Solving the wave PDE using the Fourier method

Question

I am trying to solve the wave PDE using the Fourier method and I need to animate the plots of the partial sum of order 45 -- U45(x,t) with initial conditions U45(x,t0), {t0, 0, 12, .2}.

I am confused: should I define the function f[x, t, k_] or should I do it as f[x, k_] and later access the 't'.

a[k_] := 
  Piecewise[{
    {2/(k*π) Integrate[Cos[(k*π*x)/6] (2 - x) Sin[π*x]^2, {x, 1, 3}], k != 0},
    {1/6 Integrate[(2 - x) Sin[π*x]^2, {x, 1, 3}], k == 0}}]

f[x, t, k_] := 
  a[0]*t + Sum[a[i] Cos[(i*π*x)/6] Sin[(i*π*t)/6], {i, k}]

sol = f[x, t, 45];

Anyway, the above is the piece of code I've written so far. Tried a dozen different Plot3D combinations, but the error I receive tells me I can not replace t. I am open to everything tutorials, papers, tips, plain solutions, everything.

Answer 1

In your definition of f you should have f[x_,t_,k_], x_ means "Match anything and let it be referenced by x" you might want to read about patterns in general in the documentation.

Clear[a, f];
a[k_] := Piecewise[{
   {0, k == 12},
   {(1728 (k (-144 + k^2) \[Pi] Cos[(k \[Pi])/6]
    - 18 (-48 + k^2) Sin[(k \[Pi])/6]) Sin[(k \[Pi])/3])
         /(k^3 (-144 + k^2)^2 \[Pi]^3), k != 0},
   {0, k == 0}}]

f[x_, t_, k_] := a[0]*t + Sum[a[i] Cos[(i*\[Pi]*x)/6] Sin[(i*\[Pi]*t)/6], {i, k}]

Plot3D[f[x, t, 45], {x, -1, 1}, {t, 0, 2}]

I replaced the Integrate in your definition of a to not have to calculate it over and over again, thereby speeding up the plot.