1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

252 Chapter 3 The Wave Equation

can be obtained fromφ(x−ct)andφ(x− ̄ct).(Herep=

√

ω/ 2 kand ̄cis

the complex conjugate ofc. Refer to Exercise 6 in Chapter 2, Section 10.)

28.Some nonlinear equations can also result in “traveling wave solutions,”

u(x,t)=φ(x−ct). Show thatFisher’s equation,

∂^2 u

∂x^2 =

∂u

∂t−u(^1 −u),

has a solution of this form ifφsatisfies the nonlinear differential equa-

tion

φ′′+cφ′+φ( 1 −φ)= 0.

29.Show that the functionuis a solution of the problem

∂^2 u

∂x^2 =

1

c^2

∂^2 u

∂t^2 ,^0 <x<a,^0 <t,

u( 0 ,t)= 0 , u(a,t)=sin(ωt), 0 <t,

provided that the parameters are such that sin(ωa/c)=0.

u(x,t)=sin(ωx/c)sin(ωt)

sin(ωa/c)

.

30.Ifω=πc/a, the denominator of the function in Exercise 29 is 0. Show

that, for this value ofω, a function satisfying the wave equation and the

given boundary condition is

u(x,t)=−

ct

asin

(πx

a

)

cos

(πct

a

)

−

x

acos

(πx

a

)

sin

(πct

a

)

.

31.A string in a musical instrument is typically not as flexible as assumed

in Section 1. For such a string, the displacementumay satisfy the partial

differential equation

∂^2 u

∂x^2 −

∂^4 u

∂x^4 =

1

c^2

∂^2 u

∂t^2 ,^0 <x<a,^0 <t,

wherec^2 =T/ρ,asinSection1,and=EI/T, withE=Yo u n g ’s m o d -

ulus for the material,I=second moment of area. (See an elasticity ref-

erence.)

Assuming thatu(x,t)=φ(x)T(t), carry out a separation of variable

and find the eigenvalue problem forφ. Take the boundary conditions to

be