In this paper a submerged horseshoe vortex under a free surface is discussed and the algebraic expression of the wave elevation is obtained. From this expression, some characteristics of the ship wave are described. There exists a smooth region nearθ=0°, but when the uniform stream passes the other singularities （source, sink, doublet, etc.） there is no smooth region. The mechanism of synthetic aperture radar （SAR） images of the narrow ship wakes is also explained.
The stationary phase method in conventional Lighthill＇s two-stage scheme to get the expressions of the velocity field was given up in this paper. The method that Ursell had used in deducing the elevation expression of ship wave was adopted, and an asymptotic solution of velocity field of ship waves on an inviscid fluid that is perfectly fit for the region inside and outside the critical lines was obtained. It is very convenient to be used in SAR technique.
