File:Gibbs phenomenon 10.svg

${\displaystyle \sin(x)+{\frac {1}{3}}\sin(3x)+{\frac {1}{5}}\sin(5x)+{\frac {1}{7}}\sin(7x)+{\frac {1}{9}}\sin(9x)}$

as the 5-th approximation (the first five terms) of the wave function:

${\displaystyle f\left(x\right)=\sum _{n=1}^{\infty }\,{\frac {1}{2n-1}}\sin((2n-1)x)}$

which is the Fourier series of the function:

${\displaystyle f(x)={\begin{cases}0&x=2n\pi \\\pi /4&2n\pi <x<(2n+1)\pi \\0&x=(2n+1)\pi \\-\pi /4&(2n+1)\pi <x<2(n+1)\pi \\\end{cases}}}$