Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: GENERAL AND PARTICULAR SOLUTIONS(b) The tube initially has a small transverse displacement u = a cos kx and issuddenly released from rest. Find its subsequent motion.20.20 A sheet of material of thickness w, specific heat capacity c and thermal conductivityk is isolated in a vacuum, but its two sides are exposed to fluxes ofradiant heat of strengths J 1 and J 2 . Ignoring short-term transients, show that thetemperature difference between its two surfaces is steady at (J 2 − J 1 )w/2k, whilsttheir average temperature increases at a rate (J 2 + J 1 )/cw.20.21 In an electrical cable of resistance R and capacitance C, each per unit length,voltage signals obey the equation ∂ 2 V/∂x 2 = RC∂V/∂t. This has solutions of theform given in (20.36) and also of the form V = Ax + D.(a) Find a combination of these that represents the situation after a steadyvoltage V 0 is applied at x =0attimet =0.(b) Obtain a solution describing the propagation of the voltage signal resultingfrom the application of the signal V = V 0 for 0