Miscellaneous Exercises 393
∂^2 u
∂x^2=∂
(^2) u
∂t^2
, 0 <x, 0 <t,
u( 0 ,t)=h(t), 0 <t,
u(x, 0 )= 0 ,
∂u
∂t(x,^0 )=^0 ,^0 <x,
u(x,t)bounded asx→∞.
Use the solution of the same problem as found in Section 3.6, to verify
the rule
L−^1
(
e−sxH(s))
=
{
h(t−x), t>x,
0 , t<x.25.Solve this wave problem with time-varying boundary condition, assum-
ingω=nπ,n= 1 , 2 ,....
∂^2 u
∂x^2 =∂^2 u
∂t^2 ,^0 <x<^1 ,^0 <t,
u( 0 ,t)= 0 , u( 1 ,t)=sin(ωt), 0 <t,u(x, 0 )= 0 , ∂u
∂t(x, 0 )= 0 , 0 <x< 1.26.SolvetheprobleminExercise25inthespecialcaseω=π.
27.Certain techniques for growing a crystal from a solution or a melt may
cause striations — variations in the concentration of impurities. Authors
R.T. Gray, M.F. Larrousse, and W.R. Wilcox [Diffusional decay of stria-
tions,Journal of Crystal Growth, 92 (1988): 530–542] use a material bal-
ance on a slice of a cylindrical ingot to derive this boundary value prob-
lem for the impurity concentration,C:
∂
∂x(
D(x)∂C
∂x)
−V
∂C
∂x=∂C
∂t,^0 <x,^0 <t,C( 0 ,t)=Ca+Asin( 2 πt
tC)
, 0 <t,C(x, 0 )=Ca, 0 <x.Here,Vis the crystal growth rate,tCis the striation period,Cais the
average concentration in the solid, andD(x)is the diffusivity of the im-
purity at distancexfrom the growth face (which is located atx=0). Of
course,C(x,t)is bounded asx→∞.