+dy^2$, a <metric perturbation> $h=\psi\,du^2$ has zero <trace>, zero <divergence> and $\Box h=0$, since $\partial_v\psi=0$ and $\partial_x^2\psi+\partial_y^2\psi=0$. The two coefficient <functions> describe the two <gravitational wave polarizations>.
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