Example 11.14.  Suppose that two parallel planes are perpendicular to the z plane and pass through the horizontal lines [Graphics:Images/TemperaturesMod_gr_26.gif] and [Graphics:Images/TemperaturesMod_gr_27.gif] and that the temperature is held constant at the values  [Graphics:Images/TemperaturesMod_gr_28.gif]  and  [Graphics:Images/TemperaturesMod_gr_29.gif],  respectively, on these planes.  Then [Graphics:Images/TemperaturesMod_gr_30.gif] is given by  

            [Graphics:Images/TemperaturesMod_gr_31.gif].

Figure 11.17  The temperature between parallel planes where  [Graphics:Images/TemperaturesMod_gr_45.gif].

Explore Solution 11.14.

Show that [Graphics:../Images/TemperaturesMod_gr_46.gif] is the harmonic conjugate of [Graphics:../Images/TemperaturesMod_gr_47.gif].

Enter the formula for the temperature function.  This is similar to Example 11.1, and Example 11.18.

[Graphics:../Images/TemperaturesMod_gr_48.gif]

 

 

 

 

[Graphics:../Images/TemperaturesMod_gr_49.gif]

 

 

 

Solve  [Graphics:../Images/TemperaturesMod_gr_50.gif] to get the isothermals.

[Graphics:../Images/TemperaturesMod_gr_51.gif]



[Graphics:../Images/TemperaturesMod_gr_52.gif]

 

 

Solve  [Graphics:../Images/TemperaturesMod_gr_53.gif] to get the heat flow lines.

[Graphics:../Images/TemperaturesMod_gr_54.gif]



[Graphics:../Images/TemperaturesMod_gr_55.gif]

 

 

For illustration we choose  [Graphics:../Images/TemperaturesMod_gr_56.gif].  

[Graphics:../Images/TemperaturesMod_gr_57.gif]



[Graphics:../Images/TemperaturesMod_gr_58.gif]

 

 

Use Mathematica to make a contour plot of the temperature.

[Graphics:../Images/TemperaturesMod_gr_59.gif]





[Graphics:../Images/TemperaturesMod_gr_60.gif]

[Graphics:../Images/TemperaturesMod_gr_61.gif]

Hence the temperature function  T[x,y] = 50 - 20 (-1+y)  is harmonic in the horizontal strip  [Graphics:../Images/TemperaturesMod_gr_62.gif]  and has the desired boundary values.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2006 John H. Mathews, Russell W. Howell