Question

4.3 In the derivation in Sect.4.1, suppose we include the effect of gravity by adding a term-Pg Ar to the discrete equation of motion (4.2), where g>O is the constant of gravitational acceleration. The wave equation is then modified to (4.48) 012 r2 = -8. 70 4 The Wave Equation Assume that x € (0,), with u satisfying Dirichlet boundary conditions at the end points. (a) Find an equilibrium solution uo(x) for (4.48), that satisfies the boundary condi- tions but does not depend on time. (b) Show that if u is a solution of the original wave equation (4.5), also with Dirichlet boundary conditions, then u = Ho + uz solves (4.48).

Please answer question b.

Answer 1

(b) us (0) = od first observe that us satisfies gell - C²211 = 0 with ų (0) = uo(o) e ų () = Uoll) at a sez thus u= lotus satisfies 22 (U, tuo) - c²22 (U, tho) =(2²u - c²a²uilta 242 2x² ate dx²) at² dx² satisfies deuil + Jo - 22 - I u (o)= u(o)tu, (o) with boundary condition & u.(b) = 4, (l) + Uo(l).