242 Chapter 3 The Wave Equation
Figure 5 Solution of Eqs. (1)–(4) withg(x)≡0. On the left are graphs of
fo(x+ct)(solid) and offo(x−ct)(dashed) at the times shown. On the right are
the graphs ofu(x,t)for 0<x, made by averaging the graphs on the left.
Now, given the functionsf(x)andg(x), it is a simple matter to constructfo
andGeand thus to graphu(x,t)as a function of either variable or to evaluate it
for specific values ofxandt. By way of illustration, Fig. 5 shows the solution of
Eqs. (1)–(4) as a function ofxat various times, forf(x)as shown andg(x)≡0.
Another interesting problem that can be treated by the d’Alembert method
is one in which the boundary condition is a function of time. For simplicity,
we take zero initial conditions. Our problem becomes
∂^2 u
∂x^2 =1
c^2∂^2 u
∂t^2 ,^0 <t,^0 <x, (7)
u(x, 0 )= 0 , 0 <x, (8)
∂u
∂t(x,^0 )=^0 ,^0 <x, (9)