Chapter 11  Applications of Harmonic Functions

11.1  Preliminaries

Overview

    A wide variety of problems in engineering and physics involve harmonic functions, which are the real or imaginary part of an analytic function.  The standard applications are two dimensional steady state temperatures, electrostatics, fluid flow and complex potentials.  The techniques of conformal mapping and integral representation can be used to construct a harmonic function with prescribed boundary values.  Noteworthy methods include Poisson's integral formulae; the Joukowski transformation; and Schwarz-Christoffel transformation.  Modern computer software is capable of implemeting these complex analysis methods.

    In most applications involving harmonic functions, a harmonic function that takes on prescribed values along certain contours must be found.  In presenting the material in this chapter, we assume that you are familiar with the material covered in Sections 2.5, 3.3, 5.1, and 5.2.  If you aren't, please review it before proceeding.

Example 11.1.  Find the function u(x,y) that is harmonic in the vertical strip and takes on the boundary values

                for all  y,  and
                for all  y,

along the vertical lines  ,  respectively.

Solution.  Intuition suggests that we should seek a solution that takes on constant values along the vertical lines of the form and that u(x,y) be a function of x alone; that is,

            ,    for    and for all y.

Laplace's equation,  ,  implies that  ,  which implies  ,  where m and c are constants.  The stated boundary conditions    and    lead to the solution  

            .  

The level curves    are vertical lines as indicated in Figure 11.1.

 

Figure 11.1  Level curves of the harmonic function  .

Explore Solution 11.1.

Example 11.2.  Find the function that is harmonic in the sector   and takes on the boundary values

               for  x > 0,
               for all points on the ray  .  

Solution.  Recalling that the function    is harmonic and takes on constant values along rays emanating from the origin, we see that a solution has the form  

            ,  

where a and b are constants.  The boundary conditions lead to  

            .  

The level curves    are rays emanating from the origin as indicated in Figure 11.2.

 

Figure 11.2  Level curves of the harmonic function  .  

 Explore Solution 11.2.

Example 11.3.  Find the function that is harmonic in the annulus    and takes on the boundary values  

               when  ,  and  
               when  .  

Solution.  This problem is a companion to the one in Example 11.2.  Here we use the fact that    is a harmonic function, for all  .  The solution is

            ,   

and the level curves    are concentric circles, as illustrated in Figure 11.3.

Figure 11.3  Level curves of the harmonic function    .  

Explore Solution 11.3.