An insulated metal rod of length lies along the -axis with both ends thermally insulated

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

Mathematical Methods for Physics, Ch Separation of variables in Cartesian, cylindrical and spherical geometry

Problem

An insulated metal rod of length LL lies along the xx-axis with both ends thermally insulated, so no heat escapes through them: ∂T/∂x=0\partial T/\partial x = 0 at x=0x=0 and at x=Lx=L, for all tt. The temperature T(x,t)T(x,t) obeys the diffusion equation ∂T/∂t=D ∂2T/∂x2\partial T/\partial t = D\,\partial^2T/\partial x^2 with diffusivity DD. (a) Substitute T(x,t)=X(x)G(t)T(x,t)=X(x)G(t) into the diffusion equation and separate it into two ordinary differential equations joined by a separation constant λ\lambda. Give the full argument for why a function of xx alone that is equal, for every xx and every tt, to a function of tt alone must be a constant. (b) Apply the insulated-end boundary conditions to every sign of λ\lambda in turn, and show that solutions with XX not identically zero exist only for λn=(nπ/L)2\lambda_n=(n\pi/L)^2, n=0,1,2,…n=0,1,2,\dots, with Xn(x)=cos⁡(nπx/L)X_n(x)=\cos(n\pi x/L). State which value of nn gives a mode that does not decay with time, and explain physically why that must happen. (c) The rod has length L=0.40 mL=0.40\,\mathrm{m}. Give the two lowest non-zero eigenvalues λ1\lambda_1 and λ2\lambda_2, in m−2\mathrm{m^{-2}}.

Browse the problem library