  The semidiscretization of a sound soft scattering problem modelled by the
wave equation is analyzed. The spatial treatment is done by integral equation
methods. Two temporal discretizations based on Runge-Kutta convolution
quadrature are compared: one relies on the incoming wave as input data and the
other one is based on its temporal derivative. The convergence rate of the
latter is shown to be higher than previously established in the literature.
Numerical results indicate sharpness of the analysis. The proof hinges on a
novel estimate on the Dirichlet-to-Impedance map for certain Helmholtz
problems. Namely, the frequency dependence can be lowered by one power of
$\abs{s}$(up to a logarithmic term for polygonal domains) compared to the
Dirichlet-to-Neumann map.
