The responses of a semi-circular canyon for incident SH, SV, P, and Rayleigh waves in a two-dimensional elastic half-space are investigated in time domain as well as in frequency domain. The author proposes the discrete wavenumber boundary element method, in which the direct boundary element method is used with the discrete wavenumber Green's function. This combination achieves both the efficiency in computation and the flexibility for boundary configurations. First, the validity of the method is confirmed by comparing its results with published ones in frequency domain. Then time histories of seismic motion along the surface in and around the canyon are studied for incident SH, SV, P, and Rayleigh waves with the shape of a Ricker wavelet. In all cases, the diffracted waves called the creeping waves can be seen propagating inside the canyon with P- or S-wave velocity. For SV-wave incidence, Rayleigh waves generated at the edges of the canyon carry a significant portion of energy outward, while for SH-wave incidence the direct and reflected waves play a major role. It should be noted that the amplitude fluctuation in frequency domain does not always mean that in time domain because different arrival time of wavelets results in the fluctuation in spectral amplitude even if each wavelet propagates with the same shape and amplitude.