  Following a recent proposal to describe inelastic eikonal scattering
processes in terms of gravitationally dressed elastic eikonal amplitudes, we
motivate a collinear double graviton dressing and investigate its properties.
This is derived from a generalized Wilson line operator in the worldline
formalism by integrating over fluctuations of the eikonal trajectories of
external particles in gravitationally interacting theories. The dressing can be
expressed as a product of exponential terms -- a coherent piece with
contributions to all odd orders in the gravitational coupling constant and a
term quadratic in graviton modes, with the former providing classical
gravitational wave observables. In particular, the coherent dressing involves
$\mathcal{O}(\kappa^3)$ subleading double graviton corrections to the Weinberg
soft factor. We use this dressing to derive expressions for the waveform,
radiative momentum spectrum and angular momentum. In a limiting case of the
waveform, we derive the nonlinear memory effect resulting from the emission of
nearly soft gravitons from a scattering process.
