Sandberg and Thomas [Phys. Rev. Lett. 130, 085001 (2023)0031-900710.1103/PhysRevLett.130.085001] proposed a scheme to generate ultrashort, high-energy pulses of XUV photons though dephasingless photon acceleration in a beam-driven plasma wakefield. An ultrashort laser pulse is placed in the plasma wake behind a relativistic electron bunch such that it experiences a comoving negative density gradient and therefore shifts up in frequency. Using a tapered density profile provides phase-matching between driver and witness pulses. In this paper, we give the details of the wakefield solutions and phase-matching conditions used to generate the phase-matching density profile. The short, high-density, and weak driver limits are considered. We show, explicitly, the numerical algorithm used to calculate the density profiles.