Repetition in continuous systems

A standard approach to partial differential equations (PDEs) used for more than a century is so-called linear stability analysis, in which one assumes that small fluctuations around some kind of basic solution can be treated as a superposition of waves of the form Exp[ k x] Exp[ ω t]. And at least in a linear approximation any given PDE then typically implies that ω is connected to the wavenumber k by a so-called dispersion relation, which often has a simple algebraic form. For some k this yields a value of ω that is real—corresponding to an ordinary wave that maintains the same amplitude. But for some k one often finds that ω has an imaginary part. The most common case Im[ω] > 0 yields exponential damping. But particularly when the original PDE is nonlinear one often finds that Im[ω] < 0 for some range of k—implying an instability which causes modes with certain spatial wavelengths to grow. The mode with the most negative Im[ω] will grow fastest, potentially leading to repetitive behavior that shows a particular dominant spatial wavelength. Repetitive patterns with this type of origin are seen in a number of situations, especially in fluids (and notably in connection with Kelvin–Helmholtz, Rayleigh–Taylor and other well-studied instabilities). Examples are ripples and swell on an ocean (compare page 1001), Bénard convection cells, cloud streets and splash coronas. Note that modes that grow exponentially inevitably soon become too large for a linear approximation—and when this approximation breaks down more complicated behavior with no sign of simple repetitive patterns is often seen.