1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

220 Chapter 3 The Wave Equation

whereanandbnarearbitrary.(Inotherwords,therearetwoindependentsolu-
tions.) Note, however, that there is a substantial difference between theTthat
arises here and theTthat we found in the heat conduction problem. The most
important difference is the behavior asttends to infinity. In the heat conduc-
tion problem,T(t)tends to 0, whereas hereT(t)has no limit but oscillates
periodically in agreement with our intuition.
For eachn= 1 , 2 , 3 ,...,wenowhaveproductsolutions

un(x,t)=sin(λnx)

[

ancos(λnct)+bnsin(λnct)

]

. (8)

Such solutions are calledstanding waves.Foraparticularanandbn,un(x,t)
maintains the same shape with a variable, periodic amplitude. For any choice
ofanandbn,un(x,t)is a solution of the homogeneous partial differential
equation (1) and also satisfies the boundary conditions Eq. (2). Some standing
waves are shown animated on the CD.
By the Principle of Superposition, linear combinations of theun(x,t)also
satisfy both Eqs. (1) and (2). In making our linear combinations, we need no
new constants because theanandbnare arbitrary. We have, then,

u(x,t)=

∑∞

n= 1

sin(λnx)

[

ancos(λnct)+bnsin(λnct)

]

. (9)

The initial conditions, which remain to be satisfied, have the form

u(x, 0 )=

∑∞

n= 1

ansin

(

nπx

a

)

=f(x), 0 <x<a,

∂u

∂t

(x, 0 )=

∑∞

n= 1

bnnπ

a

csin

(

nπx

a

)

=g(x), 0 <x<a.

(Here we have assumed that

∂u

∂t(x,t)=

∑∞

n= 1

sin(λnx)

[

−anλncsin(λnct)+bnλnccos(λnct)

]

.

In other words, we assume that the series forumay be differentiated term by
term.) Both initial conditions take the form of Fourier series problems: A given
function is to be expanded in a series of sines. In each case, then, the constant
multiplying sin(nπx/a)must be the Fourier sine coefficient for the given func-
tion. Thus we determine that

an=

2

a

∫a

0

f(x)sin

(nπx

a

)

dx, (10)