1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

3.2 Solution of the Vibrating String Problem 225

u(x,t)=^1

2

( ̄

fo(x−ct)+f ̄o(x+ct)

)

+^1

2

( ̄

Ge(x+ct)−G ̄e(x−ct)

)

.

Here, ̄fo(x)andG ̄e(x)are periodic with period 2a.

7. The pressure of the air in an organ pipe satisfies the equation
   ∂^2 p
   ∂x^2 =

1

c^2

∂^2 p

∂t^2 ,^0 <x<a,^0 <t,

with the boundary conditions (p 0 is atmospheric pressure)

a.p( 0 ,t)=p 0 ,p(a,t)=p 0 if the pipe is open, or

b.p( 0 ,t)=p 0 ,∂∂px(a,t)=0 if the pipe is closed atx=a.

Find the eigenvalues and eigenfunctions associated with the wave equation

for each of these sets of boundary conditions.

8. Find the lowest frequency of vibration of the air in the organ pipes referred
   to in Exercise7aandb.


9. If a string vibrates in a medium that resists the motion, the problem for
   the displacement of the string is
   ∂^2 u
   ∂x^2 =

1

c^2

∂^2 u

∂t^2 +k

∂u

∂t,^0 <x<a,^0 <t,

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

plus initial conditions. Find eigenfunctions, eigenvalues, and product so-

lutions for this problem. (Assume thatkis small and positive.)

10.For the problem in Exercise 9, find frequencies of vibration and show that
they donotform an arithmetic sequence. If we form a series solution, will
it be periodic? What happens tou(x,t)ast→∞?

11.The displacementsu(x,t)of a uniform thin beam satisfy

∂^4 u

∂x^4 =−

1

c^2

∂^2 u

∂t^2 ,^0 <x<a,^0 <t.

If the beam is simply supported at the ends, the boundary conditions are

u( 0 ,t)= 0 , ∂

(^2) u
∂x^2
( 0 ,t)= 0 , u(a,t)= 0 , ∂
(^2) u
∂x^2
(a,t)= 0.
Find product solutions to this problem. What are the frequencies of vibra-
tion?
12.Write out formulas for the first four frequencies of vibration for a thin
beam (Exercise 11) and for a string (text). Then find their values, assum-
ing that parameterscandahave values that make the lowest frequency of