Answer Happy

Question 3 (20 points) Let the z axis of a coordinate system be vertical and the x and y axes be hor- izontal. A string is attached at the points with coordinates (0,0,0) and (L,0,0) and is vibrating in the (x,z) plane, as shown in the picture (the y axis is "sticking out" of the picture). Let at the g (gravity acceleration) z= u(x,0) z= u(x) X X L L Initial position Asymptotic position moment t the position of the string be described by the equation z = u(x, t). (Remember that the string is all the time in the (x,z) plane, so that y is identically equal to zero.) The initial-boundary problem below describes the vibrations of the string is x = [0, L], te [0,00) putt = Tuxx-ru, pg, u(0, t) = 0, u(L, t) = 0 (3) u(x,0) = p(x), u₁(x,0) = y(x). Of course, you know how to solve the complete IBVP (3) to find u(x, t). However, you do not need to do this here! Instead, I we are interested in the asymptotic behavior of the function u(x, t), i.e., its form for very large values of t. (a). (5 points) Looking at the IBVP (3), we expect the motion of the string to slow down and asymptotically to stop, so that the asymptotic position of the string (shown in the picture above on the right) would be described by a function uoo (x):= lim u(x, t). 1-700 Which term in the PDE in the IBVP (3) is responsible for the fact that the vibrations of the spring will die out and the function u(x, t) will tend to u(x) in the limit t → ∞? What is the physical meaning of this term? (No calculations are needed, one sentence is enough.).
(b). (5 points) Show that the function (x) must satisfy the ordinary differential equation u(x) = P8 T (c). (5 points) What are the boundary conditions (at x = 0 and x = L) that uoo (x) must satisfy? (d). (5 points) Solve the boundary value problem for the function u(x) derived in parts (a) and (b). (e) (BONUS! 5 points) The function u(x) you found in part (c) shows that asymptotically the string is hanging down, and because of the symmetry of the problem, it is quite clear (and can be proved, but you don't need to do this) that the maximum hanging of the string, H: max uoo (x)\, x= [0,L] occurs at the middle, i.e., at x = . Find H as a function of p, T, g, and L.
Solutions of the heat and the wave equations with different BCs Consider the heat equation u₁(x, t) = Uxx (x, t) and the wave equation utt (x, t) = Uxx (x, t) on the spatial interval x = [0, 7], with homogeneous Dirichlet BCs u(0,t)=0, u(7,t)=0, or with homogeneous Neumann BCs ux(0, t) = 0, ux (π, t) = 0. The solutions of the four possible combinations (equation/BCs), namely (heat/Dirichlet), (heat/Neumann), (wave/Dirichlet), (wave/Neumann), are represented by the four expressions below, in scrambled order! u(x, t) = Σ (An cos nt + Bn sin nt) sin nx, n=1 u(x, t) = Ao+Ane-n²t cos nx, n=1 u(x, t) = Ao + (An cos nt + B₁, sin nt) cos nx, n=1 00 u(x, t) -n²t sin nx. = Σ Amer n=1