In this paper we investigate x-ray and $γ$-ray propagation in crystals having a constant strain gradient and flat or cylindrical surfaces. When a displacement field is present, we solve the Takagi-Taupin equations either by the Riemann-Green method or by a numerical method. We apply the results to study the operation of a double-crystal Laue-Laue diffractometer having a flat collimating crystal followed by a bent analyzer crystal. In particular, we examine the effect of the analyzer strain on the location of the diffraction peaks in the dispersive and non-dispersive setup, thus confirming our previously reported peak-location as being set only by the diffracting plane spacing on the analyzer entrance surface. ; Comment: 25 pages, 7 figures, 1 table