This third-order dispersion relation is a direct consequence of avoiding secular terms, when inserting the second-order Stokes solution into the third-order equations ( of the perturbation series for the periodic wave problem ).
22.
More generally, for periodic structures having discrete translational symmetry, the solutions take the form of Bloch waves, most famously in crystalline atomic materials but also in photonic crystals and other periodic wave equations.
23.
For instance, an inverse formulation of the periodic wave problem as used by Stokes with the spatial coordinates as a function of velocity potential and stream function does not converge for high-amplitude waves.
24.
Temporal filtering amounts to removing the periodic waves not of interest to us from the power spectrum, and then summing the waves back again, using the inverse Fourier transform to create a new timecourse for the voxel.
25.
In order to apply the Stokes expansion to progressive periodic waves, it is advantageous to describe them through Fourier series as a function of the wave phase " ? " ( " x ", " t " ):
26.
When a periodic wave is composed of a fundamental and only odd harmonics ( f, 3f, 5f, 7f, . . . ), the summed wave is " half-wave symmetric "; it can be inverted and phase shifted and be exactly the same.
27.
In the first, a man named Ed Starsgard ( Baby Brain ) tells Guardian that the Sheeda have been attacking humanity in periodic waves, taking everything of value ( physical and mental ) and leaving behind just enough for the survivors to rebuild for next time.
28.
Depending on the application, different definitions of " H " and " T " are used, for example : for periodic waves the wave height " H " 0 at deep water or the breaking wave height " H " b at the edge of the surf zone.
29.
But the role of scandal in American political history is thoroughly aired in the context of explaining the periodic waves of apathy and anger sparked by the usual suspects ( Watergate, Teapot Dome, etc . ) as well as the hazy ones _ I am still a little shaky on the Tilden-Hayes imbroglio.
30.
Analogue to the drivation of the cnoidal wave solution for the KdV equation, periodic wave solutions " ? " ( " ? " ), with " ? " = " x " " " ct " are considered Then the BBM equation becomes a third-order ordinary differential equation, which can be integrated twice, to obtain:
