recall that the general normal mode solution for the wave equation utt 8ut finite domain l with

In this question we have given a uniform string of length 20 meter and it is suspended from a ri

Given that F is opposed to 0y plus sine y J plus K. We have S x x equals to y square where y is

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Recall that the general normal-mode solution for the wave equation Ut^2 - uxx = 0 in a finite domain [0, L] with Dirichlet boundary conditions u(0, t) = 0 and u(L, t) = 0 can be written as: u(x, t) = Î£(A_n cos(Ï‰_n t) + B_n sin(Ï‰_n t)) sin(nÏ€x/L) where Ï‰_n = c(nÏ€/L) and the coefficients (A_n, B_n) are arbitrary. Compute the total energy E = (1/2)âˆ«[u_t^2 + u_x^2]dx in terms of the normal modes and show that it is conserved in time (You may assume that summation and integration commute). How are the normal modes in (2) changed if the boundary condition at x = 0 is changed to the Neumann form, which corresponds to a freely sliding string? Is the lowest possible frequency in this case higher or lower than before? Without repeating the calculation, is E still conserved? Revert to Dirichlet boundary conditions u(0, t) = 0 and u(L, t) = 0, and compute the solution for the "plucked string" initial conditions u(x, 0) = 0 and u_t(x, 0) = r for 0 < x < 1, and u(x, 0) = 2 - x for x > 1.

          Recall that the general normal-mode solution for the wave equation Ut^2 - uxx = 0 in a finite domain [0, L] with Dirichlet boundary conditions u(0, t) = 0 and u(L, t) = 0 can be written as:

u(x, t) = Î£(A_n cos(Ï‰_n t) + B_n sin(Ï‰_n t)) sin(nÏ€x/L)

where Ï‰_n = c(nÏ€/L) and the coefficients (A_n, B_n) are arbitrary. 

Compute the total energy E = (1/2)âˆ«[u_t^2 + u_x^2]dx in terms of the normal modes and show that it is conserved in time (You may assume that summation and integration commute). 

How are the normal modes in (2) changed if the boundary condition at x = 0 is changed to the Neumann form, which corresponds to a freely sliding string? Is the lowest possible frequency in this case higher or lower than before? Without repeating the calculation, is E still conserved? 

Revert to Dirichlet boundary conditions u(0, t) = 0 and u(L, t) = 0, and compute the solution for the "plucked string" initial conditions u(x, 0) = 0 and u_t(x, 0) = r for 0 < x < 1, and u(x, 0) = 2 - x for x > 1.

recall that the general normal mode solution for the wave equation utt 8ut finite domain l with dirichlet boundary conditions 0 at both ends can be written as urt cia cos wnt basin wnt sinku 99656

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