What happens with signal in frequency spectrum when it is time shifted in time spectrum?
Each frequency in the FT of a time shifted waveform is rotated in phase by an amount proportional to the frequency and proportional to the amount of time shift.

If you delay a pure sinusoid by 25% of its period, it's phase referenced to any fixed point in time will change by pi/2 radians.  Delay a slightly higher frequency sinusoid by the same amount of absolute time, and it's phase will change more.  So the phase change will change with frequency.

If looking at a 3d plot of the FT of a time shifted signal, it looks like taking the FT before the time shift and twisting it.  The twist will be linear, e.g. proportional to frequency.  The more time shift, the greater the amount twist per unit of frequency (or twist revolutions per graph width).