# Classification of higher order partial differential equations

For second order linear PDEs we have the classifications parabolic (e.g. heat equation), hyperbolic (e.g. wave equation), elliptic (e.g. laplace equation) and ultrahyperbolic (at least two positive and two negative Eigenvalues).

I am reading a book on finite element methods and the author states that the model for a vibrating beam

$$\rho A\dfrac{\partial^2 w}{\partial t^2}+\dfrac{\partial^2}{\partial x^2}\left(EI\dfrac{\partial^2 w}{\partial x^2} \right)+f_q=0$$

is a hyperbolic PDE.

My question: I am very confused how can the author state that this is a hyperbolic PDE if it is a 4. order PDE? I searched for classifications of higher order PDEs and didn't find anything useful. I would be glad if someone could explain me this classification.

• Relevant: Wiki : Hyperbolic PDE definition – Winther Jun 27 '16 at 10:15
• There is no classification of higher order PDE, but the terms elliptic, parabolic, and hyperbolic are still used for PDE that share some features of the corresponding classes of 2nd order PDE in two variables. – user147263 Jun 27 '16 at 10:45
• Why would you call this type of PDE hyperbolic? – MrYouMath Jun 27 '16 at 14:47