1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

3.4 One-Dimensional Wave Equation: Generalities 235

and its two initial conditions, yet to be satisfied, are

w(x, 0 )=

∑∞

n= 1

anφn(x)=f(x)−v(x), l<x<r,

∂w

∂t(x,^0 )=

∑∞

n= 1

bnλncφn(x)=g(x), l<x<r.

By employing the orthogonality of theφn, we determine that the coefficients
anandbnare given by

an=^1

In

∫r

l

[

f(x)−v(x)

]

φn(x)p(x)dx, (16)

bn=^1

Inλnc

∫r

l

g(x)φn(x)p(x)dx, (17)

where

In=

∫r

l

φ^2 n(x)p(x)dx. (18)

Finally,u(x,t)=v(x)+w(x,t)is the solution of the original problem, and
each of its parts is completely specified. From the form ofw(x,t),wecanmake
certain observations aboutu.

1.u(x,t)does not have a limit ast→∞. Each term of the series form ofw

is periodic in time and thus does not die away.

2. Except in very special cases, the eigenvaluesλ^2 nare not closely related to
   each other. So in general, ifucauses acoustic vibrations, the result will
   not be musical to the ear. (A sound would be musical if, for instance,
   λn=nλ 1 , as in the case of the uniform string.)


3. In general,u(x,t)is not even periodic in time. Although each term in the
   series forwis periodic, the terms do not have acommonperiod (except in
   special cases), and so the sum is not periodic.

EXERCISES

1. Verify the formulas for theanandbn. Under what conditions onfandg
   can we say that the initial conditions are satisfied?


2. Check the statement thatv(x)is the same for the heat conduction problem
   and for the problem considered here.


3. Identify the period ofTn(t)and the associated frequency.