Wave Equation with Neumann boundary conditions

Initial position f(x)=x(1-x), initial velocity zero

Compute the Fourier coefficients

Integrate[x(1-x),{x,0,1}]

Simplify[2 Integrate[x(1-x) Cos[n π x],{x,0,1}],Assumptions->n∈Integers]

Plot the solution

Manipulate[
Plot[(1/6)+Sum[-((2 (1+(((-1))^n)))/((n^2) (π^2)))Cos[n π x]Cos[n π t],{n,1,50}],{x,0,1},PlotRange->{0,0.3},AxesLabel->Automatic],
{t,0,10}
]

Initial position zero, initial velocity g(x)=1 for 1<x<2 and zero otherwise

Compute the Fourier coefficients

Simplify[Integrate[Cos[n (π/3) x],{x,1,2}],Assumptions->n∈Integers]

Plot the solution

Manipulate[
Plot[(t/3)+Sum[((6(Sin[((2n π)/3)]-Sin[((n π)/3)]))/((n^2) (π^2)))Cos[((n π)/3)x]Sin[((n π)/3)t],{n,1,50}],{x,0,3},PlotRange->{-1,3},AxesLabel->Automatic],
{t,0,10}
]

Plot the hyperbolic trig functions

GraphicsRow[{
Plot[Cosh[x],{x,-2,2},PlotRange->{0,4}],
Plot[Sinh[x],{x,-2,2},PlotRange->{-4,4}]
}]