Question: If we call your answer to Exercise w1 (x,z) and add in a uniform flow field of velocity V¯∞ = Ui^, where i^ is the x-direction unit vector, what is the new velocity field in the y = 0 plane in terms of U and w1 (x,z)?
Exercise: Given a horseshoe vortex of strength 0, find the induced velocity field at all points in (x,z) space where y = 0. The length of the head of the horseshoe vortex is b. There will be singularities (infinite velocities) on the vortex lines and on extensions of those lines, but they are absent from any real vortex, so we do not worry about them. This result will be applicable to every horseshoe vortex you may use in wing theory or elsewhere. Draw a top view looking straight down from the +y direction, as this will be more useful for setting up your analysis than the isometric view from below is likely to be.