Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: SEPARATION OF VARIABLES AND OTHER METHODSx = 0, and free at the other, x = L, show that the angular frequency of vibrationω satisfies( ) ( )ω 1/2 Lω 1/2 Lcosh = − sec .aa[ At a clamped end both u and ∂u/∂x vanish, whilst at a free end, where there isno bending moment, ∂ 2 u/∂x 2 and ∂ 3 u/∂x 3 are both zero. ]21.12 A membrane is stretched between two concentric rings of radii a and b (b >a).If the smaller ring is transversely distorted from the planar configuration by anamount c|φ|, −π ≤ φ ≤ π, show that the membrane then has a shape given byu(ρ, φ) = cπ ln(b/ρ)2 ln(b/a) − 4c ∑( )a m b2mπ m 2 (b 2m − a 2m ) ρ − m ρm cos mφ.m odd21.13 A string of length L, fixed at its two ends, is plucked at its mid-point by anamount A and then released. Prove that the subsequent displacement is given by∞∑[ ] [ ]8A (−1) n (2n +1)πx (2n +1)πctu(x, t) =π 2 (2n +1) sin cos,2 LLn=0where, in the usual notation, c 2 = T/ρ.Find the total kinetic energy of the string when it passes through its unpluckedposition, by calculating it in each mode (each n) and summing, using the result∞∑ 1(2n +1) = π22 8 .0Confirm that the total energy is equal to the work done in plucking the stringinitially.21.14 Prove that the potential for ρ0andcosφ