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248 Chapter 3 The Wave Equation

6.Find an analytic (integral) solution of this wave problem

∂^2 u

∂x^2

=^1

c^2

∂^2 u

∂t^2

, −∞<x<∞, 0 <t,

u(x, 0 )=f(x), ∂∂ut(x, 0 )=g(x), −∞<x<∞,

withg(x)=0and

f(x)=

{h, |x|<,

0 , |x|>.

7.Sketch the solution of the problem in Exercise 6 at timest=0,/c,2/c,

3 /c.

8.Same as Exercise 7, butf(x)=0and

g(x)=

{c, |x|<,

0 , |x|>.

9.Letu(x,t)be the solution of the problem

∂^2 u

∂x^2 =

1

c^2

∂^2 u

∂t^2 ,^0 <x,^0 <t,

u( 0 ,t)= 0 , 0 <t,

u(x, 0 )=f(x),

∂u

∂t(x,^0 )=g(x),^0 <x.

Sketch the solutionu(x,t)as a function ofxat timest=0,a/ 6 c,a/ 2 c,

5 a/ 6 c,7a/ 6 c.Useg(x)≡0and

f(x)=











3 hx

2 a

, 0 <x<^2 a

3

,

3 h(a−x)

a ,

2 a

3 <x<a,

0 , a<x.

10.Same task as in Exercise 9 but

f(x)=

{

sin(x), 0 <x<π,

0 ,π<x

andg(x)=0. Sketch at timest=0,π/ 4 c,π/ 2 c,3π/ 4 c,π/c,2π/c.

11.Letu(x,t)be the solution of the wave equation on the semi-infinite in-

terval 0<x<∞, with both initial conditions equal to zero but with the