1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Miscellaneous Exercises 253

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

∂^2 u

∂x^2 (^0 ,t)=^0 ,

∂^2 u

∂x^2 (a,t)=^0 ,^0 <t.

32.Show thatφ(x)=sin(μx)satisfies the eigenvalue problem found in Ex-
ercise 31, provided thatμ=nπ/aand
λ^2 =μ^2 +μ^4 ,
where−λ^2 =T′′(t)/c^2 T(t).

33.The values ofλ^2 are related to the frequencies of vibration of the string
mentioned in Exercise 31. Show thatλnapproachesnπ/afor any fixedn
asapproaches 0.

34.The longitudinal vibration of a thin rod has been described by Love (see
Bibliography) with the equation
∂^2 u
∂x^2 =

1

c^2

(

∂^2 u

∂t^2 −

∂^4 u

∂x^2 ∂t^2

)

, 0 <x<a, 0 <t.

Here,u(x,t)is the displacement of the points that are atxwhen there

is no motion;=νK^2 ,whereνis Poisson’s ratio andKis the radius of

gyration of a cross section of the rod. Take boundary conditions

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

and separate the variables by assumingu(x,t)=φ(x)T(t).Itwillbeuse-

ful to nameφ′′/φ=−λ^2.

35.The eigenvalue problem in Exercise 34 is routine. Once that is solved,
find the differential equation forT(t), solve it, and determine the fre-
quencies of vibration.