The wave equation is a second-order, linear partial differential equation (PDE)

models propagation of waves in a one-dimensional medium, such as a guitar string.

• One approach is to seek solutions of the form $u(t,x)=y\left(x\right)\exp (i\omega t)$ leading to the second-order equation

  ${\displaystyle y^{\text{'}\text{'}}+k^{2}y=0,k^2={\omega }^2/c^2}$

• Another approach is to seek a transformation of the independent variables that leads to a solvable PDE

  ${\displaystyle (\frac{{\partial }^2}{\partial t^2}-c^2\frac{{\partial }^2}{\partial x^2})u=(\frac{\partial }{\partial t}+c\frac{\partial }{\partial x})(\frac{\partial }{\partial t}-c\frac{\partial }{\partial x})u=\frac{{\partial }^{2}u}{\partial w\partial v}=u_{vw}=0,}$

  with the transformation of independent variables defined as

  ${\displaystyle v=x+ct,w=x-ct,}$

  leading to a solution of the form

  ${\displaystyle u(t,x)=\varphi (x+ct)+\psi (x-ct).}$

  Given the initial conditions $u(0,x)=f\left(x\right)$, $u_t(0,x)=g\left(x\right)$, one obtains the D'Alembert solution

  ${\displaystyle u(t,x)=\frac{1}{2}[f(x+ct)+f(x-ct)]+\frac{1}{2c}\underset{x-ct}{\overset{x+ct}{\int }}g\left(s\right)ds}$