222 Chapter 3 The Wave Equation
By applying the trigonometric identitysin(A)cos(B)=^1
2[
sin(A−B)+sin(A+B)]
we can expressu(x,t)as
u(x,t)=1
2
[
8 h
π^2∑∞
n= 1sin(nπ/ 2 )
n^2 sin(nπ(x−ct)
a)
+
8 h
π^2∑∞
n= 1sin(nπ/ 2 )
n^2 sin(nπ(x+ct)
a)]
.
We know that the series
8 h
π^2∑∞
n= 1sin(nπ/ 2 )
n^2 sin(
nπx
a)
actually converges to the odd periodic extension, with period 2a,ofthefunc-
tionf(x). Let us designate this extension byf ̄o(x)andnotethatitisdefinedfor
all values of its argument. Using this observation, we can expressu(x,t)more
simply as
u(x,t)=^12[ ̄
fo(x−ct)+f ̄o(x+ct)]
. (13)
In this form, the solutionu(x,t)can easily be sketched for various values
oft.Thegraphoff ̄o(x+ct)has the same shape as that off ̄o(x)but is shiftedct
units to the left. Similarly, the graph off ̄o(x−ct)is the graph of ̄fo(x)shifted
ctunits to the right. When the graphs off ̄o(x+ct)andf ̄o(x−ct)are drawn on
the same axes, they may be averaged graphically to get the graph ofu(x,t).
In Fig. 3 are graphs of ̄fo(x+ct),f ̄o(x−ct),and
u(x,t)=1
2
[ ̄
fo(x+ct)+ ̄fo(x−ct)]
for the particular example discussed here and for various values oft.Thedis-
placementu(x,t)is periodic in time, with period 2a/c. During the second
half-period (not shown), the string returns to its initial position through the
positions shown. The horizontal portions of the string have a nonzero veloc-
ity. Equation (12) can also be used to findu(x,t)for any givenxandt.For
instance, if we takex= 0. 2 aandt= 0. 9 a/c,wefindthat
u(
0. 2 a, 0. 9a
c)
=
1
2
[ ̄
fo(− 0. 7 a)+f ̄o( 1. 1 a)]
=
1
2
[
(− 0. 6 h)+(− 0. 2 h)]
=− 0. 4 h.