Superoscillations for monochromatic standing waves

For complex scalar waves, a convenient measure of the local oscillations and ('faster than Fourier') superoscillations is the phase gradient vector: the local wavevector, or weak value of the momentum operator. This vanishes for standing waves, described by real functions psi(r); for such waves, an alternative descriptor of oscillations is the local weak value of the square of one of the momentum components, i.e. K-2 (r) = -partial derivative(2)psi(r)/partial derivative x(2)/psi(r), here called the 'weak curvature'. Superoscillations correspond to places where K-2 lies outside the interval 0 <= K-2 <= 1. Two illustrations are given. First is an explicit family of real waves in dimension d = 2, with arbitrarily strong superoscillations; this could represent Neumann standing modes in a strip waveguide. Second is an exact calculation of the probability distribution of K-2 for Gaussian random real waves in d dimensions. This decays as 1/K-2(2), as a consequence of the codimension 1 of nodes (e.g. nodal lines for d = 2). The superoscillation probability varies from 0.3918 (d = 2) to 0.3041 (d = infinity).