Math — The Wave Equation (PDE)

\frac{\partial^2 u}{\partial t^2} = c^2\frac{\partial^2 u}{\partial x^2}

u(x,t) = X(x)T(t)

u_n = \sin\frac{n\pi x}{L}\cos\frac{n\pi ct}{L}

\text{superposition of modes}