This paper presents a method for analyzing long-wave propagation under conditions where the properties of the earth-ionosphere waveguide change markedly over lateral distances comparable with a wavelength or Fresnel zone. The method is most useful for extremely low frequencies (ELF). To facilitate numerical solution, the lateral wave equation is transformed into an integral equation that accounts for most full-wave properties, including diffraction around localized disturbances and reflection from lateral gradients. Approximate solutions show that full-wave methods must be used to account for lateral gradients transverse to the propagation path if the effective width of a localized ionospheric disturbance is less than about two-thirds the width of the first Fresnel zone. In that situation the lateral WKB approximation significantly overestimates the corresponding propagation anomaly when the disturbance is centered near the propagation path and underestimates the anomaly when the disturbance is centered far off path. Numerical solutions for disturbances having gradients in the direction of propagation reveal pronounced standing wave patterns if the waveguide properties change markedly over a lateral distance equal to about one-sixth of a wavelength.