3.3 d’Alembert’s Solution 231
Similarly, iff(x)≡0, sketchG(x)and its even periodic extensionG ̄e(x).
Then sketch the graphs ofG ̄e(x+ct∗)(same shape asG ̄e(x)but shiftedct∗
units to the left) and−G ̄e(x−ct∗)(graph ofG ̄e(x)shiftedct∗units to the
right and reflected in the horizontal axis). These two are then averaged graph-
ically to obtain the graph ofu(x,t∗). Check that the boundary conditions are
satisfied.
EXERCISES
- Letu(x,t)be a solution of Eqs. (2)–(5), withg(x)≡0andf(x)afunction
whose graph is an isosceles triangle of widthaand heighth.Findu(x,t)
forx= 0. 25 aand 0. 5 aand fort=0, 0. 2 a/c,0. 4 a/c,0. 8 a/c,1. 4 a/c. - Sketchu(x,t)of Exercise 1 as a function ofxfor the times given. Compare
your results with Fig. 3. - Letu(x,t)be a solution of Eqs. (2)–(5), withf(x)≡0andg(x)=αc,0<
x<a.Findu(x,t)at:x=0,t= 0. 5 a/c;x= 0. 2 a,t= 0. 6 a/c;x= 0. 5 a,
t= 1. 2 a/c. - Sketchu(x,t)of Exercise 3 as a function ofxfor timest=0, 0. 25 a/c,
0. 5 a/c,a/c. - Find the functionG(x)corresponding to (see Eq. (8))
g(x)={ 0 , 0 <x< 0. 4 a,
5 c, 0. 4 a<x< 0. 6 a,
0 , 0. 6 a<x<a.- Justify this alternate description of the functionG(x),thatisspecifiedin
Eq. (8):Gis the solution of the initial value problem
dG
dx=1
cg(x),^0 <x,
G( 0 )= 0.- Using Eq. (8) or Exercise 6, sketch the functionG(x)of Exercise 5.
- Letu(x,t)be the solution of the vibrating string problem, Eqs. (2)–(5),
withf(x)≡0andg(x)as in Exercise 5. Sketchu(x,t)as a function of
xfor timesct=0, 0. 2 a,0. 4 a,0. 5 a,a,1. 2 a. Hint: SketchG ̄e(x+ct)and
−G ̄e(x−ct); then average them graphically.