Write a matlab/python script using the pseudo-spectral method for the 1D wave equation.

1 Wave Equation The 1-D expression of the wave equation is: p(x)u(x, t) = 0,[0x6()u(x,t)], (2 (0.1],t [0,+c)) where u(x,t) is the displacement field at the position r at instant t, p(x) is the material density and (x) is the material bulk modulus. The initial conditions are: (2,0) = exp-0.1+(3-50), Oulx,0) = 0 To solve the wave equation, you can recognize that it is equivalent to p(xv (2,t) a,Ta,t) 2T (2.0) K(r),(2,t) where 24(,t) is a velocity field, K(2), 4(x,t) is a stress field. T(x,t) Write the discretized form of the system to solve for (v.T) The grid size Ar is chosen to be 0.1. The string length is L = 100. Plot your numerical results at several time steps for a homogeneous material case: p=1,k=1 (x [0, 1000) Heterogeneous material Use the same code you just wrote and investigate the evolution of the displacement field time series, when p(x) = 1, x(x) = 1 (te [0,60]) and p(x) = 1, 6(x) = 4 (1 (60,100]) 1 Wave Equation The 1-D expression of the wave equation is: p(x)u(x, t) = 0,[0x6()u(x,t)], (2 (0.1],t [0,+c)) where u(x,t) is the displacement field at the position r at instant t, p(x) is the material density and (x) is the material bulk modulus. The initial conditions are: (2,0) = exp-0.1+(3-50), Oulx,0) = 0 To solve the wave equation, you can recognize that it is equivalent to p(xv (2,t) a,Ta,t) 2T (2.0) K(r),(2,t) where 24(,t) is a velocity field, K(2), 4(x,t) is a stress field. T(x,t) Write the discretized form of the system to solve for (v.T) The grid size Ar is chosen to be 0.1. The string length is L = 100. Plot your numerical results at several time steps for a homogeneous material case: p=1,k=1 (x [0, 1000) Heterogeneous material Use the same code you just wrote and investigate the evolution of the displacement field time series, when p(x) = 1, x(x) = 1 (te [0,60]) and p(x) = 1, 6(x) = 4 (1 (60,100])