A materials engineer scans a steel plate of thickness

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Temperature, heat and the macroscopic behaviour of matter

Conditions of Validity, Ch The three equations, their characteristics and what makes a problem well posed

Problem

L=0.10\,\mathrm{m}

, whose two faces are held at a fixed reference temperature (so the temperature deviation from that reference vanishes at

x=0

and

x=L

for all time), and records today's deviation profile across the thickness. The steel has thermal diffusivity

D=1.0\times10^{-5}\,\mathrm{m^2\,s^{-1}}

. The engineer wants to know the deviation profile

t=100\,\mathrm{s}

earlier, and expands both profiles in the sine modes

\sin(n\pi x/L)

. ((a)) Compute the dimensionless group

\pi^2Dt/L^2

. ((b)) Compute the amplification factor

e^{n^2\pi^2Dt/L^2}

by which the

n

th mode's coefficient must be multiplied to run the profile backward, for

n=1,2,3

. ((c)) State which of Hadamard's three conditions (existence, uniqueness, continuous dependence) hold for this reconstruction, and why.

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