1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

236 Chapter 3 The Wave Equation

4.Althoughu(x,t)has no limit ast→∞, show that the following general-

ized limit is valid:

v(x)=Tlim→∞T^1

∫T

0

u(x,t)dt.

(Hint: Do the integration and limiting term by term.)

5.Formally solve the problem

∂

∂x

(

s(x)∂u

∂x

)

=p(x)

c^2

(∂ (^2) u
∂t^2
+γ∂u
∂t

)

+q(x)u, l<x<r, 0 <t,

with the boundary conditions Eqs. (2) and (3) and initial conditions

Eqs. (4) and (5), takingγto be constant.

6.Ve r i f y t h a tw(x,t)as given in Eq. (15) satisfies the differential equation and

the boundary conditions Eqs. (6)–(8).

7.In reference to the observations at the end of the section, prove the follow-

ing statement: The product solutions of the problem in Eqs. (6)–(8) all have

a common period in time if the eigenvalues of the problem in Eqs. (12)–

(14) satisfy the relation

λn=α(n+β),

whereβis a rational number.

8.Find a separation-of-variables solution of the problem

∂u^2

∂x^2 =

1

c^2

(∂ (^2) u
∂t^2 +γ
(^2) u

)

, 0 <x<a, 0 <t,

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

u(x,t)=f(x), ∂∂ut(x, 0 )=g(x), 0 <x<a.

Is this an instance of the problem in this section? Which of the observations

at the end of the section are valid for the solution of this problem?

3.5 Estimation of Eigenvalues

In many instances, one is interested not in the full solution to the wave equa-
tion, but only in the possible frequencies of vibration that may occur. For ex-
ample, it is of great importance that bridges, airplane wings, and other struc-
tures not vibrate; so it is important to know the frequencies at which a struc-
ture can vibrate, in order to avoid them. By inspecting the solution of the gen-
eralized wave equation, which we investigate in the preceding section, we can