We study the propagation of magnetohydrodynamic perturbations in a one-dimensional (1-D) model of the magnetotail current sheet. Starting from a general analysis of the MHD linear response of a Harris current sheet to external pressure pulse, we reconstruct the magnetic, velocity, and pressure perturbations for successive times and at different positions. The spatial-temporal evolution of the signal propagating on the discrete modes has been studied by Fruit et al. <2002>. We here describe the propagation in the continuous spectrum domain that results from the existence of a spatial singularity in the differential equation defining the modes of propagation. At t = 0, the current sheet is supposed to be excited by an external pressure pulse. We estimate the energy that propagates in the continuum domain and will thus be absorbed by the resonant absorption process. Initial pulses of different shapes and characteristic spatial and temporal scales are used. Since we perform the complete inverse Laplace and Fourier transform of the continuous modes, we get the full description of the perturbations in real space, hence, the typical temporal and spatial scales of the absorption process. For example, starting with a pulse of total amplitude of 1 Pme (magnetic pressure in the lobe) with a spatial scale of 4 Earth radii and a temporal scale of 20 s, we show that 2% of the initial plasma thermal energy are transferred to the sheet over a distance of 20 Earth radii and in less than 10 min. Pulses of larger scales have no energetic consequences.