Homework 6
114B
due Monday November 4, 2002:
Boas Chapter 12: 22.11 (Hint: first look at problem 22.6).
Chapter 13: 7.15, 7.17, 8.4
1. Solve the three dimensional wave equation inside a cube 
with the boundary conditions that u=0 on the faces of the cube. At t=0
and
Use eqn (7.6) of chapter 15 Boas.
2. Consider a damped circular membrane that is
attached to a rigid support along
its circumference at 
. Find the general form of the solution assuming
it satisfies the equation
3. Consider the one dimensional diffusion equation
on the interval 
.
(a) Given that the boundaries are insulated, that is for all time,
show that 
does not change with time.
(b) Suppose instead the boundaries are held fixed in temperature
with 
and T(x=L,t)=1. Find the temperature of the
bar as a function of x in the limit of long times.
4. Calculate the solution to
for an isosceles right
triangle whose hypotenuse is of length 
and is held at 
.
The other two sides are held at 
. Hint: First subtract 1 from
all temperatures in the problem. Now consider how the solution of this
problem can be obtained from a square. Use symmetry to argue that the
temperature along the diagonal of the square is 0 and hence corresponds
to a solution to the triangle problem. Then solve the square problem using
superposition and separation of variables.