# Why is the wave equation a second order differential?

1. Sep 22, 2015

### thegirl

I don't know if this is a silly question? Am I missing simple math? How does a wave depending on amplitude and frequency make it's equation a second order differential equation?

2. Sep 22, 2015

### blue_leaf77

Actually it's the other way around, every system which is described by a second order type differential equation in space and time supports the propagation of waves.

3. Sep 22, 2015

### UncertaintyAjay

I don't know much about second order differential equations, only about the equations for harmonic oscillators. So if any second order diff. eq describes a system that supports wave propagation, that means that the solution or solutions to any second order differential equation must have a bit that's sinusoidal right?

4. Sep 22, 2015

More correctly, shouldn't it be any system described by a hyperbolic differential equation supports wave propagation?

5. Sep 22, 2015

### vanhees71

Mh, I'd say, it should be a hyperbolic partial differential equation, because, e.g., the Laplace equation doesn't describe waves. The sign pattern in the corresponding differential operator of the linear case is important, i.e., for waves it should be a D'Alembertian rather than a Laplacian to lead to wave propgation solutions.

6. Sep 22, 2015

### blue_leaf77

Thanks boneh3ad, I missed the requirement on the coefficients.

7. Sep 22, 2015

### nasu

Actually what we call usually "wave equation" it has that form because we make a lot of approximations. You can have much more complicated equations (with higher order differential terms) describing waves that do not satisfy various approximations implicit in the simple wave equation.
So there is nothing special about second order. Is maybe the lowest order that allow for a wave solution.
Same as there is nothing special about a lot of "laws" being linear. Just the first order approximation,

8. Sep 22, 2015

### blue_leaf77

For mechanical waves, may be. But Maxwell equations are no approximation.

9. Sep 22, 2015

### nasu

I did not see Maxwell's equations mentioned in the OP. Or EM waves only.
But this is a good (interesting) point.

10. Sep 22, 2015

### blue_leaf77

I mean the EM wave equation is build from Maxwell's equations.
I can agree with you, this shows that the EM waves are an example of a truly sinusiodal disturbance in nature provided they propagate in free space.

11. Sep 22, 2015

$$\dfrac{\partial \phi}{\partial t} \pm c\dfrac{\partial \phi}{\partial x} = 0$$