The standard parabolic wave equation can be derived by introducing a narrowangle approximation to a modal representation of the field in a waveguide. Let the modal solution be given by
where the eigenfunctions satisfy the depth-separated wave equation
Here, k0 is the reference wavenumber and the index of refraction. By assuming the modal eigenvalues to cluster around k0 (a narrow-angle approximation) and to be given in the form where is small compared to unity, show that to leading order in the field solution can be written in the for where the envelope function satisfies the standard parabolic equation (6.9).