I am reading "Fundamental of Aerodynamics " by J.D.Anderson Fifth edition. Now I am at the chapter 3. If you got the book, please go to the page 286, the part that says: "when j = i, the contribution to the derivative is simply $\lambda/2$ ". How could it be $\lambda/2$ ? I thought it would be zero.
I will summarize the problem for those who don't have the book:
We have a uniform flow and a source sheet cover the the surface of a known body. We have to find out the $\lambda=\lambda(s)$ along the source sheet that when we superimpose the uniform flow with this source sheet, we obtain the streamline over surface of the body.
A source sheet is an infinite number of line sources side by side, where the strength of each line source is infinitesimally small
We approximate the source sheet by a series of straight panel over the body surface (if we take n panels, then we have to find n values of $\lambda$) and calculate the velocity at each panel and set the normal component of it equal zero.
Take a point $P$ at the middle of $i$th panel and calculate the velocity at $P$ induced by the panels, or just calculate the normal component of the velocity at P. The book said that the normal component of the velocity contributed by only $i$th panel at the mid point $P$ of this panel is $\lambda_i/2$. I don't know how to calculate this, I have tried and thought it would be zero.
@Peter: Look at the image, you can see the velocity induced at P by the panel i is in the ab-line direction (I mean the horizontal direction). Then, the normal component (vertical direction) of it will be zero