Method of images for a line source

[JyJ]

Homework Statement

Consider the curve C (image attached). C coincides with the real-z axis for $$|z| > a$$ and, in $$|z| < a$$ C coincides with the semi-circle $$|z| = a, =z > 0$$ In terms of simple singular flows, what is the image in C of a line source of strength 2πm lying at z = z_0, above C, as shown on the figure?

Homework Equations

Complex potential for the line source of strength 2πm lying at z = z_0:
$$w = mlog(z-z_0)$$

The Attempt at a Solution

This problem has to do with the topic "Method of images" where we were taught to omit the boundary wall (in this case C) and place additional line sources around: in places where flows would cancel from those line sources, there would be a wall (in this case C). In this particular problem I am having difficulties placing those additional line sources so that when they cancel we get our desired C shape.
My initial thoughts on placing those line sources was:
1) Place the 1st one at the origin inside the semicircle
2) Place the 2nd at (-z_0)
3) Place the 3rd at (z_0*) (conjugate of z_0)
4) Place the 4th at (-z_0*) (conjugate of -z_0)
5) And of course we have our given line source at z_0
If that it the case then this the complex potential:
$$w = mlog(z-z_0)+mlog(z)+ mlog(z+z_0)+ mlog(z-z_0^*) +mlog(z+z_0^*)$$
I would appreciate any advise on this! Thank you!

 

[Asim Riaz]

A:The idea behind the method of images is that you place the images in such a way that the velocity field is zero on the boundary. To do this, you need to think about the flow lines of the velocity field.Let's start by considering the flow lines of the line source at $z_0$: these are straight lines going out from $z_0$. The flow lines of the line source at $-z_0$ will be straight lines going out from $-z_0$, and similarly for the other two line sources.So, it looks like the only choice is the one you have already made. You are right that you need to place four additional line sources to get the required shape.