1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

224 Chapter 3 The Wave Equation

tare frequencies, in radians per unit time;λnc/ 2 πare frequencies in cycles
per unit time (or Hertz, if the time unit is seconds). For the vibrating string
problem, the possible frequencies of vibration are

(nπ/a)c

2 π

=nπc

2 a

.

The fact that these form an arithmetic sequence guarantees a common period
for all theun(x,t)in Eq. (8), and thusu(x,t)in Eq. (9) is a function that is
periodic in time.

EXERCISES

1.Verify that the product solution

un(x,t)=sin(λnx)

[

ancos(λnct)+bnsin(λnct)

]

satisfies the wave equation (1) and the boundary conditions, Eq. (2).
2.Sketchu 1 (x,t)andu 2 (x,t)as functions ofxfor several values oft.Assume
a 1 anda 2 =1,b 1 andb 2 =0. (The solutionsun(x,t)are calledstanding
waves.)
In Exercises 3–5, solve the vibrating string problem, Eqs. (1)–(4), with the ini-
tial conditions given.

3.f(x)=0, g(x)=1, 0 <x<a.

4.f(x)=sin

(

πx

a

)

, g(x)=0, 0 <x<a.

5.f(x)=

{

U 0 , 0 <x<a/2,

0 , a/ 2 <x<a,

g(x)=0, 0 <x<a.

(This initial condition is difficult to justify for a vibrating string, but it

may be reasonable where the unknown function is pressure in a pipe with

a membrane at the midpoint. See Miscellaneous Exercise 18 of this chapter

for some derivations.)

6.If

∑∞

n= 1

ansin

(nπx

a

)

=f ̄o(x),

∑∞

n= 1

bncos

(nπx

a

)

=G ̄e(x),

show thatu(x,t)asgiveninEq.(9)maybewritten