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Supplementary Problems

Exercise 96. Consider heat flow in a long circular cylinder where the temperature
depends only on t and on the distance r to the axis of the cylinder. Here r =
p
2
2
x + y is the cylindrical coordinate. From the three-dimensional heat equation
derive the equation u = .k(u + u /r).
r
rr
t
Exercise 97. Same problem as above in a ball except that the temperature de-
p
2
2
2
pends only on the spherical coordinate x + y + z . Derive the equation u =
t
k(u + 2u /r).
r
rr
R 2
Exercise 98. For the hydrogen atom, iff |u| dx = 1 at t = 0, show that the
same is true at all later times. (Hint: Differentiate the integral with respect to t,
taking care about the solution being complex valued.)
Exercise 99. An incompressible fluid of density ρ and negligible viscosity flows
with velocity v along a thin straight tube, perfectly light and flexible, of cross-

section A and held under tension T. Assume that small transverse displacements u
of the tube are governed by

2
2
2
∂ u ∂ u T ∂ u
2
+ 2v + (v − ) = 0.
∂t 2 ∂x∂t ρA ∂x 2
(a) Show that the general solution consists of a superposition of two waveforms
travelling with different speeds.

(b) The tube initially has a small transverse displacement u = a cos kx and is
suddenly released from rest. Find its subsequent motion.

Exercise 100. The daily and annual variations of temperature at the surface of

the earth may be represented by sine-wave oscillations with equal amplitudes and pe-
riods of 1 day and 365 days respectively. Assume that for (angular) frequency ω the
temperature at depth x in the earth is given by u(x, t) = A sin(ωt+µx) exp(−λx),
where λ and µ are constants.

(a) Use the diffusion equation to find the values of λ and µ.
(b) Find the ratio of the depths below the surface at which the amplitudes have
dropped to 1/20 of their surface values.
(c) At what time of year is the soil coldest at the greater of these depths, assum-

ing that the smoothed annual variation in temperature at the surface has a
minimum on February 1st?

Exercise 101. Consider each of the following situations in a qualitative way and
determine the equation type, the nature of the boundary curve and the type of
boundary conditions involved.
(a) a conducting bar given an initial temperature distribution and then thermally
isolated;

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