Space-time average of periodic modes ($⟨\cdot {⟩}_{x,t}$)

A space-time average over periodic modes integrates over the spatial periodic cell and one temporal period. For real spatial Fourier modes sharing a wavevector, $⟨\mathrm{Re}(\mathbf{v}{e}^{i\mathbf{k}\cdot \mathbf{x}})\times \mathrm{Re}(\mathbf{w}{e}^{i\mathbf{k}\cdot \mathbf{x}}){⟩}_x=\mathrm{Re}({\mathbf{v}}^{\ast }\times \mathbf{w})/2$. A matching sine or cosine temporal factor supplies another $1/2$. Opposite wavevectors can have nonzero spatial cross averages, so their temporal coefficients must be checked rather than discarded merely because the wavevectors are distinct.