section_path: Wave in elastic medium
section_name: Wave in elastic medium
paragraph_text: Wave in elastic medium
  
  Consider a traveling transverse wave (which may be a pulse) on a string (the medium). Consider the string to have a single spatial dimension. Consider this wave as traveling
  
  This wave can then be described by the two-dimensional functions
  
  or, more generally, by d'Alembert's formula: u (x, t) = F (x − v t) + G (x + v t). {\displaystyle u(x,t)=F(x-vt)+G(x+vt).} representing two component waveforms F {\displaystyle F} and G {\displaystyle G} traveling through the medium in opposite directions. A generalized representation of this wave can be obtained as the partial differential equation 1 v 2 ∂ 2 u ∂ t 2 = ∂ 2 u ∂ x 2. {\displaystyle {\frac {1}{v^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}={\frac {\partial ^{2}u}{\partial x^{2}}}.}
  
  General solutions are based upon Duhamel's principle.
  
  Beside the second order wave equations that are describing a standing wave field, the one-way wave equation describes the propagation of single wave in a defined direction.