Question

Find the nature of the second-order wave equation.

(a) Hyperbolic/elliptic

(b) Parabolic

(c) Hyperbolic

(d) Elliptic

The question was posed to me in an online quiz.

The query is from Classification of PDE topic in section Mathematical Behaviour of Partial Differential Equations of Computational Fluid Dynamics

Answer 1

Right answer is (c) Hyperbolic

For explanation: The second-order wave equation is

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

The general equation is in this form.

A \(\frac{\partial^2 \Phi}{\partial x^2}+B\frac{\partial ^2 \Phi}{\partial x\partial y}+C\frac{\partial^2 \Phi}{\partial y^2}+D\frac{\partial \Phi}{\partial x}+E\frac{\partial\Phi}{\partial y}+F\Phi +G=0\)

Comparing \(\frac{\partial ^2 u}{\partial t^2}-c^2\frac{\partial ^2 u}{\partial x^2}=0\) with the above equation, (let ‘y’ be ‘t’).

A=-c^2

B=0

C=1

To find the type,

d=B^2-4AC

d=4c^2

As d is positive, the second order wave equation is hyperbolic.