1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

3.6 Wave Equation in Unbounded Regions 245

EXERCISES

1. Derive a formula similar to Eq. (6) for the case in which the boundary con-
   dition Eq. (4) is replaced by
   ∂u
   ∂x

( 0 ,t)= 0 , 0 <t.

2. Derive Eq. (6) from Eq. (5) by using trigonometric identities for the prod-
   uct sin(λx)·cos(λct), and so forth, and recognizing certain Fourier inte-
   grals.


3. Sketch the solution of Eqs. (1)–(4) as a function ofxat timest=0, 1/ 2 c,
   1 /c,2/c,3/c,ifg(x)=0everywhereandf(x)is the rectangular pulse

f(x)=

{ 0 , 0 <x<1,

1 , 1 <x<2,

0 , 2 <x.

4. Same as Exercise 3, butf(x)=0andg(x)is the rectangular pulse

g(x)=

{ 0 , 0 <x<1,

c, 1 <x<2,

0 , 2 <x.

5. Sketch the solution of Eqs. (7)–(10) at timest=0,π/2,π,3π/2, 2π,5π/2,
   ifh(t)=sin(t).Takec=1.


6. Sketch the solution of Eqs. (7)–(10) at timest=0, 1/2, 3/2, and 5/2, if
   c=1and

h(t)=

{ 0 , 0 <t<1,

1 , 1 <t<2,

0 , 2 <t.

7. Use the d’Alembert solution of the wave equation to solve the problem

∂^2 u

∂x^2 =

1

c^2

∂^2 u

∂t^2 , −∞<x<∞,^0 <t,

u(x, 0 )=f(x), −∞<x<∞,

∂u

∂t(x,^0 )=g(x), −∞<x<∞.

8. The solution of the problem stated in Exercise 7 is sometimes written

u(x,t)=

1

2

(

f(x+ct)+f(x−ct)

)

+

1

2 c

∫x+ct

x−ct

g(z)dz.