Solution 4.

See text and/or instructor's solution manual.

Answer.   The transformation  ,  of Exercise 11 in Section 11.9, is known to map a horizontal flow in the upper half plane    

onto flow the upper half plane    over the dam.  

Use the details in the solution to Exercise 11 in Section 11.9, to construct the solution using the Schwarz Christoffel formula.

The derivative is   ,   integration and the boundary conditions produces  

            .  

Furthermore, the image of horizontal streamlines in the z-plane are curves in the w-plane given by the parametric equation

            ,    for    .  

Solution.   Along the x-axis use the points  ,  ,  and in the w-plane use  ,  ,  respectively, and let  .   

The exterior angles are   ,  

and the formula for the derivative is  given by the Schwarz-Christoffel formula  

                    

Integrate and get  

                    

        The images of   ,   are   ,   respectively.

Use   ,   and obtain the system of equations

                      

Which simplifies to be   

                      

The values    are solutions for this system of equations.

Therefore,   

                    .

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

        We can let Mathematica graph some of the streamlines.

                    The streamlines in the w-plane given by the parametric equation

                    ,    for    .  

We are really really done.   

Aside.  We can use Mathematica to graph   .  

                      The image of the upper half plane    under the mapping   .    

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell