A light, taut string of length is held fixed at both ends. Its transverse deflection…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Partial differential equations and boundary-value problems

Conditions of Validity, Ch Green functions: inverting the operator

Problem

A light, taut string of length L=3.0 mL = 3.0\,\text{m} is held fixed at both ends. Its transverse deflection under a unit point load at x=ξx = \xi obeys −u′′(x)=δ(x−ξ)-u''(x) = \delta(x - \xi) on [0,L][0, L] with u(0)=u(L)=0u(0) = u(L) = 0. ((a)) Construct the Green function G(x,ξ)G(x,\xi) by matching a homogeneous solution satisfying the left boundary condition to one satisfying the right boundary condition, continuous at x=ξx = \xi with the correct jump in slope. ((b)) Evaluate G(1.0 m,2.0 m)G(1.0\,\text{m}, 2.0\,\text{m}).

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