Deriving Linear Wave Equation: Step-by-Step Guide

[Oerg]

Hi can anyone show me how to derive the linear wave equation mathematically or show me a link?

I googled but unfortunately I am unable to find out anything about it. Wikipedia showed a derivation via Hooke's law, but I am not really interested since it is not a general derivation. My text also derived it the same way.

$$\frac{\partial^2 y}{\partial x^2}=\frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}$$

 

[f95toli]

What do you mean by "derive" in this case?
I suspect the derivation you have seen is probably just about as "general" as you can make it, I assume it was derived by studying a chain of particles connected via springs? This is actually a VERY general approach and is used a lot in physics (phonons in a solid would be one example).
There are an enormous amount of phenomena in physics that that lead to PDEs of the type you have written above; but that does no mean that there is any "obvious" connection between them expect for the fact that they are all periodic.
The wave equation was first derived to study mechanical waves (water etc) but if you start studying problems like e.g. traffic flow you will find that you also end up with a wave-equation (at least in the simplest case).

Note also that what you have written this is just the simplest wave equation; there are more general equations that include non-linear effects etc that can also lead to e.g. soliton solutions.

 

[Oerg]

oh, I thought that since the equation is can be applied to most kinds of simple waves, there must be a "general" derivation of the equation rather that a case specific derivation of the linear wave equation.

It seems a little weird to me that a wave equation that can be applied to different kinds of waves does not have a general method of deducing the relationship between the variables as shown in the equation.

 

[Ben Niehoff]

Hooke's Law is the most general method of deriving the wave equation. All it states is that there is a restoring force which is proportional to displacement. It turns out this simple concept applies to nearly all oscillations of any kind, as long as only small displacements are considered.

Have you studied Lagrangian mechanics yet? Or do you understand potential energy? The potential energy of the simple harmonic oscillator looks like a parabola. Using Taylor's theorem, one can expand any smooth function about a minimum point such that the lowest-order terms look like a parabola. Therefore the simple harmonic oscillator is a pretty universal model.