Question

I need help with d) please help thank you

Question 1 Wave motion appears in all branches of physics. In the lectures we considered the solution of the advection equation, a first-order hyperbolic PDE. Here we consider the solution of the wave equation: c2 where c >0 is constant. , We assume all variables have been non-dimensionalised. (a) Eq. (1) has the general solution (d'Alembert, 1747): u(x,t) F(x -ct) +G(x ct), where F and G are arbitrary functions. Consider the special cases Gi) F(x)- fo(x) and G(r) 0. (ii) F(z)=G(z) =珈(z), where and where ơ is a constant. Using only analytic methods, sketch u(x, t) for each case, at the timet 0, and at a later time. d) Implement the CTCS method for the wave equation in code. Assume c1, h0.02 and τ = 0.02, and integrate for a total time T = (1 + 1)/c. Use as an initial condition a Gaussian profile centred at where σ = 0.1·The scheme requires two initial conditions, so at n = 1, to calculate u2 requires both ui and uo. The term ui corresponds to u (z,0), and uo corresponds to u (z,-r). Try two cases with your code: (i) u(z,-r) = fe(r+CT): (iii) u(z.-r)-fe(r). Provide plots of your solution at time t = 0 and t = T in each case. Also provide a diagram shoing the values of u(x, t) over the r-t plane.

Answer 1

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