1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

448 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS

is an analytic function. The curves T(x,y) = K 1 are called isothermals and

are lines connecting points of the same temperature. The curves S (x,y) = K2

are called heat flow lines, and we can visualize the heat flowing along these
curves from points of higher temperature to points of lower temperature. The
situation is illustrated in Figure 11.16(b).
Boundary value problems for steady state temperat ures are realizat ions of
the Dirichlet problem where the value of t he harmonic function T (x, y) is inter-
preted as the temperature at the point (x, y).

• EXAMPLE 11.14 Suppose that two parallel planes are perpendicular to the

z plane and pass through the horizontal lines y = a and y = b and that t he

temperature is held constant at the values T (x, a) = T 1 and T (x, b) = T2,

respectively, on these planes. Then T is given by

T2-T1

T (x, y) = T1 + b (y - a).

-a

Solution A reasonable assumption is that the temperature at all points on the

plane passing through the line y =Yo is constant. Hence T(x,y) = t(y), where

t (y) is a function of y alone. Laplace's equation implies that t" (y) = 0, and an

argument similar to that in Example 11.1 will show that the solution T (x, y)

has the form given in t he preceding equation.

The isothermals T ( x, y) = a are easily seen to be horizontal lines. The

conjugate harmonic function is

T1 -T2

S(x, y) = b x,

-a

and the heat flow lines S (x, y) = f3 are vertical segments between the horizontal

lines. If T 1 > T2, then the heat flows along these segments from the plane

through y = a to the plane through y = b, as illustrated in Figure 11.17.

• EXAMPLE 11 .15 Find the temperature T (x, y) at each point in the upper
  half-plane Im (z) > 0 if the temperature along the x-axis satisfies

T (x, 0) = Ti, for x > 0,

y

and T(x, 0) = T2,

·.

~' -T(x,y)=a

· isothermals

for x < 0.

Figure 11.17 The temperature between parallel planes where T 1 > T2.