230 Chapter 3 The Wave Equation
At the second endpoint, a similar calculation shows thatf ̃(a+ct)+f ̃(a−ct)+G ̃(a+ct)−G ̃(a−ct)= 0.Once again, the independence offandGimplies that
f ̃(a+ct)=−f ̃(a−ct), G ̃(a+ct)=G ̃(a−ct). (12)The oddness of ̃fand evenness ofG ̃can be used to transform the right-hand
sides. Then
̃f(a+ct)=f ̃(−a+ct), G ̃(a+ct)=G ̃(−a+ct).These equations say thatf ̃andG ̃ are both periodic with period 2a,because
changing their arguments by 2adoes not change the functional value. Thus
we want ̃fto be the odd periodic extension offandG ̃to be the even periodic
extension ofG. In the notation we used in Chapter 1, the explicit expressions
forφandψare
ψ(x+ct)=^12( ̄
fo(x+ct)+G ̄e(x+ct)+A)
,
φ(x−ct)=^1
2( ̄
fo(x−ct)−G ̄e(x−ct)−A)
.
Finally, we arrive at an expression for the solutionu(x,t):u(x,t)=^1
2[ ̄
fo(x+ct)+f ̄o(x−ct)]
+^1
2
[ ̄
Ge(x+ct)−G ̄e(x−ct)]
. (13)
The CD shows an animated version of Fig. 3 (Section 3.2) using this form of
the solution.
This form of the solution of Eqs. (2)–(5) allows us to see directly how the
initial data influence the solution at later times. From a practical point of view,
it also permits us to calculateu(x,t)at anyxandtand even to sketchuas a
function of one variable for a fixed value of the other. The following procedure
is helpful in sketchingu(x,t)as a function ofxfor a fixedt=t∗, when the
initial condition (5) hasg(x)≡0. It is easily adapted to other cases.
1.Sketch the odd periodic extension off;callthisf ̄o(x).
2.Sketchf ̄o(x+ct∗)againstx;thisisjustthegraphoff ̄oshiftedct∗units to
the left.
3.Sketch ̄fo(x−ct∗)againstxon the same axes. This graph is the same as
that off ̄obut shiftedct∗units to the right.
4.Average graphically the graphs made in the two preceding steps. Check
that the boundary conditions are satisfied.