Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 80 / 3 / a

Use positive carrier wavenumber $k$ and retain waves traveling mainly in the positive $x$ direction. Factor out their rapid wave phase by writing $\psi =e^{ikx}E(x,z)$. Substitution into the Helmholtz equation is exact and givesFor a transverse Fourier mode $e^{i\nu z}$ at a small angle, $|\nu |/k\ll 1$ and its forward longitudinal wavenumber isThus the envelope varies along $x$ on scale $k/{\nu }^2$, much more slowly than the carrier wavelength, and $|E_{xx}|/|k{E}_x|=O({\nu }^2/k^2)$. Dropping this smaller term derives the parabolic wave equation,With transform $\hat{E}(x,\nu )=\int E(x,z)e^{-i\nu z}dz$, its free propagation isThis is a one-way paraxial approximation, not a replacement for backward waves or large-angle Helmholtz components. Its Fourier multiplier has unit modulus and therefore does not attenuate any admitted component.