308 Chapter 5 Higher Dimensions and Other Coordinates
6.Supposethat,insteadofboundaryconditionsEqs.(2)and(3),wehave
u(x, 0 ,t)=f 1 (x), u(x,b,t)=f 2 (x), 0 <x<a, 0 <t,( 2 ′)
u( 0 ,y,t)=g 1 (y), u(a,y,t)=g 2 (y), 0 <y<b, 0 <t.( 3 ′)
Show that the steady-state solution involves the potential equation, and
indicate how to solve it.
7.Solve the two-dimensional heat conduction problem in a rectangle if there
is insulation on all boundaries and the initial condition is
a. u(x,y, 0 )=1;
b. u(x,y, 0 )=x+y;
c.u(x,y, 0 )=xy.
8.Verify the orthogonality relation in Eq. (16) and the formula foramn.
9.Show that the separation constant−λ^2 must be negative by showing that
−μ^2 and−ν^2 must both be negative.
10.Show that the function
umn(x,y,t)=sin(μmx)sin(νny)cos(λmnct),whereμm,νn,andλmnareasinthissection,isasolutionofthetwo-
dimensional wave equation on the rectangle 0<x<a,0<y<b, with
u=0 on the boundary. The functionumay be thought of as the displace-
ment of a rectangular membrane (see Section 5.1).
11.The places whereumn(x,y,t)=0 for alltare callednodal lines.Describe
the nodal lines for(m,n)=( 1 , 2 ), ( 2 , 3 ), ( 3 , 2 ), ( 3 , 3 ).12.Determine the frequencies of vibration for the functionsumnof Exer-
cise 10. Are there different pairs(m,n)that have the same frequency if
a=b?5.4 Problems in Polar Coordinates
We found that the one-dimensional wave and heat problems have a great deal
in common. Namely, the steady-state or time-independent solutions and the
eigenvalue problems that arise are identical in both cases. Also, in solving
problems in a rectangular region, we have seen that those same features are
shared by the heat and wave equations.