A new ray approximation for calculating the diffraction attenuation due to several knife edges takes into account the wave-normal direction of the diffracted wave in the near field, using the Fresnel approximation. The theory is intended for terrestrial radio wave propagation and for other situations where obstructions are nearly collinear. The far-field form of the wave is adopted, as in asymptotic theories, but the focal line of the diffracted ray pencil coincides with the diffracting edge only in the limit as the field point moves deep into the geometric shadow. In general, the ray passes above the diffracting edges rather than through them. The theory is developed for knife edges only, but may be adaptable to other shapes of diffracting obstruction. In the examples given, the calculated attenuations are much more accurate than if the far-field approximation is assumed, but less accurate than if the complete Fresnel integration is done. In a comparison with the complete Fresnel integration, the difference turns out to be less than l dB for five or fewer knife edges.