1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

258 Chapter 4 The Potential Equation

(a) (b)

(c) (d)

Figure 1 (a)uis displacement of a membrane; the graph off(x)is an isosceles
triangle. (b)uis the temperature on a cross section of a long bar. (c)uis voltage
in a rectangular sheet of conducting material. (d)φis a velocity potential (see
Exercise 8). What arex-andy-velocities on the boundaries?

Show that the definition of a velocity potential functionφby the equations

u=−∂φ

∂x

,v=−∂φ

∂y

causes (B) to be identically satisfied and turns (A) into the potential equa-

tion. (See Section 4.7, Comments and References, at the end of this chap-

ter.)

9.For each of the diagrams in Fig. 1, (a) write out the problem in mathemat-

ical form (partial differential equation and boundary conditions); (b) pro-

vide an interpretation in words of the boundary conditions for the given

interpretation of the unknown function.