We derive exact time-domain solutions for scattering of acoustic waves by a half plane by inverse Fourier transforming the frequency-domain integral solutions. The solutions consist of a direct term, a reflected term and two diffraction terms. The diffracting edge induces step function discontinuities in the direct and reflected terms at two shadow boundaries. At each boundary, the associated diffraction term reaches a maximum amplitude of half the geometrical optics term and has a signum function discontinuity so that the total field remains continuous. We evaluate solutions for practical point source configurations by numerically convolving the impulse diffraction responses with a wavelet. We solve the associated problems of convolution with a singular, truncated diffraction operator by analytically derived correction techniques. We produce a zero-offset section and compare it to a Kirchhoff integral solution. Our exact diffraction hyperbola exhibits noticeable asymmetry, with higher amplitudes on the reflector side of the edge. Near the apex of the hyperbola the Kirchhoff solution approximates the exact diffraction term symmetric in amplitude about the reflection shadow boundary, but omits the other low amplitude term necessary to ensure continuity at the direct shadow boundary.