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We report the first comprehensive investigation of the line profile variation caused by non-radial pulsation in an oscillating magnetic chemically peculiar star. Spectrum variation of the well-known roAp star HR 3831 was detected using very high-resolution high signal-to-noise spectroscopic time-series observations and was followed through the whole rotation cycle of the star. We confirmed the outstanding diversity of pulsational behaviour of different lines in the HR 3831 spectrum and attributed this phenomenon to an interplay between extreme vertical chemical inhomogeneity of the HR 3831 atmosphere and a running pulsation wave, that is propagating towards the upper photospheric layers with increasing amplitude. The rapid profile variation of the NdIII 6145Å line, which shows the maximum pulsational disturbances in the studied wavelength region, is characterised by measuring changes of its equivalent width and the first three moments. Each of these observables exhibits almost purely sinusoidal variation through the pulsation cycle, with the amplitude and phase clearly modulated by the stellar rotation. In contrast to previous photometric pulsational measurements of roAp stars, our spectroscopic observational material admits straightforward astrophysical interpretation and, hence, opens new and exciting possibilities for direct and accurate analysis of the roAp pulsations. We demonstrate that rotational modulation of the radial velocity oscillations cannot be fully explained by an oblique axisymmetric dipole (l=1, m=0) mode, implied by the classical oblique pulsator model of roAp stars. Pulsational variation of the higher moments, in particular the line width, reveals substantial contribution of the high-order (l=3) spherical harmonics that appear due to distortion of pulsations in HR 3831 by the global magnetic field. We interpreted observations with a novel numerical model of the pulsational variation and rotational modulation of the line profile moments in roAp stars. Comparison of the observed and computed radial velocity and line width variation was used to establish parameters of the oblique pulsator model of HR 3831. Furthermore, definite detection of pulsational variation in lines of light and iron-peak elements allows the first 3-D mapping of pulsations in a non-radially oscillating star.