Harmonic quadratic plane-wave profile ($\psi =a(u)(x^2-y^2)+b(u)xy$)

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 $dudv+d{x}^2+d{y}^2$, a metric perturbation $h=\psi d{u}^2$ has zero trace, zero divergence and $\square h=0$, 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.