Preliminary Applications of Harmonic Functions


Chapter 11  Applications of Harmonic Functions

11.1  Preliminaries


    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 [Graphics:Images/ApplicationPreliminaryMod_gr_1.gif] and takes on the boundary values

            [Graphics:Images/ApplicationPreliminaryMod_gr_2.gif]    for all  y,  and
            [Graphics:Images/ApplicationPreliminaryMod_gr_3.gif]    for all  y,

along the vertical lines  [Graphics:Images/ApplicationPreliminaryMod_gr_4.gif],  respectively.


Solution.  Intuition suggests that we should seek a solution that takes on constant values along the vertical lines of the form [Graphics:Images/ApplicationPreliminaryMod_gr_6.gif] and that u(x,y) be a function of x alone; that is,

            [Graphics:Images/ApplicationPreliminaryMod_gr_7.gif],    for  [Graphics:Images/ApplicationPreliminaryMod_gr_8.gif]  and for all y.

Laplace's equation,  [Graphics:Images/ApplicationPreliminaryMod_gr_9.gif],  implies that  [Graphics:Images/ApplicationPreliminaryMod_gr_10.gif],  which implies  [Graphics:Images/ApplicationPreliminaryMod_gr_11.gif],  where m and c are constants.  The stated boundary conditions  [Graphics:Images/ApplicationPreliminaryMod_gr_12.gif]  and  [Graphics:Images/ApplicationPreliminaryMod_gr_13.gif]  lead to the solution  


The level curves  [Graphics:Images/ApplicationPreliminaryMod_gr_15.gif]  are vertical lines as indicated in Figure 11.1.


Figure 11.1  Level curves of the harmonic function  [Graphics:Images/ApplicationPreliminaryMod_gr_16.gif].

Explore Solution 11.1.


Example 11.2.  Find the function [Graphics:Images/ApplicationPreliminaryMod_gr_24.gif] that is harmonic in the sector  [Graphics:Images/ApplicationPreliminaryMod_gr_25.gif] and takes on the boundary values

            [Graphics:Images/ApplicationPreliminaryMod_gr_26.gif]   for  x > 0,
            [Graphics:Images/ApplicationPreliminaryMod_gr_27.gif]   for all points on the ray  [Graphics:Images/ApplicationPreliminaryMod_gr_28.gif].  


Solution.  Recalling that the function  [Graphics:Images/ApplicationPreliminaryMod_gr_30.gif]  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  [Graphics:Images/ApplicationPreliminaryMod_gr_33.gif]  are rays emanating from the origin as indicated in Figure 11.2.


Figure 11.2  Level curves of the harmonic function  [Graphics:Images/ApplicationPreliminaryMod_gr_34.gif].  

 Explore Solution 11.2.


Example 11.3.  Find the function [Graphics:Images/ApplicationPreliminaryMod_gr_43.gif] that is harmonic in the annulus  [Graphics:Images/ApplicationPreliminaryMod_gr_44.gif]  and takes on the boundary values  

            [Graphics:Images/ApplicationPreliminaryMod_gr_45.gif]   when  [Graphics:Images/ApplicationPreliminaryMod_gr_46.gif],  and  
            [Graphics:Images/ApplicationPreliminaryMod_gr_47.gif]   when  [Graphics:Images/ApplicationPreliminaryMod_gr_48.gif].  


Solution.  This problem is a companion to the one in Example 11.2.  Here we use the fact that  [Graphics:Images/ApplicationPreliminaryMod_gr_50.gif]  is a harmonic function, for all  [Graphics:Images/ApplicationPreliminaryMod_gr_51.gif].  The solution is


and the level curves  [Graphics:Images/ApplicationPreliminaryMod_gr_53.gif]  are concentric circles, as illustrated in Figure 11.3.

Figure 11.3  Level curves of the harmonic function    [Graphics:Images/ApplicationPreliminaryMod_gr_54.gif].  

Explore Solution 11.3.


Exercises for Section 11.1  Preliminaries


Library Research Experience for Undergraduates

Dirichlet Problem


Steady State Temperature




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Laplace's Equation and Dirichlet Problem  



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This material is coordinated with our book Complex Analysis for Mathematics and Engineering.
























(c) 2012 John H. Mathews, Russell W. Howell