1. The wave equation describes the motion of a one-dimensional wave in a domain of length...
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1. The wave equation describes the motion of a one-dimensional wave in a domain of length L with both ends fixed, which reads: a²U მt2 282U მx2 0 < x < L,t>0 where U = U(x,t) is the position of the wave at time of the point x, c is the wave velocity here assumed a real constant. We have homogeneous boundary conditions: U(0,1) = 0, U(L,t) = 0. t>0 The two initial conditions are defined as: au U(x, 0) = f(x), (x, 0) = g(x), 0 < x <L at where the first of these defines the position of the wave at time t = 0 and the second the velocity of the wave at t = 0. In this question, we define: (4лx f(x) = sin g(x) = 0. (a) Apply separation of variables to the problem defined above using U(x,t) = X(x)T(t) and show the following: X" T" = β X c²T where ẞ is a real constant. For this problem, we can choose ẞ = -1², for 1 >0. Show that the resulting ODE system for X(x) is subject to the boundary conditions: X" +1²x=0 X(0) = 0, X(L) = 0. [6] 1. The wave equation describes the motion of a one-dimensional wave in a domain of length L with both ends fixed, which reads: a²U მt2 282U მx2 0 < x < L,t>0 where U = U(x,t) is the position of the wave at time of the point x, c is the wave velocity here assumed a real constant. We have homogeneous boundary conditions: U(0,1) = 0, U(L,t) = 0. t>0 The two initial conditions are defined as: au U(x, 0) = f(x), (x, 0) = g(x), 0 < x <L at where the first of these defines the position of the wave at time t = 0 and the second the velocity of the wave at t = 0. In this question, we define: (4лx f(x) = sin g(x) = 0. (a) Apply separation of variables to the problem defined above using U(x,t) = X(x)T(t) and show the following: X" T" = β X c²T where ẞ is a real constant. For this problem, we can choose ẞ = -1², for 1 >0. Show that the resulting ODE system for X(x) is subject to the boundary conditions: X" +1²x=0 X(0) = 0, X(L) = 0. [6]
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