PDES: GENERAL AND PARTICULAR SOLUTIONS
(b) The tube initially has a small transverse displacementu=acoskxand is
suddenly released from rest. Find its subsequent motion.20.20 A sheet of material of thicknessw, specific heat capacitycand thermal con-
ductivitykis isolated in a vacuum, but its two sides are exposed to fluxes of
radiant heat of strengthsJ 1 andJ 2. Ignoring short-term transients, show that the
temperature difference betweenits two surfaces is steady at (J 2 −J 1 )w/ 2 k, whilst
their average temperature increases at a rate (J 2 +J 1 )/cw.
20.21 In an electrical cable of resistanceRand capacitanceC, each per unit length,
voltage signals obey the equation∂^2 V/∂x^2 =RC∂V /∂t. This has solutions of the
form given in (20.36) and also of the formV=Ax+D.
(a) Find a combination of these that represents the situation after a steady
voltageV 0 is applied atx=0attimet=0.
(b) Obtain a solution describing the propagation of the voltage signal resulting
from the application of the signalV=V 0 for 0<t<T,V= 0 otherwise,
to the endx= 0 of an infinite cable.
(c) Show that fortTthe maximum signal occurs at a value ofxproportional
tot^1 /^2 and has a magnitude proportional tot−^1.
20.22 The daily and annual variations of temperature at the surface of the earth may
be represented by sine-wave oscillations, with equal amplitudes and periods of
1 day and 365 days respectively. Assume that for (angular) frequencyωthe
temperature at depthxin the earth is given byu(x, t)=Asin(ωt+μx)exp(−λx),
whereλandμare constants.
(a) Use the diffusion equation to find the values ofλandμ.
(b) Find the ratio of the depths below the surface at which the two 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,
assuming that the smoothed annual variation in temperature at the surface
has a minimum on February 1st?
20.23 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;
(b) two long conducting concentric cylinders, on each of which the voltage
distribution is specified;
(c) two long conducting concentric cylinders, on each of which the charge
distribution is specified;
(d) a semi-infinite string, the end of which is made to move in a prescribed way.
20.24 This example gives a formal demonstration that the type of a second-order PDE
(elliptic, parabolic or hyperbolic) cannot be changed by a new choice of independent
variable. The algebra is somewhat lengthy, but straightforward.
If a change of variableξ=ξ(x, y),η=η(x, y) is made in (20.19), so that it
reads
A′∂^2 u
∂ξ^2+B′
∂^2 u
∂ξ∂η+C′
∂^2 u
∂η^2+D′
∂u
∂ξ+E′
∂u
∂η+F′u=R′(ξ, η),show thatB′^2 − 4 A′C′=(B^2 − 4 AC)[
∂(ξ, η)
∂(x, y)] 2
.
Hence deduce the conclusion stated above.