A time series often represents samples of a continuous function. You hope things were smooth, and the Nyquist sampling theorem states that if there were no variations that were too rapid, then you can actually reconstruct the underlying smooth function from the samples alone. To quantify this, we must introduce the concept of frequency. A is called the amplitude, f the frequency, and f the phase. The reciprocal of wavenumber is wavelength.
Geophysicists commonly refer to 'typical wavelength' when they mean 'typical spatial scale'.
Sinusoids are the 'basis functions' of Fourier analysis. If you violate this Nyquist sampling criterion, the result is aliasing. The word alias means ' a false name' and the meaning here is that higher frequencies, when sampled, appear to be lower frequencies when the samples are examined. To prevent aliasing in seismic and other recordings, we use an anti-alias filter, which makes the data smoother before sampling. We use the term 'band width' to describe the frequency range of the data, so the bandwidth of our data must not exceed that of the Nyquist frequency to avoid aliasing.